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**Expectation Maximization**

Machine Learning

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Last Time Expectation Maximization Gaussian Mixture Models

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**Today EM Proof Clustering sequential data Jensen’s Inequality**

EM over HMMs EM in any Graphical Model Gibbs Sampling

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**Gaussian Mixture Models**

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**How can we be sure GMM/EM works?**

We’ve already seen that there are multiple clustering solutions for the same data. Non-convex optimization problem Can we prove that we’re approaching some maximum, even if many exist.

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Bound maximization Since we can’t optimize the GMM parameters directly, maybe we can find the maximum of a lower bound. Technically: optimize a convex lower bound of the initial non-convex function.

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**EM as a bound maximization problem**

Need to define a function Q(x,Θ) such that Q(x,Θ) ≤ l(x,Θ) for all x,Θ Q(x,Θ) = l(x,Θ) at a single point Q(x,Θ) is concave

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**EM as bound maximization**

Claim: for GMM likelihood The GMM MLE estimate is a convex lower bound

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**EM Correctness Proof Prove that l(x,Θ) ≥ Q(x,Θ) Likelihood function**

Introduce hidden variable (mixtures in GMM) A fixed value of θt Jensen’s Inequality (coming soon…)

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EM Correctness Proof GMM Maximum Likelihood Estimation

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**The missing link: Jensen’s Inequality**

If f is concave (or convex down): Incredibly important tool for dealing with mixture models. if f(x) = log(x)

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**Generalizing EM from GMM**

Notice, the EM optimization proof never introduced the exact form of the GMM Only the introduction of a hidden variable, z. Thus, we can generalize the form of EM to broader types of latent variable models

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General form of EM Given a joint distribution over observed and latent variables: Want to maximize: Initialize parameters E Step: Evaluate: M-Step: Re-estimate parameters (based on expectation of complete-data log likelihood) Check for convergence of params or likelihood

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**Applying EM to Graphical Models**

Now we have a general form for learning parameters for latent variables. Take a Guess Expectation: Evaluate likelihood Maximization: Reestimate parameters Check for convergence

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**Clustering over sequential data**

Recall HMMs We only looked at training supervised HMMs. What if you believe the data is sequential, but you can’t observe the state.

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**EM on HMMs also known as Baum-Welch Recall HMM parameters:**

Now the training counts are estimated.

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**EM on HMMs Standard EM Algorithm Initialize**

E-Step: evaluate expected likelihood M-Step: reestimate parameters from expected likelihood Check for convergence

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EM on HMMs Guess: Initialize parameters, E-Step: Compute

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**EM on HMMs But what are these E{…} quantities?**

so… These can be efficiently calculated from JTA potentials and separators.

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EM on HMMs

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**EM on HMMs Standard EM Algorithm Initialize**

E-Step: evaluate expected likelihood JTA algorithm. M-Step: reestimate parameters from expected likelihood Using expected values from JTA potentials and separators Check for convergence

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**Training latent variables in Graphical Models**

Now consider a general Graphical Model with latent variables.

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**EM on Latent Variable Models**

Guess Easy, just assign random values to parameters E-Step: Evaluate likelihood. We can use JTA to evaluate the likelihood. And marginalize expected parameter values M-Step: Re-estimate parameters. Based on the form of the models generate new expected parameters (CPTs or parameters of continuous distributions) Depending on the topology this can be slow

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**Maximization Step in Latent Variable Models**

Why is this easy in HMMs, but difficult in general Latent Variable Models? Many parents graphical model

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Junction Trees In general, we have no guarantee that we can isolate a single variable. We need to estimate marginal separately. “Dense Graphs”

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**M-Step in Latent Variable Models**

M-Step: Reestimate Parameters. Keep k-1 parameters fixed (to the current estimate) Identify a better guess for the free parameter.

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**M-Step in Latent Variable Models**

M-Step: Reestimate Parameters. Keep k-1 parameters fixed (to the current estimate) Identify a better guess for the free parameter.

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**M-Step in Latent Variable Models**

M-Step: Reestimate Parameters. Keep k-1 parameters fixed (to the current estimate) Identify a better guess for the free parameter.

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**M-Step in Latent Variable Models**

M-Step: Reestimate Parameters. Keep k-1 parameters fixed (to the current estimate) Identify a better guess for the free parameter.

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**M-Step in Latent Variable Models**

M-Step: Reestimate Parameters. Keep k-1 parameters fixed (to the current estimate) Identify a better guess for the free parameter.

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**M-Step in Latent Variable Models**

M-Step: Reestimate Parameters. Gibbs Sampling. This is helpful if it’s easier to sample from a conditional than it is to integrate to get the marginal. If the joint is too complicated to solve for directly, sampling is a tractable approach.

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**EM on Latent Variable Models**

Guess Easy, just assign random values to parameters E-Step: Evaluate likelihood. We can use JTA to evaluate the likelihood. And marginalize expected parameter values M-Step: Re-estimate parameters. Either JTA potentials and marginals Or don’t do EM and Sample…

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Break Unsupervised Feature Selection Principle Component Analysis

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