# Improper priors

It is important to specify a prior with a sufficiently large range to cover all possible true values for the parameter, as we have seen in the tigers in the jungle example. Failure to specify a wide enough prior will curtail the posterior distribution though this will nearly always be apparent when plotting the posterior distribution and a correction can be made. It may not be apparent that the prior range is inadequate, however, when the likelihood function has more than one peak, in which case one might have extended the range of the prior to show the first peak but no further.

A Uniform prior can be used to represent uninformed knowledge about a parameter. However, if that parameter can take on any value between zero and infinity, for example, then it is not strictly possible to use the Uniform prior p(q) = c, where c is some constant, since no value of c will let the area of the distribution sum to one, and the prior is called improper. Other common improper priors include using 1/s for the standard deviation of a normal distribution and 1/s2 for the variance.

In reality, it is very rare that we have a likelihood function that does not tail off sufficiently quickly so that p(q)* l(X|q) doesn't drop to zero at the tails (which would mean that we couldn't normalize the posterior distribution), so this is not much of a problem.

Savage (1962) has suggested that an uninformed prior can be uniformly distributed over the area of interest, then slope smoothly down to zero outside the area of interest. Such a prior can, of course, be designed to have an area of one, eliminating the need for improper priors. However, the extra effort required in designing such a prior is not really necessary if one can accept using an improper prior.

It is not, of course, possible to use improper priors if one performs a Bayesian inference by simulation, because one must draw values from the prior distribution.