# A Gentle Introduction to Expectation-Maximization (EM Algorithm)

Last Updated on August 28, 2020 Maximum likelihood estimation is an approach to density estimation for a dataset by searching across probability distributions and their parameters. It is a general and effective approach that underlies many machine learning algorithms, although it requires that the training dataset is complete, e.g. all relevant interacting random variables are present. Maximum likelihood becomes intractable if there are variables that interact with those in the dataset but were hidden or not observed, so-called latent variables. […]

# A Gentle Introduction to Monte Carlo Sampling for Probability

Monte Carlo methods are a class of techniques for randomly sampling a probability distribution. There are many problem domains where describing or estimating the probability distribution is relatively straightforward, but calculating a desired quantity is intractable. This may be due to many reasons, such as the stochastic nature of the domain or an exponential number of random variables. Instead, a desired quantity can be approximated by using random sampling, referred to as Monte Carlo methods. These methods were initially used […]

# A Gentle Introduction to Markov Chain Monte Carlo for Probability

Probabilistic inference involves estimating an expected value or density using a probabilistic model. Often, directly inferring values is not tractable with probabilistic models, and instead, approximation methods must be used. Markov Chain Monte Carlo sampling provides a class of algorithms for systematic random sampling from high-dimensional probability distributions. Unlike Monte Carlo sampling methods that are able to draw independent samples from the distribution, Markov Chain Monte Carlo methods draw samples where the next sample is dependent on the existing sample, […]

# A Gentle Introduction to Maximum a Posteriori (MAP) for Machine Learning

Density estimation is the problem of estimating the probability distribution for a sample of observations from a problem domain. Typically, estimating the entire distribution is intractable, and instead, we are happy to have the expected value of the distribution, such as the mean or mode. Maximum a Posteriori or MAP for short is a Bayesian-based approach to estimating a distribution and model parameters that best explain an observed dataset. This flexible probabilistic framework can be used to provide a Bayesian […]

# What Does Stochastic Mean in Machine Learning?

Last Updated on July 24, 2020 The behavior and performance of many machine learning algorithms are referred to as stochastic. Stochastic refers to a variable process where the outcome involves some randomness and has some uncertainty. It is a mathematical term and is closely related to “randomness” and “probabilistic” and can be contrasted to the idea of “deterministic.” The stochastic nature of machine learning algorithms is an important foundational concept in machine learning and is required to be understand in […]

# How to Use an Empirical Distribution Function in Python

Last Updated on August 28, 2020 An empirical distribution function provides a way to model and sample cumulative probabilities for a data sample that does not fit a standard probability distribution. As such, it is sometimes called the empirical cumulative distribution function, or ECDF for short. In this tutorial, you will discover the empirical probability distribution function. After completing this tutorial, you will know: Some data samples cannot be summarized using a standard distribution. An empirical distribution function provides a […]

# A Gentle Introduction to the Bayes Optimal Classifier

Last Updated on August 19, 2020 The Bayes Optimal Classifier is a probabilistic model that makes the most probable prediction for a new example. It is described using the Bayes Theorem that provides a principled way for calculating a conditional probability. It is also closely related to the Maximum a Posteriori: a probabilistic framework referred to as MAP that finds the most probable hypothesis for a training dataset. In practice, the Bayes Optimal Classifier is computationally expensive, if not intractable […]