What is Expectation Maximization (EM)?
Expectation Maximization (EM) is an iterative algorithm used to find the maximum likelihood estimates of parameters in a statistical model when the model involves latent variables.
Intuitive Explanation
Imagine you have a bag of marbles, and you want to know how many of each color are in the bag. You can’t see inside the bag, but you can draw marbles out one at a time. This is analogous to having data that’s incomplete or has missing information.
The EM algorithm works like this:
- **Expectation Step:** You make an initial guess about the number of marbles of each color. Based on this guess, you can calculate the probability of drawing each color marble. This is like “expecting” what you might see based on your initial guess.
- **Maximization Step:** You use the probabilities calculated in the Expectation step to update your guess about the number of marbles of each color. You do this by finding the guess that best explains the data you have seen so far. This is like “maximizing” the likelihood of your guess being correct.
You then repeat the Expectation and Maximization steps until your guess stops changing significantly. This is how the algorithm converges to a solution.
Analogy with Coin Toss
Consider a scenario where you have two coins, A and B. You want to find the probability of getting heads (P(H)) for each coin. The catch is, you don’t know which coin you are flipping. You only observe a sequence of heads and tails. This is the incomplete data problem.
Here’s how the EM algorithm works in this scenario:
- **Expectation Step:** Assume initial probabilities for P(H) for both coins, say P(H)A = 0.6 and P(H)B = 0.4. Based on these probabilities, you calculate the probability of each observed head or tail being generated by each coin. For example, if you see a heads, you calculate the probability that it came from coin A and the probability that it came from coin B.
- **Maximization Step:** You use the probabilities calculated in the Expectation step to update the initial estimates for P(H)A and P(H)B. You find new values that maximize the likelihood of observing the data given the calculated probabilities.
You continue repeating these steps until the estimates for P(H)A and P(H)B converge to stable values.
Applications
EM is widely used in many fields including:
- Machine Learning: Mixture models, Hidden Markov Models (HMMs), K-means clustering.
- Computer Vision: Image segmentation, object recognition.
- Bioinformatics: Gene finding, phylogenetic tree reconstruction.
- Economics: Estimating economic models with missing data.
Example: K-means Clustering
In K-means clustering, the goal is to group data points into k clusters. The EM algorithm is used to find the optimal cluster centers (centroids) and assign data points to the closest clusters.
- **Expectation Step:** Assign each data point to the cluster with the nearest centroid.
- **Maximization Step:** Calculate the new centroids for each cluster based on the assigned data points.
This process is repeated until the centroids stabilize.
Limitations
- Local optima: EM can get stuck in local optima, meaning it may not find the absolute best solution.
- Computational complexity: The computation time can increase significantly with large datasets or complex models.
Summary
The Expectation Maximization algorithm is a powerful tool for dealing with incomplete data or data with hidden variables. It involves iteratively refining estimates of parameters based on expected values and maximizing the likelihood of the observed data. While it has some limitations, EM remains a fundamental algorithm in many statistical and machine learning applications.