An outline of the Bayesian decision theory

Abstract

Abstract. The Bayesian decision theory is neo-Bernoullian in that it proves, by way of a consistency derivation, that Bernoulli's utility function is the only appropriate function by which to translate, for a given initial wealth, gains and losses to their corresponding utilities. But the Bayesian decision theory deviates from Bernoulli's original expected utility theory in that it offers up an alternative for the traditional criterion of choice of expectation value maximization, as it proposes to choose that decision which has associated with it the utility probability distribution which maximizes the mean of the expectation value and the lower and upper confidence bounds.

Deriving Proper Uniform Priors for Regression Coefficients, Parts I, II, and III

“…This research into proper uniform priors was inspired by our research into spline models [5,14]. Spline models may have hundreds of regression coefficients.…”

“…The evidence (12) is used both to normalize (11) into a posterior distribution, (1), as well as to choose between competing regression models, (5) and (6). In order to evaluate the evidence (12), we rewrite (11) as [6] p ( β, y| σ, X, M, I) = C (2πσ 2 )…”

“…The power of spline models lies in the fact that even the most non-linear of functions f (x, y) will tend to become linear on its sub-domains as the size of the sub-domains tends to zero. In B-spline models the sub-domains are taken to be triangles/tetrahedra [14,16], whereas in C-spline models the sub-domains are taken to be squares/cubes [5]; see Appendix B for a discussion of C-splines.…”

“…In Appendix A we report on two small Monte Carlo studies with the C-spline regression models, in order to give the reader a sense for all the discussed evidences. Also, a collection of three simple trivariate C-spline regression models will be discussed in Appendix B, in order to provide the reader with a low-level, hands-on introduction into C-splines [5].…”