Is it possible to apply Bayes Theorem with only samples from the prior?

I was just wondering-- is it possible/practical to apply bayes' theorem without an analytical expression for the prior, only samples?

For example, say you have sufficient draws from a posterior distribution from a previous experiment via MCMC methods, and you'd like to use that posterior as the prior for a new one. You have an analytical expression for the likelihood as before, but now only samples from the (new) prior. How would you proceed?

The short answer is yes. Have a look at sequential MCMC/ particle filters.

Essentially, your prior consists of a bunch of particles ($M$). So to sample from your prior, just select a particle with probability $1/M$. Since each particle has equal probability of being chosen, this term disappears in the M-H ratio.

A big problem with particle filters is particle degeneracy. This happens because you are trying to represent a continuous distribution with discrete particles - there's no such thing as a free lunch!

• Suppose my prior on $\mu$ was N(0,1). A possible proposal would be to just select values from N(0,1) and accept/reject as necessary. In a particle representation, we could generate $M$ RNs from the N(0, 1) and select a particle at random. As M increases, the particle representation becomes as exact. – csgillespie Nov 9 '10 at 15:28
• To be concrete, let: $f(\theta_c|-)$ be the current posterior for which I have MCMC samples. $L(y|\theta)$ be the likelihood corresponding to the new data point. Thus, the new posterior I wish to sample from is: $f(\theta_n |-) \propto f(\theta_c|-) L(y|\theta)$. Does your suggestion amount to implicitly setting the proposal density for the MH algorithm to be $f(\theta_c|-)$ and then use the particle filter approach as an approximation to $f(\theta_c|-)$? – user28 Nov 9 '10 at 15:40