{"id":198950,"date":"2025-11-04T06:54:00","date_gmt":"2025-11-04T05:54:00","guid":{"rendered":"https:\/\/datascientest.com\/en\/?p=198950"},"modified":"2025-11-03T17:54:53","modified_gmt":"2025-11-03T16:54:53","slug":"all-about-markov-chain","status":"publish","type":"post","link":"https:\/\/datascientest.com\/en\/all-about-markov-chain","title":{"rendered":"What is a Markov Chain?"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"198950\" class=\"elementor elementor-198950\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t\t<div class=\"elementor-element elementor-element-fb0d621 e-flex e-con-boxed e-con\" data-id=\"fb0d621\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-cbd1cc3 elementor-widget elementor-widget-text-editor\" data-id=\"cbd1cc3\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-text-editor.elementor-drop-cap-view-stacked .elementor-drop-cap{background-color:#69727d;color:#fff}.elementor-widget-text-editor.elementor-drop-cap-view-framed .elementor-drop-cap{color:#69727d;border:3px solid;background-color:transparent}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap{margin-top:8px}.elementor-widget-text-editor:not(.elementor-drop-cap-view-default) .elementor-drop-cap-letter{width:1em;height:1em}.elementor-widget-text-editor .elementor-drop-cap{float:left;text-align:center;line-height:1;font-size:50px}.elementor-widget-text-editor .elementor-drop-cap-letter{display:inline-block}<\/style>\t\t\t\t<p><b>A model is said to possess the Markov property if its state at time T depends solely on its state at time T-1. If we observe the states in which the model exists at each moment, we have what is known as an observable Markov model. Otherwise, it is referred to as a hidden Markov model. This article illustrates these models to grasp their functionality and usefulness.<\/b><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0fb1b87 elementor-widget elementor-widget-heading\" data-id=\"0fb1b87\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-heading-title{padding:0;margin:0;line-height:1}.elementor-widget-heading .elementor-heading-title[class*=elementor-size-]>a{color:inherit;font-size:inherit;line-height:inherit}.elementor-widget-heading .elementor-heading-title.elementor-size-small{font-size:15px}.elementor-widget-heading .elementor-heading-title.elementor-size-medium{font-size:19px}.elementor-widget-heading .elementor-heading-title.elementor-size-large{font-size:29px}.elementor-widget-heading .elementor-heading-title.elementor-size-xl{font-size:39px}.elementor-widget-heading .elementor-heading-title.elementor-size-xxl{font-size:59px}<\/style><h2 class=\"elementor-heading-title elementor-size-xl\">Rigorous Mathematical Definition<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6fdae75 elementor-widget elementor-widget-image\" data-id=\"6fdae75\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<style>\/*! elementor - v3.15.0 - 02-08-2023 *\/\n.elementor-widget-image{text-align:center}.elementor-widget-image a{display:inline-block}.elementor-widget-image a img[src$=\".svg\"]{width:48px}.elementor-widget-image img{vertical-align:middle;display:inline-block}<\/style>\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-77.png\" class=\"attachment-full size-full wp-image-198953\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-77.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-77-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-77-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-77-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-fadfcd3 elementor-widget elementor-widget-text-editor\" data-id=\"fadfcd3\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">In mathematical terms, a sequence of random variables (Xn) taking values in a state space E forms a Markov chain if it adheres to the weak Markov property. This property is articulated by the following equation:<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6788fde elementor-widget elementor-widget-text-editor\" data-id=\"6788fde\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<pre>P(X_{n+1} = i_{n+1} | X_1 = i_1, ..., X_n = i_n) = P(X_{n+1} = i_{n+1} | X_n = i_n)<\/pre>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b33c865 elementor-widget elementor-widget-text-editor\" data-id=\"b33c865\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This equation implies that the likelihood of moving to the future state i_{n+1} is contingent only on the present state i_n, and not on the entire sequence of previous states.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-356117c elementor-widget elementor-widget-heading\" data-id=\"356117c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Key Elements of a Markov Chain<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-24b1214 elementor-widget elementor-widget-text-editor\" data-id=\"24b1214\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The state space E: this is the set of all conceivable states the system can occupy. In our meteorological example, E = {sun, clouds, rain}.<\/span><\/p><p><span style=\"font-weight: 400;\">Transition probabilities: P(X_{n+1} = j | X_n = i) is the probability of transitioning from state i to state j in a single step.<\/span><\/p><p><span style=\"font-weight: 400;\">Time homogeneity: if these transition probabilities are independent of time n, the chain is termed homogeneous. This is the most frequently examined scenario.<\/span><\/p><p><span style=\"font-weight: 400;\">This mathematical formalization distinctly separates Markov chains from other stochastic processes and serves as the groundwork for their entire theory.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-3f05acc e-flex e-con-boxed e-con\" data-id=\"3f05acc\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-ce63f2e elementor-widget elementor-widget-heading\" data-id=\"ce63f2e\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">History and Foundations of Markov Chains<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5c43c35 elementor-widget elementor-widget-image\" data-id=\"5c43c35\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-87.png\" class=\"attachment-full size-full wp-image-198954\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-87.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-87-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-87-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-87-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f8b4b8d elementor-widget elementor-widget-text-editor\" data-id=\"f8b4b8d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Markov chains derive their name from the Russian mathematician Andrey Andreyevich Markov (1856-1922), who presented this concept in 1902 while researching the extension of the law of large numbers to dependent quantities.<\/span><\/p><p><span style=\"font-weight: 400;\">Markov initially sought to extend probability limit theorems beyond the realm of independent random variables. His desire was to comprehend how the law of large numbers could apply to sequences of events where each event is dependent on the preceding one, but not on the entire history.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5c5529f elementor-widget elementor-widget-heading\" data-id=\"5c5529f\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">The Conceptual Innovation<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c493b6c elementor-widget elementor-widget-text-editor\" data-id=\"c493b6c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Markov&#8217;s groundbreaking idea was to model processes where the future is influenced by the past solely through the present. This property, now identified as the &#8220;Markov property,&#8221; can be intuitively described as: &#8220;the past matters only for its influence on the present state.&#8221;<\/span><\/p><p><span style=\"font-weight: 400;\">Markov himself applied his chains to the analysis of letter sequences in Russian literature, notably in Pushkin&#8217;s &#8220;Eugene Onegin,&#8221; thereby demonstrating the applicability of mathematical concepts to non-mathematical fields.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e4f988f elementor-widget elementor-widget-heading\" data-id=\"e4f988f\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Modern Developments<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2fa7bc1 elementor-widget elementor-widget-text-editor\" data-id=\"2fa7bc1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Since their conception, Markov chains have undergone extraordinary development and have become an essential instrument in various scientific and technological domains, from statistical physics to contemporary artificial intelligence.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-17f3bfb elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"17f3bfb\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-courses\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Mastering Markov models<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-af83c63 e-flex e-con-boxed e-con\" data-id=\"af83c63\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-a53f38d elementor-widget elementor-widget-heading\" data-id=\"a53f38d\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Observable Markov Model<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-059d221 elementor-widget elementor-widget-image\" data-id=\"059d221\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-85.png\" class=\"attachment-full size-full wp-image-198955\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-85.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-85-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-85-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-85-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-849c3e4 elementor-widget elementor-widget-text-editor\" data-id=\"849c3e4\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Consider the following scenario:<\/span><\/p><p><span style=\"font-weight: 400;\">You are confined in your house on a rainy day, aiming to predict the weather for the next five days.