by What is this thing called Bayesianism? A major, and notorious, problem with this approach, at least in the domain of science, concerns how to ascribe objective prior probabilities to hypotheses. What seems to be necessary is that we list all the possible hypotheses in some domain and distribute probabilities among them, perhaps ascribing the same probability to each employing the principal of indifference. But where is such a list to come from? It might well be thought that the number of possible hypotheses in any domain is infinite, which would yield zero for the probability of each and the Bayesian game cannot get started. All theories have zero probability and Popper wins the day. How is some finite list of hypotheses enabling some objective
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What is this thing called Bayesianism?
A major, and notorious, problem with this approach, at least in the domain of science, concerns how to ascribe objective prior probabilities to hypotheses. What seems to be necessary is that we list all the possible hypotheses in some domain and distribute probabilities among them, perhaps ascribing the same probability to each employing the principal of indifference. But where is such a list to come from? It might well be thought that the number of possible hypotheses in any domain is infinite, which would yield zero for the probability of each and the Bayesian game cannot get started. All theories have zero probability and Popper wins the day. How is some finite list of hypotheses enabling some objective distribution of nonzero prior probabilities to be arrived at? My own view is that this problem is insuperable, and I also get the impression from the current literature that most Bayesians are themselves coming around to this point of view.
Chalmers is absolutely right here in his critique of ‘objective’ Bayesianism, but I think it could actually be extended to also encompass its ‘subjective’ variety.
A classic example — borrowed from Bertrand Russell — may perhaps be allowed to illustrate the main point of the critique:
Assume you’re a Bayesian turkey and hold a nonzero probability belief in 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’ theorem
P(H|e) = [P(e|H)P(H)]/P(e),
where evidence e stands for “not being eaten” and P(e|H) = 1. Given there do exist other hypotheses than H, P(e) is less than 1 and a fortiori 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, for every day that goes by, the traditional Christmas dinner also gets closer and closer …
The nodal point here is — of course — that although Bayes’ theorem is mathematically unquestionable, that doesn’t qualify it as indisputably applicable to scientific questions.
Bayesian probability calculus is far from the automatic inference engine that its protagonists maintain it is. Where do the priors come from? Wouldn’t it be better in science if we did some scientific experimentation and observation if we are uncertain, rather than starting to make calculations based on people’s often vague and subjective personal beliefs? Is it, from an epistemological point of view, really credible to think that the Bayesian probability calculus makes it possible to somehow fully assess people’s subjective beliefs? And are — as most Bayesians maintain — all scientific controversies and disagreements really possible to explain in terms of differences in prior probabilities? I’ll be dipped!