What is Expectation-maximization algorithm

Expectation-maximization algorithm: an overview

The expectation-maximization (EM) algorithm is a statistical inference method used to estimate the parameters of a probability model with latent variables. It is an iterative process that alternates between the expectation (E) step and the maximization (M) step until the convergence of the likelihood function.

The EM algorithm was introduced in the late 1970s by Arthur Dempster, Nan Laird, and Donald Rubin as a general-purpose method for maximum likelihood and Bayesian estimation in the presence of missing or incomplete data. Since then, it has become one of the most widely used algorithms in machine learning, data mining, and computational biology.

Problem statement

Suppose we have a dataset {x₁, x₂, ..., x_N} consisting of N observations of a multivariate random variable X = (X₁, X₂, ..., X_D) with an unknown probability density function f(x|θ), where θ is a vector of unknown parameters to be estimated.

If we assume that the observations are independent and identically distributed (i.i.d.), the likelihood function for the dataset can be expressed as:

L(θ; x) = Π_n=1^Nf(x_n|θ)

where f(x_n|θ) denotes the probability density function of X given the parameter vector θ, evaluated at the n-th observation x_n.

However, in many real-world scenarios, some of the variables in X may be unobserved or missing, leading to incomplete data. Moreover, even if all the variables are observed, the likelihood function may be intractable or difficult to optimize due to its complexity or non-linearity.

To solve this problem, we introduce a set of latent variables Z = (Z₁, Z₂, ..., Z_M), which are unobserved variables that capture the hidden or underlying structure of the data. The joint probability density function of X and Z can be expressed as:

p(x,z|θ) = p(x|z,θ)p(z|θ)

where p(x|z,θ) = f(x|θ) if z determines x and p(z|θ) is the prior distribution of Z given θ.

The goal of the EM algorithm is to estimate the parameter vector θ by maximizing the likelihood function L(θ; x) using the observed data x, while also inferring the posterior distribution of the latent variables given the data p(z|x,θ), which is required to calculate the expected sufficient statistics used in the M step.

Algorithm

The EM algorithm is an iterative method that alternates between two steps:

• E step (expectation step): Calculate the expected sufficient statistics of the complete data log-likelihood function, based on the current estimate of the parameter vector θ.
• M step (maximization step): Find the parameter vector θ that maximizes the expected complete data log-likelihood function, based on the expected sufficient statistics calculated in the E step.

The algorithm starts by initializing the parameter vector θ to some arbitrary value. In each iteration, it performs the following steps:

• E step: Calculate the expected sufficient statistics of the complete data log-likelihood function:

Q(θ|θ^t) = E_Z|X,θ^t[log p(X,Z|θ)]

where θ^t is the current estimate of θ, and the expectation is taken with respect to the posterior distribution of Z given the observed data X and the current estimate of θ:

p(Z|X,θ^t) = p(X,Z|θ^t) / Σ_Zp(X,Z|θ^t)

• M step: Find the parameter vector θ that maximizes the expected complete data log-likelihood function:

θ^t+1 = argmax_θ Q(θ|θ^t)

The algorithm terminates when the change in the log-likelihood function between two consecutive iterations is below some threshold, or when a maximum number of iterations is reached.

Applications

The EM algorithm has been used in a wide range of applications, including:

• Clustering: EM algorithm can be used to cluster observations into groups based on their similarity, by identifying the latent variables that capture the underlying group structure.
• HMMs: EM algorithm is widely used for training hidden Markov models (HMMs), which are probabilistic models that capture the temporal dependencies among observations.
• Missing data: EM algorithm can be used to impute missing or incomplete data by estimating the posterior distribution of the missing values given the observed data.
• Maximum likelihood estimation: EM algorithm can be used to estimate the parameters of a probability model with latent variables, by maximizing the likelihood function of the observed data. This is useful in various domains, such as natural language processing, speech recognition, computer vision, and bioinformatics.

Advantages and disadvantages

The main advantages of the EM algorithm are:

• It provides a principled and general-purpose framework for maximum likelihood and Bayesian estimation in the presence of latent variables or missing data.
• It can handle a wide range of probability models, including mixtures of distributions, hidden Markov models, factor analysis, and multidimensional scaling.
• It can handle large datasets and high-dimensional feature spaces, by using efficient approximation techniques, such as variational inference, Monte Carlo methods, or stochastic gradient descent.

The main disadvantages of the EM algorithm are:

• It is sensitive to the choice of the initial parameter vector θ, which can affect the convergence and the quality of the solution.
• It can get stuck in local optima, especially when the likelihood function is non-convex or has multiple modes.
• It may require a large number of iterations to converge, especially when the dataset is noisy or the probability model is complex.

Conclusion

The expectation-maximization algorithm is a powerful and versatile method for statistical inference and modeling, with wide-ranging applications in machine learning, data mining, and computational biology. It provides a principled and general-purpose way of dealing with latent variables and missing data, and can handle a wide range of probability models with the help of efficient approximation techniques. However, it also has some limitations and challenges, such as the sensitivity to initialization, the risk of local optima, and the convergence speed. Hence, it is important to use the EM algorithm judiciously, and to combine it with other techniques when necessary.