The expectation-maximization (EM) algorithm is an iterative method used in statistics to find local maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models that depend on unobserved latent variables. The algorithm alternates between an expectation (E) step, which computes the expected log-likelihood given the current parameter estimates, and a maximization (M) step, which updates the parameters to maximize that expected log-likelihood. These updated parameters then inform the next E step, and the process repeats until convergence. EM is widely applied in fields such as Machine learning for tasks like estimating mixture models, handling missing data, and training hidden Markov models.
The core challenge EM addresses arises when a likelihood function involves both observed data and unobserved latent variables. Directly maximizing the likelihood by taking derivatives with respect to all unknowns typically yields interlocking equations that cannot be solved analytically. EM sidesteps this by iteratively solving for one set of unknowns while holding the other fixed, alternating until both converge to a fixed point. This approach is guaranteed to increase the likelihood at each iteration, though it may converge to a local maximum or saddle point rather than the global optimum.
History
The EM algorithm was formally named and explained in a classic 1977 paper by Arthur Dempster, Nan Laird, and Donald Rubin. However, the method had been proposed in special cases by earlier authors. Cedric Smith introduced a gene-counting method for estimating allele frequencies, and H.O. Hartley proposed a related approach in 1958, with further developments by Hartley and Hocking in 1977. Rolf Sundberg provided a detailed treatment for exponential families in his thesis and subsequent papers, building on collaboration with Per Martin-Löf and Anders Martin-Löf. The 1977 Dempster-Laird-Rubin paper generalized these ideas and sketched a convergence analysis, establishing EM as a major statistical tool. A correct convergence proof was later published by C. F. Jeff Wu in 1983, addressing flaws in the original analysis and extending convergence guarantees beyond exponential families.
Algorithm Description
Given observed data X, latent data Z, and unknown parameters θ, the goal is to maximize the marginal likelihood L(θ; X) = ∫ p(X, Z | θ) dZ. The EM iteration consists of two steps:
- E step: Compute the expected value of the log-likelihood function, Q(θ | θ^(t)), with respect to the conditional distribution of Z given X and the current parameter estimate θ^(t).
- M step: Find parameters θ^(t+1) that maximize Q(θ | θ^(t)).
The updated parameters are then used in the next E step, and the process repeats until the change in parameters or likelihood falls below a threshold. This procedure monotonically increases the likelihood, ensuring convergence to a stationary point.
Applications
EM is commonly used to estimate parameters of mixture models, such as Gaussian mixtures, where each observed data point is assumed to come from one of several underlying components. It also handles missing data problems, where some observations are incomplete. In Artificial intelligence, EM underpins training algorithms for hidden Markov models, which are used in speech recognition and bioinformatics. Additionally, EM can solve multiple linear regression problems with latent variables and is applied in factor analysis and clustering.
Properties and Limitations
EM is computationally efficient and easy to implement for many models, but it has limitations. It may converge to local maxima, and the final solution depends on initialization. In mixture models, EM can find singular solutions where a component has zero variance, which are nonsensical maxima. The algorithm also requires the specification of the number of latent components or states, which is often unknown. Variants such as the generalized EM algorithm and stochastic EM address some of these issues, but the basic method remains a foundational tool in statistical computing.
Related Concepts
The EM algorithm is closely related to other iterative optimization techniques in Machine learning, such as gradient-based methods like Stochastic Gradient Descent Variants and Adam (Optimizer). It is also connected to variational inference in Deep learning, where approximate posterior distributions are optimized. In Generative AI, EM-style approaches appear in training latent variable models, and its principles are foundational for understanding more advanced algorithms like Reinforcement Learning from AI Feedback (RLAIF) and Curriculum Learning.