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Parameter estimation for mixture models

In LVM are hiddner variables $z_i$ which is unobserved. Hence the posterior no longer factorize. Hence computing MAP or ML is compilcated.

Unidentifiability

The posterior may have multiple modes, and hence it does not have a unique MLE, we say that the parameter is not identifiable.

Unidentifiability can cause a problem for Bayesian inference. If we draw samples from the posterior $\theta^{(s)} \sim p(\theta|D)$ and the posterior has multiple modes, the average $\bar{\theta} = \frac{1}{S} \sum_{s=1}^S \theta^{(s)}$ is meaning less.

There are a couple of solution to Unidentifiability

1. Use MCMC

2. Use approximate MAP estimation.

EM

An common approach is to use the EM algorithm the algorithm iterates between 2 steps:

• E step, Inferring the missing values given the parameters

• M step optimizing the parameters given the “filled in” data

Recursive neural network Area and Volume