© 1977 by Biometrika Trust
A Bayesian approach to prediction using polynomials
Department of Mathematics, University of Sierra Leone Freetown
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.
Key Words: Bayesian procedure • Least squares procedure • Modal estimate • Polynomial regression • Posterior distribution • Predictive distribution • Prior distribution