Expectation-Maximization algorithm
1. Introduction
The Expectation-Maximization (EM) algorithm is a widely used statistical algorithm for maximum likelihood estimation in cases where there is missing or incomplete data. The basic idea of the EM algorithm is to iteratively estimate the missing data using the current estimates of the model parameters, and then update the model parameters using the estimated missing data.
2. General Steps
Initialization: Initialize the model parameters with some initial guesses.
Expectation step: Estimate the missing data using the current estimates of the model parameters. This is done by computing the expected value of the missing data given the observed data and the current estimates of the model parameters.
Maximization step: Update the model parameters using the estimated missing data. This is done by maximizing the likelihood function with respect to the model parameters.
Convergence check: Check if the change in the likelihood function between iterations is below a specified threshold. If the change is below the threshold, then the algorithm has converged, and the estimated model
Output: Return the estimated model parameters as the final output.
3. Toy Example
A toy example of the EM algorithm for estimating the parameters of a simple mixture model.
Assumption :
Suppose we have a dataset of 100 points that are generated from a mixture of two normal distributions. We know that the two normal distributions have different means and variances, but we don't know what those values are. Our goal is to use the EM algorithm to estimate the means and variances of the two distributions.
Steps :
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We start by making some initial guesses for the means and variances of the two normal distributions. Let's say we guess that the first distribution has a mean of 0 and a variance of 1, and the second distribution has a mean of 3 and a variance of 1.
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We estimate the probability that each data point belongs to each of the two normal distributions. We can do this using Bayes' rule, which gives us the posterior probability of each data point belonging to each of the two distributions given the current estimates of the means and variances. Let's call these posterior probabilities 'responsibilities' .
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We update our estimates of the means and variances of the two normal distributions using the responsibilities we computed in the expectation step. Specifically, we compute the weighted mean and variance of the data points for each distribution, where the weights are the responsibilities. These weighted means and variances become our new estimates of the means and variances of the two distributions.
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We compute the likelihood of the data given the current estimates of the means and variances, and check if the change in the likelihood between iterations is below a specified threshold. If it is, we stop and return the estimated means and variances as our final output. If it isn't, we go back to step 2.
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We return the final estimates of the means and variances of the two normal distributions as our output.
4. Good initial guesses
| EM can be sensitive to the initial guesses of the model parameters, and it's often a good idea to run the algorithm multiple times with different initial guesses to ensure that the estimates are stable and reliable. |
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Domain knowledge: Have prior knowledge about the range or distribution of the parameters. For example, if you know that the mean of a variable is likely to be positive, you could choose an initial value greater than 0.
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A simpler model: Have a complex model, you can start by using a simpler model with fewer parameters and use the estimates from the simpler model as initial values for the more complex model.
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Random initialization: Randomly initialize the parameters using a reasonable range or distribution. You can then run the EM algorithm multiple times with different random initializations and choose the estimates with the highest likelihood.
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Clustering: Use clustering techniques, such as k-means, to cluster the data and use the cluster centers as initial values for the means of the mixture model.
No one-size-fits-all approach for choosing initial values, and the best may depend on the specific problem and the data at hand, so try multiple methods.
