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]

ses where the variance is known or taken as a parameter. 1. INTRODUCTION Although the term `hypothesis' in statistics is synonymous with that of a probability `model' as an explanation of data, hypothesis testing is not quite the same problem as model selection. This is because usually a particular hypothesis, called the `null hypothesis', has already been selected as a favorite model and it will be abandoned in favor of another model only when it clearly fails to explain the currently available data. In model selection, by contrast, all the models considered are regarded on the same footing and the objective is simply to pick the one that best explains the data. For the Bayesians certain models may be favored in terms of a prior probability, but in the minimum description length (MDL) approach to be outlined below, prior knowledge of any kind is to be used in selecting the tentative models, which in the end, unlike in the Bayesians' case, can and will be fitted to data

The so-called denoising problem, relative to normal models for noise, is formalized such that `noise' is defined as the incompressible part in the data while the compressible part defines the meaningful information bearing signal. Such a decomposition is effected by minimization of the ideal code length, called for by the Minimum Description Length (MDL) principle, and obtained by an application of the normalized maximum likelihood technique to the primary parameters, their range, and their number. For any orthonormal regression matrix, such as defined by wavelet transforms, the minimization can be done with a threshold for the squared coefficients resulting from the expansion of the data sequence in the basis vectors defined by the matrix. keywords: linear regression, wavelet transforms, threshold, stochastic complexity, Kolmogorov sufficient statistics 1 Introduction Intuitively speaking the so-called `denoising' problem is to separate an observed data sequence x 1 ; x 2 ; ...

Stochastic complexity of a data set is defined as the shortest possible code length for the data obtainable by using some fixed set of models. This measure is of great theoretical and practical importance as a tool for tasks such as model selection or data clustering. Unfortunately, computing the modern version of stochastic complexity, defined as the Normalized Maximum Likelihood (NML) criterion, requires computing a sum with an exponential number of terms. Therefore, in order to be able to apply the stochastic complexity measure in practice, in most cases it has to be approximated. In this paper, we show that for some interesting and important cases with multinomial data sets, the exponentiality can be removed without loss of accuracy. We also introduce a new computationally efficient approximation scheme based on analytic combinatorics and assess its accuracy, together with earlier approximations, by comparing them to the exact form.

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One of the second generation of computer scientists, Chris Wallace completed his tertiary education in 1959 with a Ph.D. in nuclear physics, on cosmic ray showers, under Dr Paul George at Sydney University. Needless to say, computer science was not, at that stage, an established academic discipline. With Max Brennan 1 andJohnMaloshehaddesignedand built a large automatic data logging system for recording cosmic ray air shower events and with Max Brennan also developed a complex computer programme for Bayesian analysis of cosmic ray events on the recently installed SILLIAC computer. Appointed lecturer in Physics at Sydney in 1960 he was sent almost immediately to the University of Illinois to copy the design of ILLIAC II, a duplicate of which was to be built at Sydney. ILLIAC II was not in fact completed at that stage and, after an initial less than warm welcome by a department who seemed unsure exactly what this Australian was doing in their midst, his talents were recognized and he was invited to join their staff (under very generous conditions) to assist in ILLIAC II design 2. He remained there for two years helping in particular to design the input output channels and aspects of the advanced control unit (first stage pipeline). In the event, Sydney decided it would be too expensive to build a copy of ILLIAC II, although a successful copy (the Golem) was built in Israel using circuit designs developed by Wallace and Ken Smith. In spite of the considerable financial and academic inducements to remain in America, Wallace returned to Australia after three months spent in England familiarizing himself with the KDF9 computer being purchased by Sydney University to replace SILLIAC. Returning to the School of Physics he joined the Basser