The EM algorithm for the hierarchical normal model: For the...

The EM algorithm for the hierarchical normal model: For the model in Section $12.5$ for the coagulation data, derive the expressions (12.12) for $\mu^{\text {new }}, \sigma^{\text {new, }}$, and $\tau^{\text {new }}$.

Key Concepts

Expectation-Maximization (EM) Algorithm

A general iterative optimization method for finding maximum likelihood estimates in the presence of latent variables. It alternates between the E-step, where the expected value of the complete-data log-likelihood is computed given current parameter estimates, and the M-step, where this expectation is maximized to update the parameters.

Hierarchical Normal Model

A statistical model that structures data into multiple levels where individual observations are modeled as arising from normal distributions whose parameters (such as means and variances) are themselves drawn from higher-level normal distributions. This layered structure allows for modeling both within-group and between-group variability.

Latent Variables in Hierarchical Models

Components of the model that are not directly observed but influence the observed outcomes. In hierarchical models, these are often the group-level effects or underlying random effects. The EM algorithm leverages the conditional expectations of these latent components during the E-step.

Complete-Data Log-Likelihood

The log-likelihood function formulated by considering both the observed data and the latent variables. It provides a fuller perspective of the model and serves as the basis for deriving update equations in the EM algorithm by taking expectations with respect to the missing data.

Parameter Update Derivation

The process of deriving new estimates for the model parameters by differentiating the expected complete-data log-likelihood with respect to each parameter, setting the derivatives to zero, and solving. This systematic derivation produces the update expressions for the parameters (e.g., global mean, within-group variance, between-group variance) that ensure convergence to a local maximum of the likelihood.

Transcript

Please give Ace some feedback