Question: Calculating Average Variance in a Gaussian Mixture Model
I am working with a Gaussian Mixture (GM) model within the framework of the Expectation-Maximization (EM) algorithm. My goal is to estimate a set of variables, and I am currently facing a challenge with calculating the average variance of the posterior distribution ( P_{S|XZ} ) with respect to the marginal distribution ( P_{Z|X} ).
Problem Details
The model involves the following conditional probability for the posterior ( P_{S|XZ} ):
$$ P_{S|XZ}(s|x, z) = \frac{P_S(s) P_{Z|SX}(z|s, x)}{\sum_{\tilde{s} \in S} P_S(\tilde{s}) P_{Z|SX}(z|\tilde{s}, x)} $$
In this context, ( P_{Z|SX}(z|s, x) ) is given by a Gaussian distribution with a mean of ( SX ) and a variance of ( S^2X^2 + 1 ).
Specific Question
How can one calculate the average variance of ( P_{S|XZ} ) with respect to ( P_{Z|X} )? I am looking for a method to integrate the squared deviations of ( S ) from its mean, weighted by the posterior probability. Any suggestions or references to relevant literature would be highly appreciated.
Thank you for your assistance!