Abstract
It is a relatively well-known fact that in problems of Bayesian model selection, improper priors should, in general, be avoided. In this paper we will derive and discuss a collection of four proper uniform priors which lie on an ascending scale of informativeness. It will turn out that these priors lead us to evidences that are closely associated with the implied evidence of the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC). All the discussed evidences are then used in two small Monte Carlo studies, wherein for different sample sizes and noise levels the evidences are used to select between competing C-spline regression models. Also, there is given, for illustrative purposes, an outline on how to construct simple trivariate C-spline regression models. In regards to the length of this paper, only one half of this paper consists of theory and derivations, the other half consists of graphs and outputs of the two Monte Carlo studies.
Highlights
Using informational consistency requirements, Jaynes [1] derived the form of maximal non-informative priors for location parameters, that is, regression coefficients, to be uniform
If we are faced with a parameter estimation problem, these limits of the uniform prior are irrelevant, as we may scale the product of the improper uniform prior and the likelihood to one, which gives us a properly normalized posterior for our regression coefficients
As we discuss the Bayesian Information Criterion (BIC), the Akaike Information Criterion (AIC), and the results of a small Monte Carlo study, that these priors lead us to evidences that are closely associated with the implied evidences of the BIC and the AIC, as these evidences fill in the space between and around the BIC and AIC on a continuum of conservativeness, in terms of the number of parameters of the chosen regression analysis models
Summary
Jaynes [1] derived the form of maximal non-informative priors for location parameters, that is, regression coefficients, to be uniform. Entropy 2017, 19, 250 both known and unknown σs, after which we specify the conditions under which improper priors become problematic for model selection This specification brings us naturally to a continuum of informativeness on which priors of regression coefficients may be located. After these preliminaries, we proceed to give the derivations of the four proper uniform priors, originally derived in [2], by way of the results in [4], which are neither grossly ignorant nor grossly knowledgeable. 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]
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