What is it? A method for approximating the posterior distribution in latent variable models.

What do we need it for? If we want to learn the latent variable model given a dataset of observations, or if we want to find the posterior distribution.

Problem setup

Want to optimize

\[\log p_{\theta}(x) = \log \int p_{\theta}(x, z) dz\]

But the integral is usually intractable.

Solution

Use the ELBO

\[\log p_{\theta}(x) - \text{KL}[q_{\phi}(z|x)\|p_{\theta}(z|x)] = \underbrace{ \mathbb{E}_{q_{\phi}(z|x)}[\log p_{\theta}(x|z)] - \text{KL}[q_{\phi}(z|x) \| p(z)]}_{\text{ELBO}}\]

This is a tractable objective because we can estimate it by sampling from $q_{\phi}(z|x)$. Note that maximizing the ELBO does two things:

1. Increases the log probability of observed samples (i.e. we are learning a good latent variable model that represents the data well)
2. Reduces the “distance” between the variational distribution and the true posterior (i.e. we are learning a good approximate posterior distribution)

Learning the Model

A simple algorithm would be

• x $\gets$ get datapoint from dataset
• sample $z \sim q_{\phi}(\cdot |x)$ several times
• use set of sampled $z$ to get Monte Carlo Estimate of ELBO, call it $\mathcal{L}$.
• Take gradient of ELBO, $\nabla_{\theta}\mathcal{L}$ and $ \nabla_{\phi}\mathcal{L}$
• Take a gradient step
  • $\theta \gets \theta + \nabla_{\theta}\mathcal{L}$
  • $\phi \gets \phi + \nabla_{\phi}\mathcal{L}$
• Repeat until converged

To get less noisy estimates of the gradient, we would normally do minibatches of $x$.

After We’ve Learned

After we have learned$^*$, we can generate samples by doing the following:

• sample from our prior over the latent variable $z \sim p(z)$.
• sample from our likelihood model $x \sim p_{\theta}(x|z)$.

$^*$ Note that most latent variable models are parameterized as \(p_{\theta}(x, z) = p_{\theta}(x | z)p(z)\) That is, we have a known prior $p(z)$ over the latent variable that is not learned. Typically it is a tractable distribution like a Gaussian.