<\/span><\/p><p><span style=\"font-weight: 400;\">Not being a meteorologist, you simplify the task by assuming the weather follows a Markov model: the weather on day J depends only on the weather on day J-1.<\/span><\/p><p><span style=\"font-weight: 400;\">To further simplify, you consider only three potential weather types: sun, clouds, or rain.<\/span><\/p><p><span style=\"font-weight: 400;\">Based on past months&#8217; observations, you create the following transition diagram:<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-46180b8 elementor-widget elementor-widget-image\" data-id=\"46180b8\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"512\" height=\"472\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image1_markdown.png\" class=\"attachment-full size-full wp-image-172321\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image1_markdown.png 512w, https:\/\/datascientest.com\/en\/files\/2023\/10\/image1_markdown-300x277.png 300w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5964133 elementor-widget elementor-widget-text-editor\" data-id=\"5964133\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The <\/span><b>transition matrix<\/b><span style=\"font-weight: 400;\"> associated is<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-fd16c1c elementor-widget elementor-widget-image\" data-id=\"fd16c1c\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"512\" height=\"225\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image2_markdown.png\" class=\"attachment-full size-full wp-image-172322\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image2_markdown.png 512w, https:\/\/datascientest.com\/en\/files\/2023\/10\/image2_markdown-300x132.png 300w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-db0174f elementor-widget elementor-widget-text-editor\" data-id=\"db0174f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Recall, this matrix is interpreted as follows:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The probability of fair weather tomorrow given that it is raining today is 35%<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The probability of clouds tomorrow when clouds are present today is 25%<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">Let&#8217;s compute the probability that the weather for the upcoming five days will be \u201csun, sun, rain, clouds, sun\u201d.<\/span><\/p><p><span style=\"font-weight: 400;\">Since the weather of any given day relies solely on the weather of the previous day, it suffices to multiply the probabilities (considering that it is raining today):<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e10b2b8 elementor-widget elementor-widget-image\" data-id=\"e10b2b8\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"512\" height=\"60\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image3_markdown.png\" class=\"attachment-full size-full wp-image-172323\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image3_markdown.png 512w, https:\/\/datascientest.com\/en\/files\/2023\/10\/image3_markdown-300x35.png 300w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d2e30dd elementor-widget elementor-widget-text-editor\" data-id=\"d2e30dd\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">We can calculate this probability for every possible combination and <\/span><b>select the one with the greatest likelihood to address the problem<\/b><span style=\"font-weight: 400;\">.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c279bbe elementor-widget elementor-widget-shortcode\" data-id=\"c279bbe\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><script src=\"https:\/\/gist.github.com\/MaximeRighini\/6632773a82550e7d1f7428db4031ebca.js\"><\/script><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9f2b111 elementor-widget elementor-widget-text-editor\" data-id=\"9f2b111\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">In our case, here are the five combinations with the highest likelihood of occurring:<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bacb91c elementor-widget elementor-widget-image\" data-id=\"bacb91c\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"439\" height=\"184\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image4_markdown.png\" class=\"attachment-full size-full wp-image-172324\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image4_markdown.png 439w, https:\/\/datascientest.com\/en\/files\/2023\/10\/image4_markdown-300x126.png 300w\" sizes=\"(max-width: 439px) 100vw, 439px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9dc0997 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"9dc0997\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-courses\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Learn applied mathematics<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-74722a2 e-flex e-con-boxed e-con\" data-id=\"74722a2\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-23045e3 elementor-widget elementor-widget-heading\" data-id=\"23045e3\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Fundamental Properties of the Transition Matrix<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c0e65d2 elementor-widget elementor-widget-image\" data-id=\"c0e65d2\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-83.png\" class=\"attachment-full size-full wp-image-198956\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-83.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-83-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-83-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-83-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-05e4a3c elementor-widget elementor-widget-text-editor\" data-id=\"05e4a3c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The transition matrix used in the prior example contains important mathematical properties that need comprehension.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-66f0434 elementor-widget elementor-widget-heading\" data-id=\"66f0434\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Matrix Structure<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c39d914 elementor-widget elementor-widget-text-editor\" data-id=\"c39d914\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A transition matrix P = (p_{ij}) must meet two essential constraints:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">All entries are non-negative: p_{ij} \u2265 0 for all i,j<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Each row must add up to 1: \u03a3_j p_{ij} = 1 for all i<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">This second constraint indicates that from any state, the system must inevitably transition to one state (possibly remaining in the same state).<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-55ff4de elementor-widget elementor-widget-heading\" data-id=\"55ff4de\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Matrix Powers and Long-term Predictions<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-967dd20 elementor-widget elementor-widget-text-editor\" data-id=\"967dd20\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A remarkable property is that P^n (P to the power of n) directly provides the transition probabilities over n steps. Thus, (P^n)_{ij} denotes the probability of moving from state i to state j in exactly n steps.<\/span><\/p><p><span style=\"font-weight: 400;\">This property allows for long-term weather forecasts. For instance, if we compute P<\/span><span style=\"font-weight: 400;\">10<\/span><span style=\"font-weight: 400;\">, we can determine the transition probabilities over 10 days.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2cf44dd elementor-widget elementor-widget-heading\" data-id=\"2cf44dd\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Convergence to Equilibrium<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2183d93 elementor-widget elementor-widget-text-editor\" data-id=\"2183d93\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Under specific conditions, the successive powers of matrix P converge to a limit matrix where all rows are identical. This common row represents the system&#8217;s stationary distribution; the probability distribution towards which the system gravitates over time, irrespective of its original state.<\/span><\/p><p><span style=\"font-weight: 400;\">In our weather model, this implies that in the very long run, the likelihood of experiencing sun, clouds, or rain becomes stable and predictable.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-79ea7dc e-flex e-con-boxed e-con\" data-id=\"79ea7dc\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-a980fe5 elementor-widget elementor-widget-heading\" data-id=\"a980fe5\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Classification of States in a Markov Chain<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-42f3632 elementor-widget elementor-widget-image\" data-id=\"42f3632\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-93.png\" class=\"attachment-full size-full wp-image-198957\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-93.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-93-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-93-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-93-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a8c3728 elementor-widget elementor-widget-text-editor\" data-id=\"a8c3728\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Once the transition matrix is determined, it is vital to understand the long-term behavior of each state. This analysis allows for the classification of states based on their dynamic properties.