Why is the prior distribution of unknown probabilities uniform?

Question

Specifically the prior for the probability of an unknown binary variable, such as in the context of the rule of succession.

In every proof I've seen of the rule of succession, it starts with the assumption of a uniform prior of the probability; i.e. that all probabilities $[0,1]$ are equally likely. For example, from Wikipedia, from Stackexchange, both of which use Bayes' law. There's also a nice proof involving creating an ordered list of numbers chosen uniformly from $[0,1]$ (including the probability itself), but I haven't found any website that mentions it.

It's not obvious that a uniform distribution is the right prior for the probability – are there proofs of the rule of succession without this assumption, or explanations of why the prior should be uniform?

Answer 1

This is a matter of debate among Bayesians. The Bayesian inference section of Wikipedia's page on the Beta distribution has a good discussion of the options.

Essentially Laplace assumes that the prior probability is uniform (which is the same as a $\mathrm{Beta}(1,1)$ distribution) and hence gets the rule of succession $(s+1)/(n+2)$, but Haldane suggests using $\mathrm{Beta}(0,0)$ and hence gets simply $s/n$ for his rule of succession. Jeffreys proposes $\mathrm{Beta}(1/2,1/2)$, which would make the rule of succession $(s+1/2)/(n+1)$.