A Bayesian Approach to Prediction Using Polynomials
A. S. Young · Biometrika · 1977
We have an unknown function h(x) which we want to estimate within a finite interval. The observed values of h(x) are independent observations of a random variable y whose mean is to be approximated by a polynomial of unknown degree. The problem of estimating h(x) then translates into that of predicting y. We assume that the mean of y is a polynomial of an arbitrarily large degree and derive a prior distribution for its coefficients which expresses the belief that these coefficients will tend to decrease in absolute value as the power of x increases. A prior-posterior analysis for the coefficients is carried out from which we obtain modal estimates of them. We derive the predictive distribution of y for the case when all parameters other than the mean are known. Two examples comparing the performance of this procedure with some of the usual least squares ones are presented.