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-fb417c2 elementor-widget elementor-widget-heading\" data-id=\"fb417c2\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Transient and Recurrent States<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-04f41ca elementor-widget elementor-widget-text-editor\" data-id=\"04f41ca\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A state is termed <\/span><b>transient<\/b><span style=\"font-weight: 400;\"> if, from that state, there&#8217;s a non-zero probability of never making a return. A state is termed <\/span><b>recurrent<\/b><span style=\"font-weight: 400;\"> if, from that state, a return is certain sooner or later.<\/span><\/p><p><span style=\"font-weight: 400;\">In our weather example, should we modify the matrix to include a &#8220;storm&#8221; state, transitioning only to &#8220;rain&#8221; without return, then &#8220;storm&#8221; becomes a transient state.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ca17ea6 elementor-widget elementor-widget-heading\" data-id=\"ca17ea6\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Absorbing States<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-139bc3b elementor-widget elementor-widget-text-editor\" data-id=\"139bc3b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A state is <\/span><b>absorbing<\/b><span style=\"font-weight: 400;\"> if it is impossible to leave once it is entered. Mathematically, this means p_{ii} = 1 for that state i.<\/span><\/p><p><span style=\"font-weight: 400;\">Concrete example: in a life cycle model of an organism with states &#8220;juvenile,&#8221; &#8220;adult,&#8221; &#8220;senescent,&#8221; and &#8220;deceased,&#8221; the &#8220;deceased&#8221; state is absorbing as a dead organism cannot change state.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-85cb9eb elementor-widget elementor-widget-heading\" data-id=\"85cb9eb\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Irreducible Chains<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d370ee8 elementor-widget elementor-widget-text-editor\" data-id=\"d370ee8\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A Markov chain is <\/span><b>irreducible<\/b><span style=\"font-weight: 400;\"> if every state can communicate with every other state, meaning it is possible to reach any state from any other state in a finite series of steps.<\/span><\/p><p><span style=\"font-weight: 400;\">Our meteorological model is irreducible: the transition from sun to rain (via clouds), from rain to sun, etc., is possible. This property assures the existence of a unique stationary distribution.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f166734 elementor-widget elementor-widget-heading\" data-id=\"f166734\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Periodicity<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-79b6eeb elementor-widget elementor-widget-text-editor\" data-id=\"79b6eeb\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A state has a <\/span><b>period d<\/b><span style=\"font-weight: 400;\"> if it can only be revisited in times that are multiples of d. If d = 1, the state is termed <\/span><b>aperiodic<\/b><span style=\"font-weight: 400;\">. An irreducible and aperiodic chain always converges to its stationary distribution.<\/span><\/p><p><span style=\"font-weight: 400;\">This classification facilitates prediction of the system&#8217;s asymptotic behavior and determines which mathematical tools are applicable for analysis.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5374adb elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"5374adb\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-scientist-course\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Training in data science<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-33688bd e-flex e-con-boxed e-con\" data-id=\"33688bd\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-13cbb9e elementor-widget elementor-widget-heading\" data-id=\"13cbb9e\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Stationary Distribution and Convergence to Equilibrium<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e828359 elementor-widget elementor-widget-image\" data-id=\"e828359\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88.png\" class=\"attachment-full size-full wp-image-198958\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-acc9f6a elementor-widget elementor-widget-text-editor\" data-id=\"acc9f6a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">We previously introduced the notion of convergence to equilibrium. Let&#8217;s delve deeper into this fundamental concept.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-78ae967 elementor-widget elementor-widget-heading\" data-id=\"78ae967\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">What is a Stationary Distribution?<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a1aa96e elementor-widget elementor-widget-text-editor\" data-id=\"a1aa96e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A stationary distribution \u03c0 is a probability vector that stays unchanged by the dynamics of the chain. Mathematically, it fulfills the equation: \u03c0 = \u03c0P, where P is the transition matrix.<\/span><\/p><p><span style=\"font-weight: 400;\">In our weather model, if the stationary distribution is \u03c0 = (0.4, 0.3, 0.3) for (sun, clouds, rain), it implies that in the long term, we will have sun 40% of the time, clouds 30%, and rain 30%, no matter the current weather state.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-064c1bb elementor-widget elementor-widget-heading\" data-id=\"064c1bb\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Existence and Uniqueness<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7a9f1fb elementor-widget elementor-widget-text-editor\" data-id=\"7a9f1fb\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">For an irreducible Markov chain with a finite state space, a unique stationary distribution always exists. This outstanding property ensures the system exhibits predictable long-term behavior.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-988473c elementor-widget elementor-widget-heading\" data-id=\"988473c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">The Ergodic Theorem<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1f450df elementor-widget elementor-widget-text-editor\" data-id=\"1f450df\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The ergodic theorem is one of the most potent results of Markov chain theory. It states that, for an irreducible and aperiodic chain:<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Convergence<\/b><span style=\"font-weight: 400;\">: Regardless of the starting state, probabilities converge to the stationary distribution<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Law of Large Numbers<\/b><span style=\"font-weight: 400;\">: The portion of time spent in each state converges to its stationary probability<\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">Concretely, this means if our weather is observed over numerous years, the frequency of each type of weather experienced converges to the stationary distribution probabilities.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-953bf02 elementor-widget elementor-widget-heading\" data-id=\"953bf02\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Practical Applications of Ergodicity<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e81beac elementor-widget elementor-widget-text-editor\" data-id=\"e81beac\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This property enables the calculation of long-term averages without exhaustive simulation. For example, if a machine is likely operational with a stationary probability of 0.95, it&#8217;s understood that it will be available 95% of the time in the long term\u2014a critical factor for industrial scheduling.<\/span><\/p><p><span style=\"font-weight: 400;\">Ergodicity also backs the use of Markov chains in Monte Carlo sampling algorithms: by letting the chain operate long enough, one can sample according to the desired outcome distribution.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-4cf8775 e-flex e-con-boxed e-con\" data-id=\"4cf8775\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-349aa44 elementor-widget elementor-widget-heading\" data-id=\"349aa44\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Practical Exercises to Master Markov Chains<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e1b4d53 elementor-widget elementor-widget-image\" data-id=\"e1b4d53\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-84.png\" class=\"attachment-full size-full wp-image-198959\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-84.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-84-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-84-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-84-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bbb3ecd elementor-widget elementor-widget-text-editor\" data-id=\"bbb3ecd\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">To reinforce your understanding, here are some classic exercises with detailed solutions.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0560d4b elementor-widget elementor-widget-heading\" data-id=\"0560d4b\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Exercise 1: Ehrenfest Urn<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-aacb482 elementor-widget elementor-widget-text-editor\" data-id=\"aacb482\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">An urn contains 4 numbered balls. At each interval, a number is randomly drawn, and the corresponding ball is shifted to the other urn. The state of the system is the count of balls in the first urn.<\/span><\/p><p><i><span style=\"font-weight: 400;\">Question:<\/span><\/i><span style=\"font-weight: 400;\"> Determine the transition matrix and find the stationary distribution.<\/span><\/p><p><i><span style=\"font-weight: 400;\">Solution:<\/span><\/i><span style=\"font-weight: 400;\"> The potential states are {0,1,2,3,4}. If urn 1 contains i balls, the probability of transitioning into state i-1 is i\/4 (drawing a ball from urn 1), while transitioning into state i+1 is (4-i)\/4 (drawing a ball from urn 2).