MCMC methods to approximate conditional predictive distributions

Abstract

Sampling from conditional distributions is a problem often encountered in statistics when inferences are based on conditional distributions which are not of closed-form. Several Markov chain Monte Carlo (MCMC) algorithms to simulate from them are proposed. Potential problems are pointed out and some suitable modifications are suggested. Approximations based on conditioning sets are also explored. The issues are illustrated within a specific statistical tool for Bayesian model checking, and compared in an example. An example in frequentist conditional testing is also given.

Introduction

Many situations in statistics require use of conditional distributions of a random vector $\mathit{X}$ given that $\mathit{X}$ lies in some subset of the sample space. The conditioning subset is most often defined by a particular value of another random vector U; usually $\mathit{X}$ is a vector of observations and U is a statistic, a function of $\mathit{X}$. Perhaps, the best known scenario is that of conditioning on a sufficient statistic to eliminate nuisance parameters (as in Fisher exact test for contingency tables) but there are many others; Kiefer (1977) advocates routine use of conditional measures of statistical evaluation for carefully selected U's; Other examples are provided by the Fisher relevant subsets (Fisher, 1959). Discussions of these and other conditional approaches to statistics can be found in Barnett (1982), Berger and Wolpert (1984), Strickland et al. (2005), Berger (1985), Barndorff-Nielsen (1988), Reid (1995) and Lehman (1997), along with many references. Some recent particular applications requiring evaluation of these type of conditional distributions include Diaconis and Sturmfles (1998), Browne (2006) and Caffo and Booth (2001), and the conditional predictive distributions of Bayarri and Berger (2000). We next give an example taken from Reid (1995).

Gamma example: Suppose that we want to test some hypothesis about the shape parameter $\alpha$ of a gamma distribution with density:

$$f(x\mid \alpha ,\beta )=\frac{{\beta }^{\alpha }}{\Gamma (\alpha )}{x}^{\alpha -1}\exp (-\beta x)\text{.}$$

In this case, for a random sample $\mathit{x}=\left(x_1,\dots ,x_n\right)$ the minimal sufficient statistic for $(\alpha ,\beta )$ is $\left(s_1,s_2\right)=\left(\sum \log x_i,\sum x_i\right)$. It can be shown that the conditional distribution of $s_1$ given $s_2$, $f\left(s_1\mid s_2,\alpha ,\beta \right)$ does not depend on the nuisance parameter $\beta$, which gets, thus, eliminated. Also, in this case, tests of hypothesis about $\alpha$ based on this $f\left(s_1\mid s_2,\alpha ,\beta \right)=f\left(s_1\mid s_2,\alpha \right)$ are (unconditionally) uniformly most powerful among the class of unbiased test. It is thus natural to base tests about $\alpha$ on this conditional distribution. However, explicitly deriving and working with $f\left(s_1\mid s_2,\alpha \right)$ is challenging, to say the least.

Conditional distributions, except for trivial, standard, situations, cannot be obtained in closed form. In these cases, inferences have traditionally relied on asymptotic approximations. These approximations, however, may not be appropriate for small, or even moderate sample sizes, and nowadays it has become standard to base inferences on samples generated from the target distributions.

When the target conditional distributions are easy to generate from, either directly or indirectly, samples are easily and conveniently generated with usual Monte Carlo (MC) methods. However, most distributions for $\mathit{X}$ and functional forms for U result in very complicated conditional distributions and simple simulation methods will not suffice. Markov chain Monte Carlo (MCMC) methods (see, for example, Robert and Casella, 1999) have proved to be an excellent tool to simulate from these challenging distributions.

In addition, the peculiarities of conditional distributions, in which regions of high probability can vary wildly with the conditioning values, make implementation of MC and MCMC methods a bit tricky. In particular, we have to guarantee the moves to such high probability regions, and the algorithms have to be suitably adapted to these difficult conditions.

In this paper, we propose different versions of most usual MCMC algorithms—the Metropolis–Hastings algorithm and the Gibbs sampler algorithm, to simulate from conditional distributions. Our particular application is that of model checking from a Bayesian point of view, but the general ideas are applicable for simulating from conditional distributions in many other settings. As an illustration, in Section 6, we address the conditional frequentist testing of the gamma example above.

In the next section, we briefly review both MCMC algorithms. Our specific statistical problem is presented in Section 3, and in Section 4 the algorithms are applied and suitably modified for our problem. In Section 5, we develop a particular application. After the conditional frequentist example in Section 6, we conclude with a comparison of the different algorithms and some comments.