Sampling from the prior

There are a number of reasons to run an analysis in which you sample only from the prior before running the full analysis. One reason to do this is to confirm that a prior is proper. Here we mean proper in a strict mathematical sense, that is, that the prior integrates to unity (or any finite constant).

Although there are some situations in which improper priors can be argued for (Berger and Bernardo 1992), having an improper prior typically results in an improper posterior and if that is the case then the results of the MCMC analysis will be meaningless, and its statistical properties undefined. Often the observed behaviour of the chain will be that some parameters meander to either very large or very small values and never converge to a steady-state target distribution. For example, a uniform prior with bounds 0 and +∞ for a molecular clock rate will result in the clock rate wandering to extreme values when an attempt is made to sample the prior in the absence of data. Even if no problems are evident when sampling the posterior, certain analyses rely on a proper prior and will return invalid results regardless of whether the posterior appears to be sampled correctly (an example of a method that requires a proper prior is path sampling for model comparison, see Section 1.5.7). For this reason, you should chose proper priors unless you know what you are doing.

Another reason to sample from the prior is to make sure that the various priors do not produce an unexpected joint prior in combination, or if they do, to check that the resulting prior is close enough to the practitioner's intentions. Especially when calibrations are used this can be an issue (Heled and Drummond 2012, 2013) since calibrations are priors on a part of a tree and a prior for the full tree is usually also specified (like Yule or coalescent). This means there are overlapping priors on the same parameter, which can produce unexpected results. Another situation occurs where a calibration with an upper bound on an ancestral clade sets an upper bound on the age of all the descendant clades, since none of them can exceed the age of their direct ancestor.