Identity Trick

Identity Trick #

I think the slides from the MLSS 2018 meeting is the only place that I have encountered anyone actually explicitly mentioning this Identity trick.

Given an integral problem:

\[p(x) = \int p(x|z)p(z)dz\]

I can multiply by an arbitrary distribution which is equivalent to 1.

\[p(x)=\int p(x|z) p(z) \frac{q(z)}{q(z)}dz\]

Then I can regroup and reweight the integral

\[p(x) = \int p(x|z)\frac{p(z)}{q(z)}q(z)dz\]

This results in a different expectation that we initially had

\[p(x) = \underset{q(z)}{\mathbb{E}}\left[ p(x|z)\frac{p(z)}{q(z)} \right]\]

Examples: