I am performing Approximate Bayesian Computation (ABC) in phylogenetic research. I have a few questions for which I am looking for a useful answer (as I am a biologist, but not a statistician).

(1) How can I combine the posterior distributions of several ABC runs?

I need to calculate the joint posterior distribution to get the estimate I am looking for.

(2) How can I calculate the estimate?

Usually, the maximum likelihood estimator (MLE) would be the peak of the joint posterior. However, I need to differentiate between a "pointy" peak and a peak in a flat curve, i.e. the "quality" of my estimate.

My aim is to compare multiple joint posteriors that are the result of different parameter settings.

As I am conducting simulations with known parameters, I know the true value that is estimated through ABC. Now I need to calculate the difference between the estimate from ABC and the true value.

(3) How can I calculate the difference between estimate and true value?

Simply taking the distance between true value and MLE seems a bit "wasteful" to me, as the "quality" described above is not taken into account, i.e. a peak that is pointy and close to the true value is better than a peak that is flat and close to the true value.

Your help on this is much appreciated!

Cheers,

Patrick