Confirmation Theory

Here’s a rough go at why I think SOME of the attempts to reconcile Bayesianism with traditional problems in confirmation theory are missing the point.

Not all Bayesians are missing the points below, of course. Far from it. As far as I can see Dorling, for example, is not missing the point at all … but he is buying in to a debate that is partly irrelevant, in my opinion.

The following is an extract from some other work of mine and still needs a lot of thought before it becomes a paper. I suspect I’ve failed to do justice to my opponents, but I don’t know in exactly what way(s). One day it might make a paper, so any comments are welcome.

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Bayesian statistical inference is based on two prem-is-es:

(i)Bayes’s Theorem is applicable to any statement of conditional probability.

(ii)The set of posterior probabilities tells us everything we can know (or, gives us the best idealisation of everything we can know, for the purposes of statistical modelling and inference) about uncertain propositions, given the mathematical model and observations available to a particular doxastic agent at a particular time.

I would define Bayesianism as any position that agrees with those two premises. This definition of Bayesianism is uncontentious except within one school of thought. A number of philosophers, relatively recently, have proposed a school of inference called “Bayesian confirmation theory[=”=] which, I will claim, we need to distinguish from Bayesianism simpliciter. According to these recent Bayesian confirmation theorists, there is some function of the data which tells us to what extent data confirm a hypothesis and, crucially, this”confirmation[=“=] function need not be (and often is not) a function of the posterior probability distribution. Steel Steel-2003 has recently shown that some such functions are incompatible with Bayesianism as defined above. Since Bayesian confirmation theorists see a need for a theory of confirmation not based on the posterior distribution, they are generally not Bayesians according to the bulk of the literature on Bayesianism, despite their name.make things even more confusing, they {}

Most Bayesians see no need for a single confirmation function separate from the posterior probability function. These Bayesians — for example, Howson and Urbach Howson1993 — note that if the posterior $\pr(h|e)$ is greater than $\pr(h)$ then e confirms h, but they do not claim to have told us to what numerical

There is a small literature on confirmation functions written by orthodox Bayesians. These works argue that the confirmation function obtained by dividing the posterior by the prior (resulting in the likelihood ratio — essentially the same quantity as is foregrounded by the likelihood principle) is the most desirable. I will discuss the rationale for this while discussing Barnard’s views later. [That section not reproduced here, since it’s tangential … or is it?] , who is rather proud of being prolific, counts 33 publications in which he has made this point, and says that “[w]hat I say thirty-three times is true[=“=] \cite[.159]{Good-1983.} Orthodox Bayesians, regardless of whether they accept this argument or not, rarely give it any importance. Consequently, the school of thought that says that Bayesianism as formulated by premises (i) and (ii) above is already complete is proceeding almost independently of the school of thought that says that Bayesianism needs to be supplemented by a confirmation function.

There is no good reason for this schism in the use of the term “Bayesian[=”=]., such a schism, to the extent that it exists, is counterproductive, because it confuses our reading of the literature. Worse, it {} One resolution of the problem would be to use the words as they are currently used by both schools of thought, on the clear understanding that”Bayesian confirmation theory[=“=] is not always Bayesian; but I believe this resolution is not feasible. We can no more expect people to bear in mind that “Bayesian confirmation theory[=”=] is not always Bayesian than we can expect people to remember that George W.~Bush’s”environmental[=“=] legislation is not environmental. Instead, it would be best if the “Bayesian confirmation theorists[=”=] would drop the tag”Bayesian[=“=]. I would like to emphasise that to say that “Bayesian confirmation theory[=“=] is misnamed, as I do, is not to disparage it; it is only to wish on it a separate existence.

I propose that we should (ideall, and as far as possible, given the confused history) restrict the use of the word “Bayesian[=“=] to its only unconfusing meaning, which is the one given to it by the founders of Bayesian theory and the