Although Bayesian analysis has been in use since Laplace, the Bayesian method of model--comparison has only recently been developed in depth. In this paper, the Bayesian approach to regularisation and model--comparison is demonstrated by studying the inference problem of interpolating noisy data. The concepts and methods described are quite general and can be applied to many other problems. Regularising constants are set by examining their posterior probability distribution. Alternative regularisers (priors) and alternative basis sets are objectively compared by evaluating the evidence for them. `Occam's razor' is automatically embodied by this framework. The way in which Bayes infers the values of regularising constants and noise levels has an elegant interpretation in terms of the effective number of parameters determined by the data set. This framework is due to Gull and Skilling. 1 Data modelling and Occam's razor In science, a central task is to develop and compare models to a...

this paper is to describe some of the relationships among the different streams and to try to clarify some of the important differences in their assumptions and development. Other studies mentioning the relationships appear in [1, Section IV, pp. 1038--1039], [2, sections 5.2, 5.5] and [3, p. 465]

This paper continues the introduction to minimum encoding inductive inference given by Oliver and Hand. This series of papers was written with the objective of providing an introduction to this area for statisticians. We describe the message length estimates used in Wallace's Minimum Message Length (MML) inference and Rissanen's Minimum Description Length (MDL) inference. The differences in the message length estimates of the two approaches are explained. The implications of these differences for applications are discussed.

This paper continues the introduction to minimum encoding inference given by Oliver and Hand. This series of papers were written with the objective of providing an introduction to this area for statisticians. We examine the relationship between Bayesianism and Minimum Message Length (MML) inference. We argue that MML augments Bayesian methods by providing a sound Bayesian method for point estimation which is invariant under non-linear transformations. We explore the issues of invariance of estimators under non-linear transformations, the role of the Fisher Information matrix in MML inference, and the apparent similarity between MML and the adoption of a Jeffreys' Prior. We then compare MML to an approximate method of Bayesian Model Class Selection. Despite apparent similarities in their expressions, the properties of the two approaches can be different.

Information Theory,

Investigators interested in model order estimation have tended to divide themselves into widely separated camps; this survey of the contributions of Schwarz, Wallace, Rissanen, and their coworkers attempts to build bridges between the various viewpoints, illuminating connections which may have previously gone unnoticed and clarifying misconceptions which seem to have propagated in the applied literature. Our tour begins with Schwarz's approximation of Bayesian integrals via Laplace's method. We then introduce the concepts underlying Rissanen 's minimum description length principle via a Bayesian scenario with a known prior; this provides the groundwork for understanding his more complex non-Bayesian MDL which employs a "universal" encoding of the integers. Rissanen's method of parameter truncation is contrasted with that employed in various versions of Wallace's minimum message length criteria.

this paper. 129 in encoding y using q(y) is \Gamma ln q(y) + ln p l (yjx(y)) = ln