EE-148 Expectation Maximization

1. EE-148Expectation Maximization Markus Weber 5/11/99

2. Overview • Expectation Maximization is a technique used to estimate probability densities under missing (unobserved) data. • Density Estimation • Observed vs. Missing Data • EM

3. Probability Density EstimationWhy is it important? • It is the essence of • Pattern Recognition  estimate p(class|observations) from training data • Learning Theory (includes pattern recognition) • Many other methods use density estimation • HMM • Kalman Filters

4. Probability Density Estimationhow does it work? • Given: samples, {xi} • Two major philosophies: Parametric • Provide a parametrized class of density functions, e.g. • Gaussians: p(x) = f(x, mean, Cov) • Mixture of Gaussians: p(x) = f(x, Mean, Cov, • Estimation means to find the parameters which best model the data. Measure? Maximum Likelihood! • Choice of class reflects prior knowledge Non-Parametric • Density is modeled explicitly through the samples, e.g. • Parzen Windows (Rosenblatt, ‘56; Parzen, ‘62): Make a histogram and convolve with kernel (could be Gaussian) • K-nearest-neighbor • Prior knowledge less prominent

5. Maximum Likelihood • The standard method (besides Bayesian inference) for parametric density estimation • Def. Likelihood of a parameter (independent samples): • Oftentimes one uses negative log-likelihood: • We can use this as an error function while estimating . • For a Gaussian, the sample mean and covariance are the ML estimates!

6. Missing Data Problems • Occur whenever part of the data is unknown • intrinsically inaccessible, e.g.: • Constellation models: p(X,N,d,h,O)  d,h not known!  p(O|X,N)? • Gaussian mixture models: which cluster does a data point belong to? • data is lost/erroneous e.g.: • Some faulty/noisy process has generated the data. • You erased the wrong file and part of your data is gone. • If the missing data is correlated in any way with the observed, we can hope to extract information about the missing data from the observed. • If the missing data is independent from the observed, everything is lost.

7. p(z|y,) p(y|z,) z y missing z missing y Example Problem • Samples, xi RN, from a joint Gaussian • In some xi, some dimensions are lost/unobserved. We know where data is missing. • How can we estimate theparameters of the Gaussian? • How can we replace themissing data? • Essential EM ideas: • If we had an estimate of the joint density, the conditional densities would tell us how the missing data is distributed. • If we had an estimate of the missing data distribution, we could use it to estimate the joint density. • There is a way to iterate the above two steps which will steadily improve the overall likelihood. p(y,z|)

8. Expectation Maximization • Task: Estimate p(y,z|,) (Gaussian) from the available data. • We want to do maximum likelihood density estimation, although we do not know all the data. If we knew the missing data, we would minimize the negative log-likelihood (o = observed, m = missing): • Since we do not know the missing data, Xm, we want to minimize the likelihood of the observed data, Xo: • We propose an iterative solution (1,2,...) • But for now we are still stuck with a log of an integral.

9. Some Rewriting • We write pn(.) for p(.|n). • We can rewrite the expression for the likelihood:and similarly:Now we use Jensen’s inequality:and obtain

10. The Gist • This shows that, if we minimize Q(, ) with respect to the second argument, the likelihood can only increase. • We still need to show that after convergence (at a minimum of Q) we have reached a minimum of thelikelihood. We will skip this part here. • To summarize: At every iteration, we need to minimize the following expression with respect to n • This corresponds to our initial intuition: We minimize the expectation of the joint likelihood, where expectation is computed using conditional of the previously estimated density.

11. Specific Example • This is what we need to minimize: • Concrete: the Gaussian example, zT = (xo, xm)T: • Update rule for : • Update rule for  is equally simple to derive.

12. Solution to Example Problem(for proof see separate handout) • Compute at each iteration:

13. Demo

14. Other Applications of EM • Estimating mixture densities • Learning constellation models from unlabeled data • Many problems can be formulated in an EM framework: • HMMs • PCA • Latent variable models • “condensation” algorithm (learning complex motions) • many computer vision problems