Combining multiple posterior distributions

Question

I am new to Bayesian statistics, and thus have problems to come up with a solution for the following problem:

Using Approximate Bayesian Computation (ABC), I generate a posterior distribution from one set of observed summary statistics, and a set of about a million simulated summary statistics. The prior distribution is the variable value of the model parameter used in the simulations. The posterior distribution is used to obtain point estimates (e.g., mode, mean, etc.) of the underlying model parameter.

The biological problem I am working on is that the true parameter value cannot be estimated from one observation only. Hence, I have N observed data sets, and obtain N posterior distributions. In ABC, I use the same simulated data for each observation data set.

Now, my problem is to combine the N posterior distributions in a way to estimate the true parameter value, thus capturing the information contained in all N densities.

Any ideas?

Answer 1

I think you are getting the problem wrong and this has nothing to do with ABC. If your posterior distribution is such that the parameter $\theta$ is not entirely identifiable from one dataset, $x_1$, but is identifiable from a $N$ sample, $x_1,\ldots,x_N$, what you need to consider is the "single" posterior conditional on all of those datasets: $$ \pi(\theta|x_1,\ldots,x_N) $$ not a product of posteriors (which is not defined from a Bayesian viewpoint anyway).

This major point being straightened, the ABC implementation of your problem means that you need to run the loop

1. Generate theta from the prior
2. Generate N dataset x[1:N] from the sampling distribution with parameter theta
3. Accept theta if x[1:N] is close enough to the observed N dataset

because your data is then the $N$ dataset.

You mention a million summary statistics: do you mean a summary statistic of dimension one million or one million replications of the summary statistic?