New Foundations for Imprecise Bayesianism
My dissertation examines two kinds of statistical tools for taking prior information into account, and investigates what reasons we have for using one or the other in different sorts of inference and decision problems.
Chapter 1 describes a new objective Bayesian method for constructing `precise priors'. Precise prior probability distributions are statistical tools for taking account of your `prior evidence' in an inference or decision problem. `Prior evidence' is the wooly hodgepodge of information that you come to the table with. `Experimental evidence' is the new data that you gather to facilitate inference and decision-making. I leverage this method to provide the seeds of a solution to the problem of the priors, the problem of providing a compelling epistemic rationale for using some `objective' method or other for constructing priors. You ought to use the proposed method, at least in certain contexts, I argue, because it minimizes your need for epistemic luck in securing accurate `posterior' (post-experiment) beliefs.
Chapter 2 addresses a pressing concern about precise priors. Precise priors, some Bayesians say, fail to adequately summarize certain kinds of evidence. As a class, precise priors capture improper responses to unspecific and equivocal evidence. This motivates the introduction of imprecise priors. We need imprecise priors, or sets of distributions to summarize such evidence. I argue that, despite appearances to the contrary, precise priors are, in fact, flexible enough to capture proper responses to unspecific and equivocal evidence. The proper motivation for introducing imprecise priors, then, is not that they are required to summarize such evidence. We ought to search for new epistemic reasons to introduce imprecise priors.
Chapter 3 explores two new kinds of reasons for employing imprecise priors. We ought to adopt imprecise priors in certain contexts because they put us in an unequivocally better position to secure epistemically valuable posterior beliefs than precise priors do. We ought to adopt imprecise priors in various other contexts because they minimize our need for epistemic luck in securing such posteriors. This points the way toward a new, potentially promising epistemic foundation for imprecise Bayesianism.