Michaelmas 2009This thesis is dedicated to the late

This paper presents a new explanation of how preferring the simplest theory compatible with experience assists one in finding the true answer to a scientific question when the answers are theories or models. Science is portrayed as an infinite game between science and nature. Simplicity is a structural invariant reflecting sequences of theory choices nature could force the scientist to produce. It is demonstrated that among the methods that converge to the truth in an empirical problem, the ones that do so with a minimum number of reversals of opinion prior to convergence are exactly the ones that prefer simple theories. The idea explains not only simplicity tastes in model selection, but aspects of theory testing and the unwillingness of natural science to break symmetries without a reason. In natural science, one typically faces a situation in which several (or even infinitely many) available theories are compatible with experience. Standard practice is to choose the simplest theory among them and to cite “Ockham’s razor ” as the excuse (figure

Provenance of correlations in psychological data There are few truisms in the field of psychology, but one of them is surely that measurement error is found in all experiments. Data are inevitably produced that do not perfectly reflect the logic imposed by the experimental design. To the extent that a psychological experiment succeeds in measuring something or in making some sort of distinction, the data will partially reflect the design, and this leads to a way of thinking about data that is found throughout all the experimental sciences: data � signal � noise. This innocent equation almost always contains an implicit but critical assumption: that the noise may be regarded as independent samples from some distribution— typically taken to be the Gaussian distribution. In this way, the residual error is conceived of as a featureless background of white noise in which the interesting part, the treatment means, are more or less buried. Often this conception of data is justified. Whenever there is random assignment to cells and each participant contributes a single datum, errors may be expected to be uncorrelated. However, in all of sensory psychophysics and most of cognitive psychology, single individuals respond to entire blocks of trials in a given experimental session. Here, the residual error will develop correlations by virtue of the circumstance that the response history was laid down by a nervous system that has memory. In many situations, these correlations are little more than a Preparation of this article was supported by NIMH Grants R01-

For decades, much research has been devoted to developing and comparing variable selection methods, but primarily for the classical case of independent observations. Existing variable-selection methods can be adapted to cluster-correlated observations, but some adaptation is required. For example, classical model fit statistics such as AIC and BIC are undefined if the likelihood function is unknown (Pan, 2001). Little research has been done on variable selection for generalized estimating equations (GEE, Liang and Zeger, 1986) and similar correlated data approaches. This thesis will review existing work on model selection for GEE and propose new model selection options for GEE, as well as for a more sophisticated marginal modeling approach based on quadratic inference functions (QIF, Qu, Lindsay, and Li, 2000), which has better asymptotic properties than classic GEE. The focus is on selection using continuous penalties such as LASSO (Tibshirani, 1996) or SCAD (Fan and Li, 2001) rather than the older discrete penalties such as AIC and BIC. The

Phenotypic evolution of two-element populations with proportional selection and normally distributed mutation is considered. Trajectories of the expected location of the population in the space of population states are investigated. The expected location of the population generates a discrete dynamical system. The study of its fixed points, their stability and time to convergence is presented. Fixed points are located in the vicinity of optima and saddles. For large values of the standard deviation of mutation, fixed points become unstable and periodical orbits arise. In this case, fixed points are also moved away from optima. The time to convergence to fixed points depends not only on the mutation rate, but also on the distance of the points from unstability. Results show that a population spends most time wandering slowly towards the optimum with mutation as the main evolution factor.

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What has science actually achieved? A theory of achievement should (1) define what has been achieved, (2) describe the means or methods used in science, and (3) explain how such methods lead to such achievements. Predictive accuracy is one truth-related achievement of science, and there is an explanation of why common scientific practices (of trading off simplicity and fit) tend to increase predictive accuracy. Akaike’s explanation for the success of AIC is limited to interpolative predictive accuracy. But therein lies the strength of the general framework, for it also provides a clear formulation of many open problems of research.

Studying how individuals compare two given quantitative stimuli, say d1 and d2, is a fundamental problem. One very common way to address it is through ratio estimation, that is to ask individuals not to give values to d1 and d2, but rather to give their estimates of the ratio p = d1/d2. Several psychophysical theories (the best known being Stevens ’ power-law) claim that this ratio cannot be known directly and that there are cognitive distortions on the apprehension of the different quantities. These theories result in the so-called separable representations (Narens 1996, Luce 2002), which include Stevens ’ model as a special case. In this paper we propose a general statistical framework that allows for testing in a rigorous way whether the separable representation theory is grounded or not. We conclude in favor of it, but strongly reject Stevens ’ model. As a byproduct, we provide estimates of the psychophysical functions of interest.

We study the problem of model fitting in the framework of nested probabilistic families. Our criteria are: (i) sparsity of the identified representation, (ii) its ability to fit the (finite length) data set available. As we show in this paper, current methodologies, often taking the form of penalized versions of the data likelihood, cannot simultaneously satisfy these requirements, as the examples presented clearly demonstrate. On the contrary, maximization of the Bayesian model posterior, even without assumption of a complexity penalizing prior, is able to select models with appropriate complexity, enabling sound determination of its parameters in a second step. 1.1 Problem formulation 1.

This paper is an introduction to model selection intended for nonspecialists who have knowledge of the statistical concepts covered in a typical first