Markov chain Monte Carlo without likelihoods

Abstract

Many stochastic simulation approaches for generating observations from a posterior distribution depend on knowing a likelihood function. However, for many complex probability models, such likelihoods are either impossible or computationally prohibitive to obtain. Here we present a Markov chain Monte Carlo method for generating observations from a posterior distribution without the use of likelihoods. It can also be used in frequentist applications, in particular for maximum-likelihood estimation. The approach is illustrated by an example of ancestral inference in population genetics. A number of open problems are highlighted in the discussion.O ne of the basic problems in Bayesian statistics is the computation of posterior distributions. We imagine data D generated from a model M determined by parameters , the prior density of which is denoted by ( ). We assume unless otherwise stated that the data are discrete. The posterior distribution of interest is f( ͉D), which is given bywhere ‫(ސ‬D) ϭ ͐ ‫(ސ‬D͉ ) ( )d is the normalizing constant. In most scientific contexts, explicit formulae for such posterior densities are few and far between, and we usually resort to stochastic simulation to generate observations from f. Perhaps the simplest approach for this is the rejection method: A1. Generate from (⅐). A2. Accept with probability h ϭ ‫(ސ‬D͉ ); return to A1. There are many variations on this theme. Of particular relevance here is the case in which the likelihood ‫(ސ‬D͉ ) cannot be computed explicitly. One obvious approach then is:The success of this approach depends on the fact that the underlying stochastic model M is easy to simulate. This approach can be useful when computation of the likelihood is possible but time-consuming.The practicality of algorithms such as these depends crucially on the size of ‫(ސ‬D), because the probability of accepting an observation is proportional to ‫(ސ‬D). In cases where the acceptance rate is too small, one might resort to approximate methods such as: This approach requires selection of a suitable metric as well as a choice of . As 3 ϱ it generates observations from the prior. If ϭ 0, an observation DЈ is accepted only if DЈ ϭ D, and then accepted observations come from the density f( ͉D). The choice of therefore reflects a tension between computability and accuracy. The method is still honest in that, for a given and , we are generating independent and identically distributed observations from f( ͉ (D, DЈ) Յ ).When D is high-dimensional or continuous, this approach can be impractical as well, and then the comparison of DЈ with D can be made by using lower-dimensional summaries of the data. The motivation for this approach is that if the set of statistics S ϭ (S 1 , . . . , S p ) is sufficient for , in that ‫(ސ‬D͉S, ) is independent of , then f( ͉D) ϭ f( ͉S). The normalizing constant ‫(ސ‬S) is typically larger than ‫(ސ‬D), resulting in more acceptances. In practice it will be hard, if not impossible, to identity a suitable set of sufficient statistics, and we then might resort to ...

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