Spatial Nonparametric Bayesian Models

The prior distribution is an essential ingredient of any Bayesian analysis, and it plays a major role in determining the final results. As such, Bayesians attempt to use prior distributions that have certain properties. Perhaps the main property is a desire to accurately reflect prior information, i.e., information external to the experiment at hand. We would supplement this vague property with a second equally vague property. The posterior distribution should exhibit behavior that is qualitatively acceptable. The second property for prior distributions is vague, but carries with it several implications. An immediate implication is that we should dispense with parametric Bayesian models in all but the simplest of settings! This perhaps surprising implication follows from an examination of various cases. As a case in point, consider a survival analysis setting where there is a follow-up period of limited duration. With large samples, one could hope to learn the survival distribution over the follow-up period, but there is no hope of learning the exact distribution of survival times beyond the follow-up period. In Bayesian terms, with large samples, the posterior distribution within the follow-up period would be concentrated near the actual survival distribution while the posterior distribution beyond the follow-up period would not concentrate. In the limit, as the experiment tended to one of infinite size, the posterior distribution would ideally tend to a degenerate (and correct) distribution within the follow-up period but would not tend to a degenerate distribution beyond the follow-up period. In a similar fashion, if event times are recorded on a scale of limited precision (for example, in terms of months), one might hope to learn the survival distribution on the monthly scale, but would have no hope of learning the exact distribution on a finer scale. Parametric prior distributions will typically provide degenerate inference over the entire survival distribution and to infinite precision with only limited precision, limited follow-up data. Often, one only needs a discrete observation space consisting of p+1 possible values in order to obtain a degenerate posterior distribution in the limit for a p dimensional parametric model.