Seonglae ChoNot a mathematical proposal, just a real approximation
Self-normalized importance sampling
Weight is probability of target divided by proposal distribution
Small setback: for the particular case where our function to integrate is the posterior , we can only evaluate up to a constant.
When we define unnormalized weights, which can be used to approximate the Evidence.
Use the unnormalized importance weights with the same sampled value from , to estimate both the numerator and the denominator.
And then, we could normalize unnormalized importance weight into , then
After that we can compute Posterior with Delta function for point mass
Likelihood Weighting
Using the prior as the proposal distribution , we can weight samples by their likelihood. This provides a softer approach compared to Rejection Sampling, which uses hard rejection/acceptance when sampling latent variables.
In Importance sampling practice, a good proposal must be close to the posterior which might be quite different from the prior. MCMC algorithms draw samples from a target distribution by performing a biased Random Walk over the space of latent variables .
Annealed Importance Sampling (AIS)
Instead of using a fixed proposal distribution, samples are drawn through a process of gradually changing distributions, transitioning to the target distribution through intermediate distributions MCMC
Consider the data according to the probability of trajectory occurrence, allowing mathematical use of Off-policy data to utilize more data while weighing by importance
