A moment of indecision

Abstract

We consider inference from a single observation X about a location parameter 0. That is, given 0 = 0, the density function of X has the form f (x -0) for some known function f. The prior distribution for 0 has density g(6). It is well known that if f and g are both normal densities then the posterior density is normal with a variance independent of x and less than the variance of either f or g. Very different behaviour can occur if either component is not normal. Dawid (1973) studied the limiting behaviour of the posterior distribution as the observation x tends to infinity. He showed that for certain forms of f and g, depending on the thickness of their tails, one component of the posterior will in the limit completely dominate the other. Then the limiting posterior density is either the prior density g(6) or the likelihood f(x -0). In particular he showed that if one of these components is a normal density and the other a t density then the normal component will dominate. O'Hagan (1979) proved that the normal density cannot be dominated by any other. Dawid's work showed that with a thick-tailed f or g the limiting behaviour of the posterior distribution given an extreme observation could be very different from the case when both f and g are normal. In the present note we consider the approach to that limit and prove that under certain circumstances the posterior variance undergoes a sharp peak before settling down to its limiting value. We interpret this phenomenon intuitively as a manifestation of indecision; the observation is large enough for the prior and the data to conflict, i.e. the marginal probability of X being in a neighbourhood of its observed value is very small, but not large enough for that conflict to be clearly resolved by one source of information dominating the other. Box (1980) uses such conflict between prior and data as a means for criticizing the model, but we will suppose that the model is not questioned.