Infinite hierarchies and prior distributions. (English) Zbl 1006.62017

Roberts, Gareth O.; Rosenthal, Jeffrey S.

Bernoulli 7, No. 3, 453-471 (2001).

From the introduction: Suppose that we had independent data from an \(\text{Exp} (\theta_0^{-1})\) distribution. In a Bayesian framework, we suppose that a priori \(\theta_0\sim \text{Exp} (\theta_1^{-1})\), and that with uncertainty on the hyperparameter \(\theta_1\), we might also give it a prior, \(\text{Exp} (\theta_2^{-1})\) say. In fact, at each level of the hierarchy we can hedge our bets by imposing a further level of prior uncertainty. Suppose we impose \(N\) levels of the hierarchy by fixing the hyperparameter \(\theta_N\) and sequentially setting \(\theta_i\sim \text{Exp} (\theta^{-1}_{t+1})\), \(i=N-1\), \(N-2,\dots, 1,0\). In terms of the data, the only thing that matters is the marginal prior of \(\theta_0\), obtained (if it were possible) by integrating out the hierarchical parameters.
In such a situation, it is natural to consider the prior distribution of \(\theta_0\) as \(N\to\infty\). In this case and many others, no proper distributional limit exists, but the limit can sometimes still be described in terms of an improper prior distribution. This paper constructs such improper prior distributions, as limits of marginal priors produced from hierarchical structures of certain types. Of particular interest is the fact that, in some cases at least, the improper prior distribution produced by this limiting operation is invariant to the distributions imposed in the construction of the hierarchy, thus supporting the use of such prior distributions as canonical non-informative priors.

MSC:

62F15	Bayesian inference
62A01	Foundations and philosophical topics in statistics

Keywords:

hierarchial priors; non-informative priors; martingale location family; stable laws; ergodic Markov chains; limit of priors