H. R. N. van Erp

Deriving proper uniform priors for regression coefficients, part II

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 derive a proper and parsimonious uniform prior for regression coefficients. We then use this prior to derive the corresponding evidence values of the regression models under consideration. By way of these evidence values one may proceed to compute the posterior probabilities of the competing regression models.

Constructing Cartesian Splines

We introduce here Cartesian splines or, for short, C-splines. C-splines are piecewise polynomials which are defined on adjacent Cartesian coordinate systems and are Cr continuous throughout. The Cr continuity is enforced by constraining the coefficients of the polynomial to lie in the null-space of some smoothness matrix H . The matrix-product of the null-space of the smoothness matrix H and the original polynomial base results in a new base, the so-called Cspline base, which automatically enforces Cr continuity throughout. In this article we give a derivation of this C-spline base as well as an algorithm to construct C-spline models.

Inquiry Calculus and the Issue of Negative Higher Order Informations

In this paper, we will give the derivation of an inquiry calculus, or, equivalently, a Bayesian information theory. From simple ordering follow lattices, or, equivalently, algebras. Lattices admit a quantification, or, equivalently, algebras may be extended to calculi. The general rules of quantification are the sum and chain rules. Probability theory follows from a quantification on the specific lattice of statements that has an upper context. Inquiry calculus follows from a quantification on the specific lattice of questions that has a lower context. There will be given here a relevance measure and a product rule for relevances, which, taken together with the sum rule of relevances, will allow us to perform inquiry analyses in an algorithmic manner.

Bayesian logistic regression analysis

In this paper we present a Bayesian logistic regression analysis. It is found that if one wishes to derive the posterior distribution of the probability of some event, then, together with the traditional Bayes Theorem and the integrating out of nuissance parameters, the Jacobian transformation is an essential added ingredient. The application of the product rule gives the posterior of the unknown logistic regression coefficients. The Jacobian transformation then maps the posterior of these regression coefficients to the posterior of the corresponding probability of some event and some nuisance parameters. Finally, by way of the sumrule the nuissance parameters are integrated out.

An outline of the Bayesian decision theory

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.

Bayesian decision theory: A simple toy problem

We give here a comparison of the expected outcome theory, the expected utility theory, and the Bayesian decision theory, by way of a simple numerical toy problem in which we look at the investment willingness to avert a high impact low probability event. It will be found that for this toy problem the modeled investment willingness under the Bayesian decision theory is minimally three times higher compared to the investment willingness under either the expected outcome or the expected utility theories, where it is noted that the estimates of the latter two theories seem to be unrealistically low.