Bayesian Methodology in Science

Abstract

The previous chapter noted shortcomings of the H-D method. No natural theory of confirmation flows easily from it and one has to be grafted on to it. It can only deal with one hypothesis at a time for testing. It is not able to readily compare a number of hypotheses to see how well they are doing with respect to evidence.(1) It has no way of taking into account the initial plausibility of the hypothesis under evaluation. It cannot easily accommodate statistical hypotheses for testing; though inferences can be made from them, it is not clear how evidence is to bear on them. Moreover, there is a large theory of statistical testing that seems to be by-passed by the H-D method. It offers no solution to the Quine-Duhem problem due to an incomplete account of confirmation. These, and other problems, have led methodologists to consider other methodologies.(2)Is there a theory of method that overcomes these problems? There is! And it is one in which the good features of the H-D method come out as a special case. Moreover, it contains as a natural feature many of the plausible confirmation and other methodological principles that we added to the H-D method. This is the theory of Bayesian confirmation (named after one of the eighteenth century founders of probability, Thomas Bayes). This theory has had much discussion in the philosophy of science over the last several decades. Nevertheless, it is one that has made little impact on discussions within science education which has remained in the backwaters of the 1970s in the debates between Popper, Kuhn, Lakatos and Feyerabend. We will see that Bayesianism incorporates in a natural way many of their more plausible ideas, and much else as well. This chapter is intended as a brief primer to probabilistic thinking and Bayesianism within scientific method in order to lift science education out of its methodological "time-warp". It also suggests further research in this area not undertaken here.In earlier Sections 1.2 and 2.2, we considered the ideas of belief, and degrees of belief. Building on this basis we introduce one of the main constraints that turns mere degree of belief into rational degree of belief. Degrees of belief are said to be coherent or rational if and only if the degrees of belief are distributed in accordance with the probability calculus, thereby enabling us to treat degrees of belief as probabilities. This is a central result that we will not prove, but it can be found in most books on Bayesianism (such as Earman (1992), Howson and Urbach (1993)). Here we wish to eschew the technical side of Bayesianism which can be formidably off-putting for those trying to approach the subject for the first time. We will pass over technicalities in order to get at the core of this theory of method. Unfortunately not all technicalities can be by-passed; but they will be kept to a minimum.(3)Section 9.1 sets out some of the basic principles of probability, as applied to belief, that leads to Bayes' Theorem. In Section 9.2 Bayes' Theorem is introduced in some of its various forms. Section 9.3 considers one of the characteristic rules of Bayesianism, the rule of conditionalization. This is in effect a rule about updating the probabilities of our hypotheses with the growth of evidence. It is this principle, along with others, that goes a long way towards realising a theory of rational probable belief that rivals theories of knowledge of the sort envisaged by Russell and discussed in Section 4.6. In Section 9.4 Bayesianism is applied to some of the principles of method mentioned in previous chapter.Bayesianism captures, in a systematic way, some of the principles of method advocated by other methodologists. This is a theme further developed in Section 9.4 that shows how many of Kuhn's own methodological claims (made after he wrote The Structure of Scientific Revolutions) can fit into a Bayesian framework. Section 9.5 mentions some of the problems that confront Bayesianism and how they might be solved. It also introduces, but does not discuss in detail, vexing matters that arise concerning subjectivist, or extreme personalist Bayesianism, less subjective or tempered personalism, and objective Bayesianism.In some cases it is useful to show that some of the claims made on behalf of Bayesianism can actually be proved. We will not mention any proofs here; the reader will be referred to some of the standard accounts of Bayesianism. We should also make it clear that we do not think that all is rosy in the Bayesian garden; it is not without its problems. ( In fact neither author of this book is a committed Bayesian.) But these are part of the frontier of research into much current thinking about methodology which cannot be avoided and which needs to be extended to science education.Section 9.6 draws out some important implications of Bayesianism for science education. So far as we know, this is a new area hitherto uninvestigated by science educators, and we realize that we have barely scratched the surface. We hope, therefore, others find our efforts to incorporate the Bayesian approach into science education worthwhile and give it the attention it deserves, an attention which we believe to be well overdue.Two final points. Bayesianism provides a natural way of understanding one aspect of constructivist pedagogy. As will be seen, Bayesians, like constructivists, start with the beliefs of learners. But they also consider the degree of belief that the learners have for each particular belief they hold. Importantly there is a rationality requirement for learning on the basis of acquiring new information that leads them to revise their degrees of belief appropriately. This is not something emphasised in constructivism, but it is something we have emphasised under the rubric of critical inquiry in learning. Though what Bayesians propose is something of an ideal model for learning, it is suggestive as to what goes on. The import of this will become clear in what follows. The second point is that there are several "interpretations" of probability discussed in most texts. These include the classical theory of probability, the frequency theory, theories of probability as a logic of partial entailment (largely due to Keynes and Carnap), theories of objective chance or propensity, and so on. The theory of rational subjective belief is also a further interpretation of the axioms of probability. Anyone interested in probability should also consider these theories. Since they are not germane to our task here, they will be omitted. We will focus exclusively on the interpretation of degrees of belief as probability.