## The Bayesian folly

16 February, 2018 at 18:18 | Posted in Economics | 1 CommentAssume you’re a Bayesian turkey and hold a nonzero probability belief in the hypothesis H that “people are nice vegetarians that do not eat turkeys and that every day I see the sun rise confirms my belief.” For every day you survive, you update your belief according to Bayes’ Rule

P(H|e) = [P(e|H)P(H)]/P(e),

where evidence e stands for “not being eaten” and P(e|H) = 1. Given that there do exist other hypotheses than H, P(e) is less than 1 and so P(H|e) is greater than P(H). Every day you survive increases your probability belief that you will not be eaten. This is totally rational according to the Bayesian definition of rationality. Unfortunately — as Bertrand Russell famously noticed — for every day that goes by, the traditional Christmas dinner also gets closer and closer …

Neoclassical economics nowadays usually assumes that agents that have to make choices under conditions of uncertainty behave according to Bayesian rules — that is, they maximize expected utility with respect to some subjective probability measure that is continually updated according to Bayes theorem. If not, they are supposed to be irrational.

Bayesianism reduces questions of rationality to questions of internal consistency (coherence) of beliefs, but — even granted this questionable reductionism — do rational agents really have to be Bayesian? As I have been arguing repeatedly over the years, there is no strong warrant for believing so.

The nodal point here is — of course — that although Bayes’ Rule is *mathematically* unquestionable, that doesn’t qualify it as indisputably applicable to *scientific* questions. As one of my favourite statistics bloggers — Andrew Gelman — puts it:

The fundamental objections to Bayesian methods are twofold: on one hand, Bayesian methods are presented as an automatic inference engine, and this raises suspicion in anyone with applied experience, who realizes that different methods work well in different settings … Bayesians promote the idea that a multiplicity of parameters can be handled via hierarchical, typically exchangeable, models, but it seems implausible that this could really work automatically. In contrast, much of the work in modern non-Bayesian statistics is focused on developing methods that give reasonable answers using minimal assumptions.

The second objection to Bayes comes from the opposite direction and addresses the subjective strand of Bayesian inference: the idea that prior and posterior distributions represent subjective states of knowledge. Here the concern from outsiders is, first, that as scientists we should be concerned with objective knowledge rather than subjective belief, and second, that it’s not clear how to assess subjective knowledge in any case.

Beyond these objections is a general impression of the shoddiness of some Bayesian analyses, combined with a feeling that Bayesian methods are being oversold as an all-purpose statistical solution to genuinely hard problems. Compared to classical inference, which focuses on how to extract the information available in data, Bayesian methods seem to quickly move to elaborate computation. It does not seem like a good thing for a generation of statistics to be ignorant of experimental design and analysis of variance, instead of becoming experts on the convergence of the Gibbs sampler. In the short term this represents a dead end, and in the long term it represents a withdrawal of statisticians from the deeper questions of inference and an invitation for econometricians, computer scientists, and others to move in and fill in the gap …

Bayesian inference is a coherent mathematical theory but I don’t trust it in scientific applications. Subjective prior distributions don’t transfer well from person to person, and there’s no good objective principle for choosing a noninformative prior (even if that concept were mathematically defined, which it’s not). Where do prior distributions come from, anyway? I don’t trust them and I see no reason to recommend that other people do, just so that I can have the warm feeling of philosophical coherence …

As Brad Efron wrote in 1986, Bayesian theory requires a great deal of thought about the given situation to apply sensibly, and recommending that scientists use Bayes’ theorem is like giving the neighborhood kids the key to your F-16 …

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As a mathematician, I kind of agree with you and Andrew. But I wish you would distinguish between the mathematics of Bayesian probability, which I regard as 100% reliable, and the pseudo-science, which I regard as dangerous.

All mathematics is about mathematics alone, and has nothing directly to say about reality. Any claim that any particular mathematics is applicable to a particular real domain needs a proper scientific justification. In my experience it is always appropriate to apply Bayesian methods, but it is dangerous to suppose that the result is reliable, except where such a view has a reasonable justification.

In the Turkey example there is a further problem that ‘the probability that my hypothesis is false’ is not even well defined, and it is certainly an abuse of the mathematical version of Bayes’ theorem to apply it as Russell does.

Comment by Dave Marsay— 16 February, 2018 #