<\/span><\/p><p><span style=\"font-weight: 400;\">The stationary distribution is aligned with a binomial distribution: \u03c0(i) = C(4,i)\/16, hence \u03c0 = (1\/16, 4\/16, 6\/16, 4\/16, 1\/16).<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-03e3175 elementor-widget elementor-widget-heading\" data-id=\"03e3175\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Exercise 2: Random Walk on a Graph<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2c1e230 elementor-widget elementor-widget-text-editor\" data-id=\"2c1e230\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A pawn maneuvers on a square with vertices A, B, C, D. At each step, it moves to an adjoining vertex with equal probability.<\/span><\/p><p><i><span style=\"font-weight: 400;\">Question:<\/span><\/i><span style=\"font-weight: 400;\"> Is the chain irreducible? Periodic? Calculate the average return time.<\/span><\/p><p class=\"whitespace-normal break-words\"><i><span style=\"font-weight: 400;\">Solution:<\/span><\/i><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Irreducible: Yes, one can move between any vertices<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Periodic: Yes, with a period of 2 (an even number of steps is needed to return to the starting point)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Average return time: 4 steps per vertex<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-4e1cbac elementor-widget elementor-widget-heading\" data-id=\"4e1cbac\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Exercise 3: Lifespan Model<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0948b8d elementor-widget elementor-widget-text-editor\" data-id=\"0948b8d\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">An organism progresses through the states: juvenile (J), mature (M), senescent (S), deceased (D). The transition matrix reads as:<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-02ab908 elementor-widget elementor-widget-code-highlight\" data-id=\"02ab908\" data-element_type=\"widget\" data-widget_type=\"code-highlight.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"prismjs-default copy-to-clipboard \">\n\t\t\t<pre data-line=\"\" class=\"highlight-height language-javascript line-numbers\">\n\t\t\t\t<code readonly=\"true\" class=\"language-javascript\">\n\t\t\t\t\t<xmp>J    M    S    D\nJ [0.6  0.4  0    0  ]\nM [0    0.7  0.3  0  ]\nS [0    0    0.8  0.2]\nD [0    0    0    1  ]<\/xmp>\n\t\t\t\t<\/code>\n\t\t\t<\/pre>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c218159 elementor-widget elementor-widget-text-editor\" data-id=\"c218159\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><i><span style=\"font-weight: 400;\">Question:<\/span><\/i><span style=\"font-weight: 400;\"> Compute the probability that a juvenile organism reaches the senescent stage.<\/span><\/p><p><i><span style=\"font-weight: 400;\">Solution:<\/span><\/i><span style=\"font-weight: 400;\"> The probability needed is 0.4 \u00d7 (0.3\/0.3) = 0.4, as any mature organism inevitably hits the senescent stage.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8a22729 elementor-widget elementor-widget-heading\" data-id=\"8a22729\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Exercise 4: Absorption Time<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3fa26ce elementor-widget elementor-widget-text-editor\" data-id=\"3fa26ce\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Revisiting the previous exercise, calculate the life expectancy of a juvenile organism.<\/span><\/p><p><i><span style=\"font-weight: 400;\">Solution:<\/span><\/i><span style=\"font-weight: 400;\"> Using absorption time equations, the life expectancy of a juvenile approximates 8.33 time units.<\/span><\/p><p><span style=\"font-weight: 400;\">These exercises demonstrate theoretical concepts within tangible scenarios and prepare you to model your problems with Markov chains.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-83a0dc6 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"83a0dc6\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-courses\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Discover our training courses<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-d381a4a e-flex e-con-boxed e-con\" data-id=\"d381a4a\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-bd22fe8 elementor-widget elementor-widget-heading\" data-id=\"bd22fe8\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Extension to Continuous-Time Markov Chains<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-83a465d elementor-widget elementor-widget-image\" data-id=\"83a465d\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-80.png\" class=\"attachment-full size-full wp-image-198960\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-80.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-80-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-80-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-80-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7a9f908 elementor-widget elementor-widget-text-editor\" data-id=\"7a9f908\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">To this point, our study focused on discrete-time Markov chains, where transitions happen at regular intervals (day 1, day 2, etc.). In numerous real-world cases, changes between states can occur at any time: this belongs to <\/span><b>continuous-time Markov processes<\/b><span style=\"font-weight: 400;\">.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-28d253a elementor-widget elementor-widget-heading\" data-id=\"28d253a\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Fundamental Conceptual Differences<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-eeb6af7 elementor-widget elementor-widget-text-editor\" data-id=\"eeb6af7\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Rather than transition matrices, <\/span><b>infinitesimal generators<\/b><span style=\"font-weight: 400;\"> are employed to express instantaneous transition rates. If q_{ij} symbolizes the rate of transition from state i to state j, then the probability of a transition occurring over a small interval dt roughly equates to q_{ij} \u00d7 dt.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-26dae1c elementor-widget elementor-widget-heading\" data-id=\"26dae1c\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Concrete Example: Call Center<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c4e82c1 elementor-widget elementor-widget-text-editor\" data-id=\"c4e82c1\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Consider a call center where calls arrive following a Poisson process with a rate of \u03bb = 5 calls\/hour, each being attended to in an average of 10 minutes (service rate \u03bc = 6 calls\/hour).<\/span><\/p><p><span style=\"font-weight: 400;\">States signify the number of calls being handled. Transitions include:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">State n \u2192 State n+1 with rate \u03bb (incoming call)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">State n \u2192 State n-1 with rate n\u03bc (completion of call handling)<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1ff73df elementor-widget elementor-widget-heading\" data-id=\"1ff73df\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Continuous Chapman-Kolmogorov Equations<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a30406f elementor-widget elementor-widget-text-editor\" data-id=\"a30406f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The system&#8217;s dynamics adhere to the differential equation: dP(t)\/dt = P(t) \u00d7 Q<\/span><\/p><p><span style=\"font-weight: 400;\">where Q is the generator matrix and P(t) provides transition probabilities at time t.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-efea88a elementor-widget elementor-widget-heading\" data-id=\"efea88a\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Stationary Distribution in Continuous Time<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2dada8c elementor-widget elementor-widget-text-editor\" data-id=\"2dada8c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">In equilibrium, the condition becomes: \u03c0 \u00d7 Q = 0, where \u03c0 represents the stationary distribution. This equation elucidates flow equilibrium: for each state, the rate of inflow equates to the outflow rate.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f5faca6 elementor-widget elementor-widget-heading\" data-id=\"f5faca6\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Practical Applications<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-be29b2c elementor-widget elementor-widget-text-editor\" data-id=\"be29b2c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>System Reliability<\/b><span style=\"font-weight: 400;\">: Equipment failures can arise at any time (rate \u03bb) and require repairs (rate \u03bc). The continuous-time process suitably models these unpredictable phenomena.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Epidemiology<\/b><span style=\"font-weight: 400;\">: SIR (Susceptible-Infected-Recovered) models utilize continuous infection and recovery rates for epidemic evolution forecasts.<\/span><\/li><li><b>Quantitative Finance<\/b><span style=\"font-weight: 400;\">: Changes in asset prices follow continuous processes, depicted by diffusions (stochastic extensions of Markov processes).<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-cbe5850 elementor-widget elementor-widget-heading\" data-id=\"cbe5850\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Advantages of Continuous Time<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ad029fb elementor-widget elementor-widget-text-editor\" data-id=\"ad029fb\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This approach authentically mirrors numerous natural phenomena and frequently permits more elegant analytical calculations. It also bridges to more complex models such as diffusion processes and stochastic differential equations.