## Bayesian rationality — nothing but a probabilistic version of irrationalism

19 August, 2016 at 09:26 | Posted in Economics, Theory of Science & Methodology | 9 Comments

The initial choice of a prior probability distribution is not regulated in any way. The probabilities, called subjective or personal probabilities, reflect personal degrees of belief. From a Bayesian philosopher’s point of view, any prior distribution is as good as any other. Of course, from a Bayesian decision maker’s point of view, his own beliefs, as expressed in his prior distribution, may be better than any other beliefs, but Bayesianism provides no means of justifying this position. Bayesian rationality rests in the recipe alone, and the choice of the prior probability distribution is arbitrary as far as the issue of rationality is concerned. Thus, two rational persons with the same goals may adopt prior distributions that are wildly different …

Bayesian learning is completely inflexible after the initial choice of probabilities: all beliefs that result from new observations have been fixed in advance. This holds because the new probabilities are just equal to certain old conditional probabilities …

According to the Bayesian recipe, the initial choice of a prior probability distribution is arbitrary. But the probability calculus might still rule out some sequences of beliefs and thus prevent complete arbitrariness.

Actually, however, this is not the case: nothing is ruled out by the probability calculus …

Thus, anything goes … By adopting a suitable prior probability distribution, we can fix the consequences of any observations for our beliefs in any way we want. This result, which will be referred to as the anything-goes theorem, holds for arbitrarily complicated cases and any number of observations. It implies, among other consequences, that two rational persons with the same goals and experiences can, in all eternity, differ arbitrarily in their beliefs about future events …

From a Bayesian point of view, any beliefs and, consequently, any decisions are as rational or irrational as any other, no matter what our goals and experiences are. Bayesian rationality is just a probabilistic version of irrationalism. Bayesians might say that somebody is rational only if he actually rationalizes his actions in the Bayesian way. However, given that such a rationalization always exists, it seems a bit pedantic to insist that a decision maker should actually provide it.

Max Albert

1. As a mathematician, it seems to me that there are circumstances where probability theory can be properly applied, and that in such circumstances the use of Bayes’ rule does tend to lead to people with different (but not Cromwellian) priors converging on the same ‘true’ results. As I see it, the logical content of Max’s remarks is that you can always use Bayes’ rule backwards to justify any claim you want. This seems to me a critique of rationalization, not probability theory.

The difficult I have is that there are many people who claim to be experts in some field who apply Bayes’ rule without seeming to pay any regard to whether or not it applies in the particular circumstances. In the UK the development of these methods is sometimes credited to the railways, where it generally works fine. But there are exceptions. Sometime these can be anticipated (such as when a new line is opened). In this case it seems perverse (of others) to say that it is irrational to take any account of the exception. It is not that Bayes’ theorem is somehow ‘wrong’ or invalid, it is just that it implies that Bayes’ rule is no longer appropriate in its usual (simple) form.

• Dave, on this I think we totally agree. Bayes’ theorem per se is not the problem. The problem emerges when you try to apply it to situations and contexts where it is not appropriate. And that is the problem with Bayesianism.

• Lars, it wasn’t that I thought we disagreed, but that I was concerned that someone unfamiliar with your views coming across this page might dismiss Max and you as misunderstanding what it is that you seek to criticise. There are Bayesians who operate much as you (not Max) describe and whose work I think is valuable, but only because they always have in mind (if not in speech or on paper) the proper mathematics. The problem is in people who may think they are following their example but lack their understanding.

I also have a concern that even the best Bayesians seem very ‘academic’ and lack an appreciation of the uncertainties of the ‘real’ world and hence of the damage that naïve applications of the theory can do. The ones I know think that Keynes’ approach is technically better but its sophistication is not needed. (I think it very much needed, but lacks an accessible exposition in modern language – and possibly with less mathematics.)

Regards.

2. So when is or is not Bayesian analysis appropriate?

• Henry, in my view it is always appropriate to do a Bayesian analysis. But …

The devil is in interpreting the result. Garbage in, garbage out!

(If you don’t find Lars’ reply more helpful, I might be tempted to try to characterise situations where the caveats matter. Or see djmarsay.wordpress.com .)

3. Unfortunately most Bayesians behave exactly as Max Albert says. With bad priors, you can come to most any result in one step. More sophisticated Bayesian analysis iterates the analysis, revising the probability with a series of doses of “information”. This seems very similar to what frequentists do with additional samples of data. After sufficient iterations, both Bayesians and frequentists ought to converge quite closely to the same final probability. This might be a very long time for sufficiently outlandish priors. However, I’ve never seen this worked out anywhere, and I’ve looked at the work of such deep thinkers as Finetti. And I don’t have the math skills (e.g. measure theory) to prove it myself.

4. p.s. …now that I’ve read the whole article rather than the quoted section, my reaction is “what a mess.” Poorly edited with misspellings (“antidote”) and missing references, jumping around between mathematical and philosophical arguments and psychological evidence. I can’t speak to the philosophy but the math is inconsistent, switching between probabilities and certainties from step to step. Likewise, there’s nothing more than an assertion that the “critical” part of any argument from “critical rationality” is a consistent argument. Deductive reasoning rests on axioms and inference rules. With bad axioms, just like with bad Bayesian priors, you can make critical arguments that are quite bogus. People who criticize mainstream economics are likely to be quite familiar with the results of bad assumptions.

• Keynes’ Treatise, while technically superior, is also a mess. Could anyone suggest a readable and reliable account?

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