Prior Probability - Improper Priors

Improper Priors

If Bayes' theorem is written as

then it is clear that the same result would be obtained if all the prior probabilities P(A_i) and P(A_j) were multiplied by a given constant; the same would be true for a continuous random variable. If the summation in the denominator converges, the posterior probabilities will still sum (or integrate) to 1 even if the prior values do not, and so the priors may only need to be specified in the correct proportion. Taking this idea further, in many cases the sum or integral of the prior values may not even need to be finite to get sensible answers for the posterior probabilities. When this is the case, the prior is called an improper prior. However, the posterior distribution need not be a proper distribution if the prior is improper. This is clear from the case where event B is independent of all of the A_j.

Some statisticians use improper priors as uninformative priors. For example, if they need a prior distribution for the mean and variance of a random variable, they may assume p(m, v) ~ 1/v (for v > 0) which would suggest that any value for the mean is "equally likely" and that a value for the positive variance becomes "less likely" in inverse proportion to its value. Many authors (Lindley, 1973; De Groot, 1937; Kass and Wasserman, 1996) warn against the danger of over-interpreting those priors since they are not probability densities. The only relevance they have is found in the corresponding posterior, as long as it is well-defined for all observations. (The Haldane prior is a typical counterexample.)