How is the Expectation-Maximization algorithm used
For problems involving latent or hidden variables, the expectation-maximization (EM) approach is an elegant technique that maximizes the likelihood function.
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Updated November 2026. This was first published on 22 April 2022. The idea is the same. Any number below is what the original piece said then. It is not a new count.
Key takeaways
- For problems involving latent or hidden variables, the expectation-maximization (EM) approach is an elegant technique that maximizes the likelihood function.
- From the name alone, it is clear that it performs two functions: one is an expectation, and the other is maximizing.
- The main goal of EM algorithms is to examine the problem of calculating the maximum likelihood estimates (MLE) of a statistical model when latent variables are present and data is missing or incomplete.
Introduction #
For problems involving latent or hidden variables, the expectation-maximization (EM) approach is an elegant technique that maximizes the likelihood function. From the name alone, it is clear that it performs two functions: one is an expectation, and the other is maximizing.
The main goal of EM algorithms is to examine the problem of calculating the maximum likelihood estimates (MLE) of a statistical model when latent variables are present and data is missing or incomplete. If the data has already been observed on some randomly picked variables, it would be easier to compute.
Let us look at how the combination of expectation and maximization may assist determine the number of clusters to be generated, but first, we must grasp the idea of the latent variable.
What is the latent variable? #
A latent variable is a random variable that is not visible during the training or testing phases. These characteristics are not measurable on a numerical scale.
The missing values in the dataset might be supplied with all of the techniques in the book. they will still introduce uncertainty into any probabilistic model derived from it. To calculate the degree of uncertainty in predictions.
All of the parameters are directly caused by the latent variable. The final model is now much easier to work with and has the same efficiency without sacrificing the model’s flexibility. One disadvantage of latent variables is that they are more difficult to train.
What is an EM algorithm? #
In statistical models, the EM approach may be used to find the local maximum likelihood estimates/parameters (MLE) or maximum a posteriori (MAP) estimates/parameters for latent variables (unobservable variables inferred from observable variables).
Or, to put it another way, the EM method in machine learning uses observable examples of latent variables to predict values in instances that are unobservable for learning, and it continues until the values converge.
Many unsupervised clustering techniques in the field of machine learning use this approach as a building element.
The goal is to allocate each data point to a cluster with a high degree of probability. The method calculates the probability for each cluster in this situation. The parameters for each cluster are updated by maximizing, resulting in a weighted mean, variance, and covariance matrix for the cluster points.
EM algorithm’s functioning #
The following is a high-level description of how the EM algorithm works.
There is data X, and it was discovered that there is unobserved (latent) data. For these parameters, compute the log-likelihood estimation. The log of the likelihood of detecting the data, as well as the latent variable assignments, are given the parameters. Given a set of parameters, the model now calculates the conditional distribution. As a result, calculating the log-likelihood. Given the conditions, this is the log of the likelihood of witnessing our data (without assuming an assignment for the latent variables).
EM Algorithm’s Properties #
The EM algorithm provides a number of useful properties.
It has numerical stability, and EM iteration improves the likelihood. It exhibits reliable-universal convergence under underlying-favorable circumstances. It is quite simple to program and search a little storage slot since it is implemented analytically and computationally. Simply said, it is easier to control convergence and programming eros by watching the continuous growth in probability, which can be easily estimated. The answers to the M-steps are usually found in closed form. Estimating missing data can be done in a way that is universally recognized.
Gaussian Mixture Model #
The Gaussian Mixture Model is a machine learning model that employs the expectation-maximization principle. A Gaussian Mixture is made up of multiple Gaussians, each of which is represented by the number ‘k,’ which is a subset of the total number of clusters to be generated.
The cluster’s center is defined by the data’s mean. The cluster width is determined by the covariance. The magnitude of a Gaussian function is determined by its mixing probability (big or small).
Gaussian Mixture Model and the EM algorithm (GMM) #
A mixture model is defined as a model that consists of an undetermined/unspecified blend of many probability distribution functions, and in general, a learning method is used to estimate the parameters of the probability distributions that fit appropriately to the density of a training dataset.
Similarly, a Gaussian Mixture Model is a sort of mixture model that requires the estimate of mean and standard deviation parameters for each of the Gaussian probability distributions.
For instance, the dataset contains a large number of data points that will be created by two processes, each with a Gaussian probability distribution.
Because the data is merged and the distributions are identical, it is difficult to tell which distributions a particular point belongs to. The methods used to generate the data points are indeed latent variables. In this circumstance, the EM method is a good methodology for estimating the parameters of the distributions.
The E-step of this method finds the value of the process latent variable for each data point, while the M-step optimizes the parameters of the probability distributions to capture data density.
The technique is repeated until a sufficient set of latent values and a maximum probability that matches the data is discovered.
Questions and answers #
What is the latent variable? #
A latent variable is a random variable that is not visible during the training or testing phases. These characteristics are not measurable on a numerical scale.
What is an EM algorithm? #
In statistical models, the EM approach may be used to find the local maximum likelihood estimates/parameters (MLE) or maximum a posteriori (MAP) estimates/parameters for latent variables (unobservable variables inferred from observable variables).