Posterior Statistic in Tracer vs Path and Stepping Stone Sampling Statistic

[dpe...@asu.edu]

Hi all, 

In tracer the statistic at the top is labeled "posterior".  What does this represent?  I thought it was the log likelihood of the model but it seems like that is determined using path and stepping stone sampling.  I'm pretty sure I'm missing something obvious since likelihood + prior = posterior but I'm new to Bayesian analysis.  An answer that contrasts it with ps/ss would be awesome.

Thanks much, Dan

[Alexei Drummond]

Hi Dan,

The “posterior” statistic it is the log of the posterior probability for each state in the chain, P(X_i | D), for parameters X in step i given data D.

The “likelihood” statistic is the likelihood of the data for each state in the chain, P(D | X_i). So if you average this statistic over the MCMC chain you get the average probability of the data, over the parameter space *weighted by the posterior* (since the MCMC chain is a random sample from the posterior).

In contrast the marginal likelihood, P(D), is the average probability of the data over the parameter space *weighted by the prior*. So you need to arrive at it using a different technique, like path sampling.

Cheers

Alexei

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[Andrew Rambaut]

The value labelled ‘posterior’ in the output of BEAST is the log product of all the prior densities and the data likelihoods (i.e., the sum of the prior and the likelihood columns in the trace file). Technically it is the posterior probability of the model given the data * the probability of the data (i.e., the normalizing constant - the one that is intractable to calculate and is the reason we do MCMC in the first place). It is has been erroneously labelled ‘posterior’ since the beginning of BEAST. Basically it is logged in a normal MCMC run simply to provide diagnostic information about convergence and mixing.  

Andrew

[dpe...@asu.edu]

Thanks very much.  It makes sense that the P(data) must be embedded in one of the 3 tracer terms.

So posterior (via tracer) = P(model|data) * P(data)

Could I then separate them using the result of the path sampling?

That posterior (via tracer) / path_sampling_log_marginal_likelihood = P(data)  ?

Thanks again.