Latent variable models

• able to represent more complex distributions by integrating over 2 simple distributions

e.g. p(x|z) is conditional normal

p(z) is normal

z is a Categorial Discrete RV (1-3) corresponding to the clusters

x is a Continuous 2D (mean and variance) RV

(1) Reason

Since we need to take the integral over all the gaussian mixture elements for each gradient step, it is completely intractable and has bad properties.

Guess which group $z$ the data point $x_i$ is in, and pretend it’s the right one

Law of Total Probability

integral over z → expectation

Summary

We wish to take the maximum log-likelihood of $p_\theta(x_i)$. However, it is intractable in latent variable models. Therefore, we maximize the evidence lower bound instead

sample z from $q_i(x_i)$ and use samples to find gradient of ELBO

then update $q_i$ to maximize ELBO

Update on $q_i$ (which is approximated by Gaussians) is difficult since there are so many parameters ⬇️

Amortized Variational Inference

We will use policy gradient to take grad of the expected portion of ELBO

Problem: It has very high variance

Solution: Reparameterize

Now we are actually using the derivatives of $r$, so we will have a lower variance