Choice of priors - a closer look

So far, we have considered mainly likelihood-dominated situations, in which the prior pdf can be included in the normalization constant. But one should be careful about the possibility of uncritically use uniform priors, as a `prescription,' or as a rule, though the rule might be associated with the name of famous persons. For instance, having made $N$ interviews to infer the proportion $\theta$ of a population that supports a party, it is not reasonable to assume a uniform prior of $\theta$ between 0 and 1. Similarly, having to infer the rate $r$ of a Poisson process (such that $\lambda = r \,T$, where $T$ is the measuring time) related, for example, to proton decay, cosmic ray events or gravitational wave signals, we do not believe, strictly, that $p(r)$ is uniform between zero and infinity. Besides natural physical cut-off's (for example, very large proton decay $r$ would prevent Life, or even stars, to exist), $p(r) = \mbox{constant}$ implies to believe more high orders of magnitudes of $r$ (see Astone and D'Agostini 1999 for details). In many cases (for example the mentioned searches for rare phenomena) our uncertainty could mean indifference over several orders of magnitude in the rate $r$. This indifference can be parametrized roughly with a prior uniform $\ln r$ yielding $p(r) \propto 1/r$ (the same prior is obtainable using invariant arguments, as it will be shown in a while).

As the reader might imagine, the choice of priors is a highly debated issue, also among Bayesians. We do not pretend to give definitive statements, but would just like to touch on some important issues concerning priors.