Bayesian Nonparametric Prediction and Statistical Inference

The problem of Bayesian nonparametric prediction and statistical inference is formulated and discussed. A solution is proposed based upon A n and H n as in Hill (1968, 1988). This solution gives rise to the posterior distribution of the number of species (or more generally, the number of distinct values) in a finite population of Hill (1968, 1979), and to the posterior distribution of percentiles of the population of Hill (1968). Next, the meaning of parameters in the subjective Bayesian theory of Bruno de Finetti is discussed in connection both with H n and with conventional parametric models. It is argued that the usual sharp distinction between prediction and parametric inference is largely illusory. The finite version of de Finetti’s theorem is emphasized for the practice of statistics, with the infinite case used only to obtain approximations and insight.