Improper priors

If Bayes' theorem is written as

then it is clear that it would remain true 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. The posterior probabilities will still sum (or integrate) to 1 even if the prior values do not, and so the priors only need 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. 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. Since

this would be an improper prior both for the mean and for the variance.