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-6efbac9 e-flex e-con-boxed e-con\" data-id=\"6efbac9\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0045e4a elementor-widget elementor-widget-heading\" data-id=\"0045e4a\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Markov Chain Monte Carlo Methods (MCMC)\n<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-32603eb elementor-widget elementor-widget-image\" data-id=\"32603eb\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-82.png\" class=\"attachment-full size-full wp-image-198961\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-82.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-82-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-82-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-82-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1a79de5 elementor-widget elementor-widget-text-editor\" data-id=\"1a79de5\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Markov chains introduce a revolutionary approach in statistical sampling methods, specifically when distributions defy direct sampling.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-935cab7 elementor-widget elementor-widget-heading\" data-id=\"935cab7\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">The Fundamental Principle of MCMC<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-98afc22 elementor-widget elementor-widget-text-editor\" data-id=\"98afc22\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The brilliant approach is to construct a Markov chain whose stationary distribution is precisely the distribution we aim to sample. Allowing this chain to evolve extensively results in samples that closely match the target distribution.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3fdf9a6 elementor-widget elementor-widget-heading\" data-id=\"3fdf9a6\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">The Metropolis-Hastings Algorithm<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c0f102c elementor-widget elementor-widget-text-editor\" data-id=\"c0f102c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">This universal algorithm facilitates sampling any distribution \u03c0(x), even when known only up to a normalization constant.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b9a0b88 elementor-widget elementor-widget-heading\" data-id=\"b9a0b88\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Operating Principle:<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5a3cf64 elementor-widget elementor-widget-text-editor\" data-id=\"5a3cf64\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ol><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">From the current state x_t, propose a new state y according to a proposal distribution q(y|x_t)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Calculate the acceptance ratio: \u03b1 = min(1, [\u03c0(y)q(x_t|y)] \/ [\u03c0(x_t)q(y|x_t)])<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Adopt y with probability \u03b1, otherwise remain at x_t<\/span><\/li><\/ol><p><span style=\"font-weight: 400;\">This process guarantees that the emergent chain&#8217;s stationary distribution matches \u03c0(x) exactly.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-e3344a0 elementor-widget elementor-widget-heading\" data-id=\"e3344a0\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Applications in Bayesian Statistics<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ca5e52e elementor-widget elementor-widget-text-editor\" data-id=\"ca5e52e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">In Bayesian inference, often the aim is to sample the posterior distribution of parameters. With data D and parameters \u03b8 having a prior \u03c0(\u03b8), the posterior distribution follows:<\/span><\/p><p><span style=\"font-weight: 400;\">\u03c0(\u03b8|D) \u221d \u03c0(D|\u03b8) \u00d7 \u03c0(\u03b8)<\/span><\/p><p><span style=\"font-weight: 400;\">MCMC enables sampling this distribution even when it lacks a straightforward analytical form, marking a revelation in modern statistical analysis.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9412d46 elementor-widget elementor-widget-heading\" data-id=\"9412d46\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Gibbs Sampling<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-697874c elementor-widget elementor-widget-text-editor\" data-id=\"697874c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A special case of Metropolis-Hastings for multivariate distributions. Instead of updating all parameters together, each parameter is updated individually while conditioned on others.<\/span><\/p><p><span style=\"font-weight: 400;\">This method is especially effective in graphical models like Bayesian networks, where many conditional distributions exhibit simple forms<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f9e6860 elementor-widget elementor-widget-heading\" data-id=\"f9e6860\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Convergence Diagnostics<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6c3909f elementor-widget elementor-widget-text-editor\" data-id=\"6c3909f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A major challenge in MCMC is determining when the chain has \u201cforgotten\u201d its initial state and is sampling from the true stationary distribution. Diagnostics include:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Time trace visualization<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Gelman-Rubin criterion (comparison of multiple chains)<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Autocorrelation tests<\/span><span style=\"font-weight: 400;\"><br \/><\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bc958e5 elementor-widget elementor-widget-heading\" data-id=\"bc958e5\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Contemporary Applications<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-818252a elementor-widget elementor-widget-text-editor\" data-id=\"818252a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Artificial Intelligence<\/b><span style=\"font-weight: 400;\">: Bayesian neural networks use MCMC to quantify prediction uncertainty \u2014 a crucial feature for high-stakes applications.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Computational Epidemiology<\/b><span style=\"font-weight: 400;\">: Epidemic spread models with uncertain parameters are calibrated using MCMC based on observed data.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Quantitative Finance<\/b><span style=\"font-weight: 400;\">: The estimation of stochastic volatility models and the pricing of complex options rely heavily on MCMC methods.<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">MCMC perfectly illustrates how a fundamental mathematical concept \u2014 Markov chains \u2014 has become a revolutionary computational tool for modern data science.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-40e168f elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"40e168f\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-courses\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Learning to make predictions<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-10a53aa e-flex e-con-boxed e-con\" data-id=\"10a53aa\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-3a4e25b elementor-widget elementor-widget-heading\" data-id=\"3a4e25b\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Markov Chains in the Modern Data Science Ecosystem<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d3c58df elementor-widget elementor-widget-image\" data-id=\"d3c58df\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88.png\" class=\"attachment-full size-full wp-image-198958\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-88-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-c1fafc5 elementor-widget elementor-widget-text-editor\" data-id=\"c1fafc5\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">In the age of big data and artificial intelligence, Markov chains have found new fields of application that go far beyond their original theoretical framework.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8078232 elementor-widget elementor-widget-heading\" data-id=\"8078232\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Natural Language Processing (NLP)<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-bebda5f elementor-widget elementor-widget-text-editor\" data-id=\"bebda5f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Markov-based language models were the precursors to modern architectures like GPT. They model the probability of a word based on the preceding ones, enabling:<\/span><\/p><ul><li><b>Automatic text generation<\/b><span style=\"font-weight: 400;\">: By chaining transition probabilities, coherent sentences can be produced<\/span><\/li><li><b>Spell checking<\/b><span style=\"font-weight: 400;\">: Markov chains detect improbable character sequences<\/span><\/li><li><b>Text classification<\/b><span style=\"font-weight: 400;\">: Each class (e.g. spam\/non-spam) is modeled with its own Markov chain<\/span><span style=\"font-weight: 400;\"><br \/><\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b9dfc90 elementor-widget elementor-widget-heading\" data-id=\"b9dfc90\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Recommendation Systems<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b824477 elementor-widget elementor-widget-text-editor\" data-id=\"b824477\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Platforms like Netflix or Spotify use Markov chains to model user journeys:<\/span><\/p><ul><li><span style=\"font-weight: 400;\">States: viewed content<\/span><\/li><li><span style=\"font-weight: 400;\">Transitions: probabilities of moving from one item to another<\/span><\/li><li><span style=\"font-weight: 400;\">Goal: predict the next piece of content likely to interest the user<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0d6e961 elementor-widget elementor-widget-heading\" data-id=\"0d6e961\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Time Series Analysis in