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

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

Examples:

• Importance Sampling

• Manipulate Stochastic gradients

• Derive Probability bounds

• RL for policy corrections