Philosophy and the Practice of Bayesian Statistics in the Social Sciences

Abstract

A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypothetico-deductivism. We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science.Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework. The usual story-which we don't likeIn so far as I have a coherent philosophy of statistics, I hope it is "robust" enough to cope in principle with the whole of statistics, and sufficiently undogmatic not to imply that all those who may think rather differently from me are necessarily stupid. If at times I do seem dogmatic, it is because it is convenient to give my own views as unequivocally as possible. (Bartlett, 1967, p. 458) Schools of statistical inference are sometimes linked to approaches to the philosophy of science. "Classical" statistics-as exemplified by Fisher's p-values, Neyman-Pearson hypothesis tests, and Neyman's confidence intervals-is associated with the hypotheticodeductive and falsificationist view of science. Scientists devise hypotheses, deduce implications for observations from them, and test those implications. Scientific hypotheses can be rejected (that is, falsified), but never really established or accepted in the same way. Mayo (1996) presents the leading contemporary statement of this view. 1In contrast, Bayesian statistics or "inverse probability"-starting with a prior distribution, getting data, and moving to the posterior distribution-is associated with an inductive approach of learning about the general from particulars. Rather than testing and attempted falsification, learning proceeds more smoothly: an accretion of evidence is summarized by a posterior distribution, and scientific process is associated with the rise and fall in the posterior probabilities of various models; see Figure 1 for a schematic illustration. In this view, the expression p(θ|y) says it all, and the central goal of Bayesian inference is computing the posterior probabilities of hypotheses. Anything not contained in the posterior distribution p(θ|y) is simply irrelevant, and it would be irrational (or incoherent) to attempt falsification, unless that somehow shows up in the posterior. The goal is to learn about general law...

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