{"id":10568,"date":"2026-02-26T15:06:02","date_gmt":"2026-02-26T09:36:02","guid":{"rendered":"https:\/\/learninglabb.com\/?p=10568"},"modified":"2026-02-26T15:07:44","modified_gmt":"2026-02-26T09:37:44","slug":"maximum-likelihood-estimation-in-machine-learning","status":"publish","type":"post","link":"https:\/\/learninglabb.com\/maximum-likelihood-estimation-in-machine-learning\/","title":{"rendered":"Maximum Likelihood Estimation in Machine Learning: A Simple Guide with Examples\u00a0"},"content":{"rendered":"\n<p>Maximum likelihood estimation in machine learning forms the backbone of how models learn from data by finding parameters that make observations most probable. This method powers many algorithms you&nbsp;encounter&nbsp;daily, from predicting customer behaviour in Indian e-commerce to analysing healthcare data in tech hubs.&nbsp;&nbsp;<\/p>\n\n\n\n<p>In this blog, we break it down step by step for anyone starting out.&nbsp;This blog covers what is maximum likelihood estimation in depth, followed by the maximum likelihood estimation formula presented as clear equation images for easy reference.&nbsp;The blog&nbsp;then examines&nbsp;specific cases such as maximum likelihood estimation for&nbsp;Poisson distribution and maximum likelihood estimation exponential distribution, complete with derivations&nbsp;and more\u2026&nbsp;<\/p>\n\n\n\n<p>Read on\u2026&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>What is Maximum Likelihood Estimation?<\/strong>&nbsp;<\/h2>\n\n\n\n<p><em>What is maximum likelihood estimation?<\/em>\u00a0At its heart, it\u00a0represents\u00a0a statistical method designed to estimate the parameters of a <a href=\"https:\/\/learninglabb.com\/probability-distributions-in-data-science-types\/\" target=\"_blank\" rel=\"noreferrer noopener\">probability distribution<\/a> by maximising a likelihood function that measures how well those parameters explain the observed data. Picture a scenario where you analyse rainfall data from Kerala monsoons: MLE\u00a0determines\u00a0the rate parameter that best matches recorded amounts, treating data as the fixed truth while adjusting model assumptions.\u00a0<a href=\"https:\/\/www.researchgate.net\/publication\/228440352_An_Introduction_to_Maximum_Likelihood_Estimation_and_Information_Geometry\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/www.researchgate.net\/publication\/228440352_An_Introduction_to_Maximum_Likelihood_Estimation_and_Information_Geometry<\/a><\/p>\n\n\n\n<p>Ronald Fisher, the pioneer behind this in the 1920s, described it as &#8220;the method which gives mathematically the most powerful test&#8221; for hypotheses, a view echoed in modern statistics texts. Unlike moment matching which equates sample moments to theoretical ones, MLE directly optimises probability, making it more precise for complex models.&nbsp;<\/p>\n\n\n\n<p>Key advantages include:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Adaptability to various data types, from discrete events like app downloads to continuous variables like stock prices.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Foundation for supervised learning where labels guide parameter tuning.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Handles multivariate cases, vital for India&#8217;s big data challenges in telecom or finance.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p><em>Ask yourself<\/em>: If your dataset shows outliers, does MLE still perform? It does, but with caveats we discuss later. In machine learning pipelines, libraries like scikit-learn embed MLE implicitly, powering classifiers used by startups in Hyderabad&#8217;s tech parks.&nbsp;<\/p>\n\n\n\n<p>This method gained traction in India through courses at IITs, where students apply it to local problems like traffic flow prediction in Delhi.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>The Maximum Likelihood Estimation Formula<\/strong>&nbsp;<\/h2>\n\n\n\n<p>The maximum likelihood estimation formula defines the estimator as the value of \u03b8 that maximises the likelihood function based on observed data. For independent observations x\u2081 through x\u2099 from a distribution f(x|\u03b8), it takes the form shown below.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"726\" height=\"63\" src=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-1.png\" alt=\"\" class=\"wp-image-10570\" srcset=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-1.png 726w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-1-300x26.png 300w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-1-200x17.png 200w\" sizes=\"(max-width: 726px) 100vw, 726px\" \/><\/figure>\n\n\n\n<p>Direct maximisation of products becomes cumbersome with large datasets, so practitioners take the natural logarithm to create the log-likelihood.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"687\" height=\"78\" src=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image.png\" alt=\"\" class=\"wp-image-10569\" srcset=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image.png 687w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-300x34.png 300w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-200x23.png 200w\" sizes=\"(max-width: 687px) 100vw, 687px\" \/><\/figure>\n\n\n\n<p>Maximising \u2113(\u03b8) yields the same result as L(\u03b8) since the log function increases monotonically. To solve, compute the partial derivative with respect to \u03b8, set it to zero, and verify the second derivative is negative for a maximum.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"531\" height=\"72\" src=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-2.png\" alt=\"\" class=\"wp-image-10571\" srcset=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-2.png 531w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-2-300x41.png 300w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-2-200x27.png 200w\" sizes=\"(max-width: 531px) 100vw, 531px\" \/><\/figure>\n\n\n\n<p>In code, optimisation routines minimise the negative log-likelihood using gradient descent, a staple in TensorFlow or&nbsp;PyTorch. For Indian developers, this translates to faster model training on cloud platforms like AWS Mumbai region.&nbsp;<\/p>\n\n\n\n<p><strong><em>Why focus on logs?<\/em><\/strong>&nbsp;They convert multiplications to sums, stabilise numerical computations, and simplify derivatives,&nbsp;essential for high-dimensional parameters in deep learning.