# Reusing past MCMC samples from a similar distribution

I have two graphical models that have significant overlap in the random variables they are modeling. Is it possible to use samples from one graphical model to avoid sampling as much from the other graphical model?

As an example can I use samples from an empirical bayes approach in estimating the distribution in a full bayesian treatment? That is can similar models pool their samples?

If this is possible? Does this procedure allow me to use less samples when trying to simulate the full bayesian treatment? Has this been proved in a paper?

1. If you have an MCMC outcome from an empirical Bayes approach, this means you have simulated a sample of parameters for a given (posterior) distribution, $\pi_1$. If you want to get the answer for the "full Bayesian treatment", it means you are changing the target (posterior) distribution, to $\pi_2$. (The fact that one is empirical and the other fully Bayes does not matter at this level.) Therefore, if you weight your original sample $\theta_1,\ldots,\theta_T$ with importance weights $$\omega_i=\pi_2(\theta_i)/\pi_1(\theta_i),$$ you get a weighted sample from $\pi_2$ by a standard importance sampling argument. Further, the ratio $\pi_2(\theta_i)/\pi_1(\theta_i)$ is actually the ratio of the priors, because the likelihoods are the same. (This technique only operates if you can numerically compute both priors or find some unbiased estimator of the ratio.)
• You have to be more precise in explaining how "they share random variables": if they have exactly the same random variables $X_1,\ldots,X_p$, then, as I tried to explain above, you can recyle your sample from the first graphical model via an importance sampling step. – Xi'an Nov 25 '11 at 8:04