IoT<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-56edd9c elementor-widget elementor-widget-text-editor\" data-id=\"56edd9c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">In the <a href=\"https:\/\/datascientest.com\/en\/all-about-internet-of-things\">Internet of Things<\/a>, sensors generate sequential data streams. Markov chains enable:<\/span><\/p><ul><li><b>Anomaly detection<\/b><span style=\"font-weight: 400;\">: identifying unusual measurement sequences<\/span><\/li><li><b>Predictive maintenance<\/b><span style=\"font-weight: 400;\">: anticipating failures in industrial equipment<\/span><\/li><li><span style=\"font-weight: 400;\"><b>Energy optimization<\/b><span>: modeling consumption patterns<\/span><\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d149111 elementor-widget elementor-widget-heading\" data-id=\"d149111\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Cybersecurity and Fraud Detection<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-19aec53 elementor-widget elementor-widget-text-editor\" data-id=\"19aec53\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Security analysts use Markov chains to:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Intrusion detection<\/b><span style=\"font-weight: 400;\">: model normal user behavior and flag deviations<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Malware analysis<\/b><span style=\"font-weight: 400;\">: system call sequences reveal malicious software signatures<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Behavioral authentication<\/b><span>: identify users based on their navigation patterns<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b8d202e elementor-widget elementor-widget-heading\" data-id=\"b8d202e\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Digital Marketing and Web Analytics<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f32178a elementor-widget elementor-widget-text-editor\" data-id=\"f32178a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Markov chains enhance the web user experience by enabling:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Customer journey analysis<\/b><span style=\"font-weight: 400;\">: understand conversion paths on e-commerce sites<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Sequential A\/B testing<\/b><span style=\"font-weight: 400;\">: assess the impact of changes on browsing behavior<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Marketing attribution<\/b><span>: measure the effectiveness of different acquisition channels<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9df60da elementor-widget elementor-widget-heading\" data-id=\"9df60da\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Computational Bioinformatics<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f9192f2 elementor-widget elementor-widget-text-editor\" data-id=\"f9192f2\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Beyond classic DNA analysis, modern applications include:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Gene expression analysis<\/b><span style=\"font-weight: 400;\">: model the temporal evolution of gene activity<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Genomic epidemiology<\/b><span style=\"font-weight: 400;\">: trace the spread of viral variants through sequence analysis<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Personalized medicine<\/b><span>: predict disease progression using sequential biomarkers<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1b16b8f elementor-widget elementor-widget-heading\" data-id=\"1b16b8f\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Integration with Machine Learning<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8158c0c elementor-widget elementor-widget-text-editor\" data-id=\"8158c0c\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Markov chains are increasingly blended with modern techniques:<\/span><\/p><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Feature engineering<\/b><span style=\"font-weight: 400;\">: extract explanatory variables from time series data<\/span><span style=\"font-weight: 400;\"><br \/><\/span><b><\/b><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Reinforcement learning<\/b><span style=\"font-weight: 400;\">: Markov decision processes underpin learning by interaction<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Recurrent neural networks<\/b><span>: LSTMs and GRUs generalize Markov chains with adaptive memory<\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">These applications show that Markov chains, far from being a purely academic concept, are a foundational methodology for extracting value from sequential data in the modern digital economy.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-37d836c e-flex e-con-boxed e-con\" data-id=\"37d836c\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-2f79796 elementor-widget elementor-widget-heading\" data-id=\"2f79796\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Hidden Markov Model<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-cee1fba elementor-widget elementor-widget-image\" data-id=\"cee1fba\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-96.png\" class=\"attachment-full size-full wp-image-198962\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-96.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-96-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-96-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-96-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d17c806 elementor-widget elementor-widget-text-editor\" data-id=\"d17c806\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">We keep the same assumptions as in the previous section.<\/span><\/p><p><span style=\"font-weight: 400;\">Now imagine a weather-obsessed psychopath has locked you in a windowless room with only a computer and a lamp. <strong>Each day, the lamp lights up in a certain color depending on the weather.<\/strong> Your captor gives you the following observation matrix:<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f45163d elementor-widget elementor-widget-image\" data-id=\"f45163d\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"512\" height=\"252\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image5_markdown.png\" class=\"attachment-full size-full wp-image-172325\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image5_markdown.png 512w, https:\/\/datascientest.com\/en\/files\/2023\/10\/image5_markdown-300x148.png 300w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-fb0d3fb elementor-widget elementor-widget-text-editor\" data-id=\"fb0d3fb\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">For example, if it&#8217;s raining, the lamp has a 70% chance of turning green and a 30% chance of turning blue.<\/span><\/p><p><span style=\"font-weight: 400;\">You\u2019ll be allowed to leave if you can determine the weather for the next five days based solely on the lamp\u2019s color. You build a<strong> Hidden Markov Model<\/strong> to solve the task.<\/span><\/p><p><span style=\"font-weight: 400;\">You stay locked in for 5 days and observe the following lamp colors:<\/span><\/p><p style=\"padding-left: 40px;\"><b>blue, blue, red, green, red<\/b><\/p><p><span style=\"font-weight: 400;\">You remember that it was raining the day before you were locked up.<\/span><\/p><p><span style=\"font-weight: 400;\">You then write a Python script that returns the most likely weather sequence:<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-cf075b8 elementor-widget elementor-widget-shortcode\" data-id=\"cf075b8\" data-element_type=\"widget\" data-widget_type=\"shortcode.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-shortcode\"><script src=\"https:\/\/gist.github.com\/MaximeRighini\/04208ab8412a1dd41fd5eebfeee906d2.js\"><\/script><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a512192 elementor-widget elementor-widget-image\" data-id=\"a512192\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"422\" height=\"183\" src=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image6_markdown.png\" class=\"attachment-full size-full wp-image-172326\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2023\/10\/image6_markdown.png 422w, https:\/\/datascientest.com\/en\/files\/2023\/10\/image6_markdown-300x130.png 300w\" sizes=\"(max-width: 422px) 100vw, 422px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-be67f3a elementor-widget elementor-widget-text-editor\" data-id=\"be67f3a\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><b>Note:<\/b><span style=\"font-weight: 400;\"> You could also implement the Viterbi algorithm, which would return the most probable sequence \u2014 in this case: <\/span><span style=\"font-weight: 400;\">[&#8216;Cloudy&#8217;, &#8216;Cloudy&#8217;, &#8216;Sunny&#8217;, &#8216;Rainy&#8217;, &#8216;Sunny&#8217;]<\/span><span style=\"font-weight: 400;\">.<\/span><\/p><p><span style=\"font-weight: 400;\">This prediction approach is interesting but still basic. If you want to learn how to make more impressive predictions \u2014 with machine learning algorithms, for example \u2014 <\/span><a href=\"https:\/\/datascientest.com\/rdv\"><span style=\"font-weight: 400;\">get in touch with us online to learn more about our data science programs<\/span><\/a><span style=\"font-weight: 400;\">!<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d9007d8 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"d9007d8\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-courses\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Learn more about Markov chains<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-42499ef e-flex e-con-boxed e-con\" data-id=\"42499ef\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5aff48b elementor-widget elementor-widget-heading\" data-id=\"5aff48b\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Deep Dive into Hidden Markov Models (HMMs)<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1a8c320 elementor-widget elementor-widget-image\" data-id=\"1a8c320\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-95.png\" class=\"attachment-full size-full wp-image-198963\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-95.