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:
- Importance Sampling
- Manipulate Stochastic gradients
- Derive Probability bounds
- RL for policy corrections