Guides

Guides inject the approximation to the posterior into the model definition. When the (variational) inference algorithm runs, the difference between the approximation and the exact posterior is minimized. When sampling from the model, variables are sampled from the (distributions in the) guide.

Reparametrization Gradient

Continuous random variables from suitable distributions can be recast in a “reparametrized” form. A sample is obtained by sampling a fixed-parameter distribution, followed by a deterministic transformation specified by the parameters of the target distribution.

The variance of the gradient obtained for the parameters is reduced further by using the “Path Derivative”, as detailed in Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference by Roeder, Wu and Duvenaud.

Black-Box Variational Inference

Dealing with discrete random variables is necessary to support a general programming language, with control flow depending on sampled variables. For discrete variables reparametrization is not possible and we resort to Black-Box Variational Inference. This suffers from large variance on estimates of the gradient of the posterior probability with respect to the variational parameters.

Two ways of reducing the variance are

• Rao-Blackwellization
• Control variates

The first (Rao-Blackwellization) is implemented by limiting, for each variable, the posterior to elements in its Markov blanket. The prior probability links the variable to its parents, likelihoods of observations and prior probabilities of downstream variables link it to children and their (other) parents. This procedure eliminates irrelevant other variables as sources of variance.

A simple control variate (moving average of the log-posterior) is used to reduce variance of the gradient further.

For further reading, see Black Box Variational Inference by Ranganath, Gerrish and Blei.