&nbsp;&nbsp;<\/p>\n\n\n\n<p>Common&nbsp;challenges&nbsp;include assuming independence; when violated, use adjusted forms like for time series.&nbsp;<\/p>\n\n\n\n<p><strong>Step-by-Step Derivation<\/strong>&nbsp;<\/p>\n\n\n\n<p>Follow this structured derivation to compute MLE for any distribution.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"683\" height=\"1024\" src=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-6-683x1024.png\" alt=\"Maximum likelihood estimation in machine learning\" class=\"wp-image-10575\" srcset=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-6-683x1024.png 683w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-6-200x300.png 200w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-6-768x1151.png 768w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-6.png 903w\" sizes=\"(max-width: 683px) 100vw, 683px\" \/><\/figure>\n\n\n\n<p>As per statistical theory, &#8220;MLE achieves the highest possible accuracy asymptotically,&#8221; per Cram\u00e9r&#8217;s work in the 1940s. In practice, Indian researchers at IISc Bengaluru use this for genomic data analysis.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Maximum Likelihood Estimation for Poisson Distribution<\/strong>&nbsp;<\/h2>\n\n\n\n<p>Maximum likelihood estimation for&nbsp;Poisson distribution proves ideal for count data such as daily COVID cases in a city or transaction volumes in Paytm.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"903\" height=\"465\" src=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-5.png\" alt=\"\" class=\"wp-image-10574\" srcset=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-5.png 903w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-5-300x154.png 300w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-5-768x395.png 768w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-5-200x103.png 200w\" sizes=\"(max-width: 903px) 100vw, 903px\" \/><\/figure>\n\n\n\n<p>Applications:&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Fraud detection: Model rare events per hour.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Queueing theory: Customer arrivals at IRCTC booking.&nbsp;<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>GLM extensions in R for overdispersion handling.&nbsp;<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Maximum Likelihood Estimation Exponential Distribution<\/strong>&nbsp;<\/h2>\n\n\n\n<p>Maximum likelihood estimation exponential distribution fits lifetimes, service times, or inter-arrival gaps, relevant for India&#8217;s renewable energy sector analysing panel failures.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"903\" height=\"498\" src=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-4.png\" alt=\"\" class=\"wp-image-10573\" srcset=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-4.png 903w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-4-300x165.png 300w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-4-768x424.png 768w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-4-200x110.png 200w\" sizes=\"(max-width: 903px) 100vw, 903px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Maximum Likelihood Estimation Example: Coin Flips and Logistic<\/strong>&nbsp;<\/h2>\n\n\n\n<p>Start with a basic maximum likelihood estimation example: 10 flips, 7 heads. Bernoulli model.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"903\" height=\"228\" src=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-3.png\" alt=\"\" class=\"wp-image-10572\" srcset=\"https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-3.png 903w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-3-300x76.png 300w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-3-768x194.png 768w, https:\/\/learninglabb.com\/wp-content\/uploads\/2026\/02\/image-3-200x50.png 200w\" sizes=\"(max-width: 903px) 100vw, 903px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>On A Final Note\u2026<\/strong>&nbsp;<\/h2>\n\n\n\n<p>Maximum likelihood estimation in machine learning delivers a robust framework for parameter learning, from basic distributions to advanced neural architectures. Experiment with datasets from UCI repository to solidify skills, after all the&nbsp;path to expertise lies in practice.&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>FAQs<\/strong>&nbsp;<\/h2>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1772015564501\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><strong>What is maximum likelihood estimation in simple terms?<\/strong><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>A way to pick model parameters that make your data most likely, like estimating rainfall rate from monsoon logs.\u00a0<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1772015582613\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><strong>What is the maximum likelihood estimation formula?<\/strong>\u00a0<\/h3>\n<div class=\"rank-math-answer \">\n\n<p>Maximise \u2211 log f(x_i|\u03b8) for independent data.\u00a0<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Maximum likelihood estimation in machine learning forms the backbone of how models learn from data by finding parameters that make observations most probable. This method powers many algorithms you&nbsp;encounter&nbsp;daily, from predicting customer behaviour in Indian e-commerce to analysing healthcare data in tech hubs.&nbsp;&nbsp; In this blog, we break it down step by step for anyone [&hellip;]<\/p>\n","protected":false},"author":10,"featured_media":10578,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_eb_attr":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[21,69],"tags":[],"class_list":["post-10568","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-data-science","category-data-analytics"],"_links":{"self":[{"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/posts\/10568","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/comments?post=10568"}],"version-history":[{"count":3,"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/posts\/10568\/revisions"}],"predecessor-version":[{"id":10580,"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/posts\/10568\/revisions\/10580"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/media\/10578"}],"wp:attachment":[{"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/media?parent=10568"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/categories?post=10568"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/learninglabb.com\/wp-json\/wp\/v2\/tags?post=10568"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}