Bayesian Model Uncertainty and Foundations

This dissertation contains research on Bayesian model uncertainty and foundations of statistical inference.

In Chapter 2, we study the properties of constrained empirical Bayes (EB) priors on regression coefficients. Unrestricted EB procedures can have undesirable properties when their ``estimates'' correspond to hyperparameters that would be seen as overly informative in an actual Bayesian analysis. For that reason, we propose constraining EB procedures so that they are at least as vague as proper Bayesian lower bounds (which can be either informative or ``noninformative''). The main emphasis of the chapter is studying the properties of a constrained EB prior that has Zellner's g-prior with g=n as its lower bound. We show that it avoids some of the pitfalls of unconstrained EB priors and the lower bound, and see that it behaves similarly to the Bayesian Information Criterion (BIC).

In Chapter 3, we take a close look at ``information inconsistency.'' Information inconsistency is said to occur when there is overwhelming evidence in favor of a hypothesis in finite sample sizes, but the Bayes factor in its favor is finite. In Chapter 3, we investigate when it occurs (and when it does not) in normal linear models. Our conclusion is that conjugate priors are usually information-inconsistent, but thick-tailed priors and empirical Bayes procedures avoid the issue. The chapter also includes a discussion of the different formalizations of information inconsistency that have appeared in the literature, which are not equivalent.

In Chapter 4, we turn to ``limit consistency,'' which is an asymptotic property of two-sample tests. Suppose the sample size of one of the groups goes to infinity while the sample size of the other one stays fixed. According to our definition, limit consistency occurs if, under this asymptotic regime, the decision rule of the two-sample test converges to the decision rule of the one-sample test we would have performed had we known the parameters of the group with ``infinite'' data. In Chapter 4, we study limit consistency in the context of comparing whether two normal means are equal. We conclude that parametrizations where the 2 groups have common parameters are generally limit-consistent when the prior on the common parameters is flat.

Finally, the goal of Chapter 5 is discussing 2 articles that cast doubt on the correctness and applicability of Birnbaum's theorem, which implies that statisticians that wish to respect the sufficiency and conditionality principle must accept the likelihood principle. This result, which was proved in 1962, is still highly controversial because some statisticians believe that sufficiency and conditionality are appealing, but the likelihood principle is not (for example, the likelihood principle precludes the use of p-values, which are highly popular in common statistical practice). In Chapter 5, we provide counterarguments to the criticisms and put them in historical context.