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-95-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-95-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-95-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0cc5b49 elementor-widget elementor-widget-text-editor\" data-id=\"0cc5b49\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Our \u201cweather-obsessed psychopath\u201d example offers a playful introduction to HMMs, but these models deserve a more technical breakdown, as they are among the most powerful tools in statistical learning.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d2abd38 elementor-widget elementor-widget-heading\" data-id=\"d2abd38\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Mathematical Structure of an HMM<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9b1c068 elementor-widget elementor-widget-text-editor\" data-id=\"9b1c068\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">A Hidden Markov Model consists of three core components:<\/span><\/p><ol><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Hidden states<\/b><span style=\"font-weight: 400;\">: the unobservable sequence of states (in this case, the actual weather)<\/span><span style=\"font-weight: 400;\"><br \/><\/span><b><\/b><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Observations<\/b><span style=\"font-weight: 400;\">: what we can measure (the color of the lamp)<\/span><span style=\"font-weight: 400;\"><br \/><\/span><b><\/b><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Two probability matrices<\/b><span style=\"font-weight: 400;\">:<\/span><span style=\"font-weight: 400;\"><br \/><\/span><b><\/b><ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>A<\/b><span style=\"font-weight: 400;\">, the transition matrix between hidden states<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>B<\/b><span>, the emission\/observation matrix<\/span><\/li><\/ul><\/li><\/ol>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-48999e0 elementor-widget elementor-widget-heading\" data-id=\"48999e0\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">The Three Fundamental HMM Problems<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-218235e elementor-widget elementor-widget-text-editor\" data-id=\"218235e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">HMM theory revolves around three central questions:<\/span><\/p><p><b>Problem 1: Evaluation<\/b><span style=\"font-weight: 400;\"> What is the probability of observing a given sequence?<\/span><span style=\"font-weight: 400;\"><br \/><\/span><span style=\"font-weight: 400;\">The <\/span><b>Forward algorithm<\/b><span style=\"font-weight: 400;\"> efficiently computes this likelihood while avoiding combinatorial explosion.<\/span><\/p><p><b>Problem 2: Decoding<\/b><span style=\"font-weight: 400;\"> What is the most probable sequence of hidden states?<\/span><span style=\"font-weight: 400;\"><br \/><\/span><span style=\"font-weight: 400;\">The <\/span><b>Viterbi algorithm<\/b><span style=\"font-weight: 400;\"> solves this using dynamic programming. In our weather example, it determines the most likely sequence of weather conditions given the observed lamp colors.<\/span><\/p><p><b>Problem 3: Learning <\/b><span style=\"font-weight: 400;\">How can the model parameters be estimated from observations?<\/span><span style=\"font-weight: 400;\"><br \/><\/span><span style=\"font-weight: 400;\">The <\/span><b>Baum-Welch algorithm<\/b><span style=\"font-weight: 400;\"> (a variant of the EM algorithm) iteratively optimizes matrices <\/span><b>A<\/b><span style=\"font-weight: 400;\"> and <\/span><b>B<\/b><span style=\"font-weight: 400;\"> to fit the data.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-593d2a6 elementor-widget elementor-widget-heading\" data-id=\"593d2a6\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Advanced Applications of HMMs<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5410445 elementor-widget elementor-widget-text-editor\" data-id=\"5410445\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Automatic Speech Recognition: <\/b><span style=\"font-weight: 400;\">Phonemes are the hidden states, and audio spectrograms are the observations. HMMs model how each phoneme generates specific acoustic features.<\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Biological Sequence Analysis:<\/b><span style=\"font-weight: 400;\"> In bioinformatics, HMMs identify genes in DNA: coding and non-coding regions are the hidden states, while nucleotides (A, T, G, C) are the observations.<\/span><\/li><li><b>Quantitative Finance:<\/b><span style=\"font-weight: 400;\"> Market regimes (bull\/bear) are hidden, and asset returns are observed. HMMs detect regime shifts to adjust investment strategies accordingly.<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-0a2d9c0 elementor-widget elementor-widget-heading\" data-id=\"0a2d9c0\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Limitations and Extensions<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-ce86c52 elementor-widget elementor-widget-text-editor\" data-id=\"ce86c52\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Classic HMMs assume that observations are conditionally independent given the states. <strong>Hierarchical HMMs<\/strong> and <a href=\"https:\/\/datascientest.com\/en\/recurrent-neural-network-what-is-it\">recurrent neural networks<\/a> lift this limitation to enable more sophisticated modeling.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-7793900 e-flex e-con-boxed e-con\" data-id=\"7793900\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5d21bb4 elementor-widget elementor-widget-heading\" data-id=\"5d21bb4\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Real-World Applications of Markov Chains<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-22815a1 elementor-widget elementor-widget-image\" data-id=\"22815a1\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-89.png\" class=\"attachment-full size-full wp-image-198964\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-89.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-89-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-89-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-89-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-059ef6f elementor-widget elementor-widget-text-editor\" data-id=\"059ef6f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Beyond weather prediction, Markov chains power many real-world systems:<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9a14777 elementor-widget elementor-widget-heading\" data-id=\"9a14777\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">PageRank and Search Engines<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-5f0f3b8 elementor-widget elementor-widget-text-editor\" data-id=\"5f0f3b8\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Google\u2019s PageRank algorithm uses a Markov chain to rank web pages. Each page is a state, and hyperlinks define transition probabilities. A page\u2019s importance is its probability in the stationary distribution of the chain.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9aa4ecb elementor-widget elementor-widget-heading\" data-id=\"9aa4ecb\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Finance and Insurance<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d1736c2 elementor-widget elementor-widget-text-editor\" data-id=\"d1736c2\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Car insurance companies use Markov chains in bonus-malus systems. The state represents a driver\u2019s bonus class, and transitions depend on the number of accidents reported annually. This modeling enables actuarial calculation of insurance premiums.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-98a501a elementor-widget elementor-widget-heading\" data-id=\"98a501a\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Bioinformatics and Genetics<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3c3e44e elementor-widget elementor-widget-text-editor\" data-id=\"3c3e44e\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p class=\"whitespace-normal break-words\"><span style=\"font-weight: 400;\">In DNA sequence analysis, Markov chains model the relationships between successive nucleotides (A, T, G, C). Hidden Markov Models (HMMs) are especially useful for gene identification, protein structure prediction, and detecting coding regions in the genome.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-38c0353 elementor-widget elementor-widget-heading\" data-id=\"38c0353\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Industrial Reliability<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-04f5b20 elementor-widget elementor-widget-text-editor\" data-id=\"04f5b20\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">In industry, Markov chains model technical systems with states such as \u201coperational,\u201d \u201cfailed,\u201d or \u201cunder maintenance.\u201d This allows for equipment availability calculations, preventive maintenance planning, and repair cost optimization.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-2d24042 elementor-widget elementor-widget-heading\" data-id=\"2d24042\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Modern Artificial Intelligence<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6f3b4e8 elementor-widget elementor-widget-text-editor\" data-id=\"6f3b4e8\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\"><a href=\"https:\/\/datascientest.com\/en\/all-about-chatbots\">Chatbots<\/a> and text generation systems use Markov chains to predict the next word in a sentence. Smartphone predictive keyboards apply the same principle to suggest words based on typing context.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a03c5dc elementor-widget elementor-widget-heading\" data-id=\"a03c5dc\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Queueing Theory<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3f535de elementor-widget elementor-widget-text-editor\" data-id=\"3f535de\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">Call centers, bank counters, and computer servers are modeled using Markov chains, where states represent the number of customers waiting. This helps optimize resource allocation and reduce wait times.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-a22d1c7 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"a22d1c7\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-courses\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Learn to code an LLM<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-63ca9fe e-flex e-con-boxed e-con\" data-id=\"63ca9fe\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-cf35125 elementor-widget elementor-widget-heading\" data-id=\"cf35125\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h2 class=\"elementor-heading-title elementor-size-xl\">Further Reading: Essential References<\/h2>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-b406803 elementor-widget elementor-widget-image\" data-id=\"b406803\" data-element_type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"1536\" height=\"1024\" src=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-91.png\" class=\"attachment-full size-full wp-image-198965\" alt=\"\" srcset=\"https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-91.png 1536w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-91-300x200.png 300w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-91-1024x683.png 1024w, https:\/\/datascientest.com\/en\/files\/2025\/08\/output1-91-768x512.png 768w\" sizes=\"(max-width: 1536px) 100vw, 1536px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8589c9f elementor-widget elementor-widget-text-editor\" data-id=\"8589c9f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<p><span style=\"font-weight: 400;\">The theory of Markov chains is a rich mathematical field supported by an extensive academic literature. Here are the must-have resources to deepen your understanding.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-877f9f6 elementor-widget elementor-widget-heading\" data-id=\"877f9f6\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Core Reference Texts<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6bb9fed elementor-widget elementor-widget-text-editor\" data-id=\"6bb9fed\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Norris, J.R. \u2013 <\/b><b><i>Markov Chains<\/i><\/b><b> (Cambridge University Press, 1997)<\/b><b><br \/><\/b><span style=\"font-weight: 400;\"> Widely regarded as <\/span><i><span style=\"font-weight: 400;\">the<\/span><\/i><span style=\"font-weight: 400;\"> modern reference on Markov chains. Offers a rigorous yet accessible treatment of the theory. Essential for building a strong mathematical foundation.<\/span><span style=\"font-weight: 400;\"><br \/><br \/><\/span><\/li><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Sericola, Bruno \u2013 <\/b><b><i>Cha\u00eenes de Markov \u2013 Th\u00e9orie, algorithmes et applications<\/i><\/b><b> (Hermes\/Lavoisier, 2013)<\/b><b><br \/><\/b><span style=\"font-weight: 400;\"> A leading French-language reference, especially valued for its practical applications and detailed algorithms.<\/span><span style=\"font-weight: 400;\"><br \/><br \/><\/span><\/li><li><b>Baldi, Paolo et al. \u2013 <\/b><b><i>Martingales et cha\u00eenes de Markov. Th\u00e9orie \u00e9l\u00e9mentaire et exercices corrig\u00e9s<\/i><\/b><b> (Hermann, 2001)<\/b><b><br \/><\/b><span style=\"font-weight: 400;\"> Ideal for learners, with numerous progressive exercises and solutions.<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9e61bee elementor-widget elementor-widget-heading\" data-id=\"9e61bee\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Domain-Specific References<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-f291720 elementor-widget elementor-widget-text-editor\" data-id=\"f291720\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>M\u00e9l\u00e9ard, Sylvie \u2013 <\/b><b><i>Mod\u00e8les al\u00e9atoires en \u00e9cologie et \u00e9volution<\/i><\/b><b> (Springer, 2016)<\/b><b><br \/><\/b><span style=\"font-weight: 400;\"> Focused on biological and ecological applications, featuring population models based on Markov chains.<\/span><span style=\"font-weight: 400;\"><br \/><br \/><\/span><\/li><li><b>Bremaud, Pierre \u2013 <\/b><b><i>Markov Chains. Gibbs Fields, Monte Carlo Simulation, and Queues<\/i><\/b><b> (Springer, 1998)<\/b><b><br \/><\/b><span style=\"font-weight: 400;\"> A technical reference covering queueing theory and Monte Carlo methods via Markov chains.<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-138b5ef elementor-widget elementor-widget-heading\" data-id=\"138b5ef\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Historical Foundational Papers<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-569dc56 elementor-widget elementor-widget-text-editor\" data-id=\"569dc56\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li><b>Markov, A.A. \u2013 <\/b><b><i>Extension of the Law of Large Numbers to Dependent Quantities<\/i><\/b><b> (1906)<\/b><b><br \/><\/b><span style=\"font-weight: 400;\"> The original paper introducing the concept of Markov chains. Available in translation through the <\/span><i><span style=\"font-weight: 400;\">Electronic Journal for History of Probability and Statistics<\/span><\/i><span style=\"font-weight: 400;\">.<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-d7bcff2 elementor-widget elementor-widget-heading\" data-id=\"d7bcff2\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Additional Learning Resources<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-7bb681f elementor-widget elementor-widget-text-editor\" data-id=\"7bb681f\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Grinstead, Charles M. &amp; Snell, J. Laurie \u2013 <\/b><b><i>Introduction to Probability<\/i><\/b><b> (AMS)<\/b><b><br \/><\/b><span style=\"font-weight: 400;\"> Features an excellent chapter on Markov chains, with a clear pedagogical approach and online interactive exercises.<\/span><span style=\"font-weight: 400;\"><br \/><br \/><\/span><\/li><li><b>Online exercises <\/b><span style=\"font-weight: 400;\">The University of Nice Sophia Antipolis offers interactive, corrected exercises via the WIMS platform \u2014 ideal for hands-on practice.<\/span><\/li><\/ul>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-3ef6f72 elementor-widget elementor-widget-heading\" data-id=\"3ef6f72\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">Modern Digital Resources<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-9aa8f1b elementor-widget elementor-widget-text-editor\" data-id=\"9aa8f1b\" data-element_type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<ul><li style=\"font-weight: 400;\" aria-level=\"1\"><b>Philippe Gay \u2013 <\/b><b><i>Markov et la Belle au bois dormant<\/i><\/b><b> (Images des Maths, 2014)<\/b><b><br \/><\/b><span style=\"font-weight: 400;\"> An outstanding popular science article explaining modern challenges of Markov chains in an accessible and engaging way.<\/span><span style=\"font-weight: 400;\"><br \/><br \/><\/span><\/li><\/ul><p><span style=\"font-weight: 400;\">These references will help you progress from introductory concepts to advanced topics, depending on your learning goals and areas of application.<\/span><\/p>\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-8f74e21 elementor-align-center button-principale elementor-widget elementor-widget-button\" data-id=\"8f74e21\" data-element_type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-grow\" href=\"https:\/\/datascientest.com\/en\/data-courses\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t<span class=\"elementor-button-text\">Discover our training courses<\/span>\n\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-d2102bc e-flex e-con-boxed e-con\" data-id=\"d2102bc\" data-element_type=\"container\" data-settings=\"{&quot;content_width&quot;:&quot;boxed&quot;}\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-968ddac elementor-widget elementor-widget-html\" data-id=\"968ddac\" data-element_type=\"widget\" data-widget_type=\"html.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t<script type=\"application\/ld+json\">\r\n{\r\n  \"@context\": \"https:\/\/schema.org\",\r\n  \"@type\": \"FAQPage\",\r\n  \"mainEntity\": [\r\n    {\r\n      \"@type\": \"Question\",\r\n      \"name\": \"Rigorous Mathematical Definition\",\r\n      \"acceptedAnswer\": {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"A sequence of random variables forms a Markov chain if the probability of moving to the next state depends only on the current state, not on the full history.\"\r\n      }\r\n    },\r\n    {\r\n      \"@type\": \"Question\",\r\n      \"name\": \"Key Elements of a Markov Chain\",\r\n      \"acceptedAnswer\": {\r\n        \"@type\": \"Answer\",\r\n        \"text\": \"The state space defines all possible states; 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If we observe the states in which the model exists at each moment, we have what is known as an observable Markov model. Otherwise, it is referred to as a hidden Markov [&hellip;]<\/p>\n","protected":false},"author":45,"featured_media":198951,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_theme","format":"standard","meta":{"content-type":"","inline_featured_image":false},"categories":[2425],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is a Markov Chain?<\/title>\n<meta name=\"description\" content=\"A model is said to have the Markov property if its state at time T depends only on its state at time T-1. If we can observe the states in which the model is at each time, we speak of an observable Markov model. Otherwise, we speak of a hidden Markov model. 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