this paper a Bayesian approach to the calibration of computer models. We represent the unknown inputs as a parameter vector `. Using the observed data we derive the posterior distribution of `, which in particular quantifies the `residual uncertainty' about

Data mining is on the interface of Computer Science and Statistics, utilizing advances in both disciplines to make progress in extracting information from large databases. It is an emerging field that has attracted much attention in a very short period of time. This article highlights some statistical themes and lessons that are directly relevant to data mining and attempts to identify opportunities where close cooperation between the statistical and computational communities might reasonably provide synergy for further progress in data analysis.

this paper we review several of these methods, and subsequently compare them in the context of two examples, the first a simple regression example, and the second a much more challenging hierarchical longitudinal model of the kind often encountered in biostatistical practice. We find that the joint model-parameter space search methods perform adequately but can be difficult to program and tune, while the marginal likelihood methods are often less troublesome and require less in the way of additional coding. Our results suggest that the latter methods may be most appropriate for practitioners working in many standard model choice settings, while the former remain important for comparing large numbers of models, or models whose parameters cannot be easily updated in relatively few blocks. We caution however that all of the methods we compare require significant human and computer effort, suggesting that less formal Bayesian model choice methods may offer a more realistic alternative in many cases.

The problem of variable selection is one of the most pervasive model selection problems in statistical applications. Often referred to as the problem of subset selection, it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors, but there is uncertainty about which subset to use. This vignette reviews some of the key developments which have led to the wide variety of approaches for this problem. 1

. Motion segmentation involves identifying regions of the image that correspond to independently moving objects. The number of independently moving objects, and type of motion model for each of the objects is unknown a priori. In order to perform motion segmentation, the problems of model selection, robust estimation and clustering must all be addressed simultaneously. Here we place the three problems into a common Bayesian framework; investigating the use of model averaging-representing a motion by a combination of models---as a principled way for motion segmentation of images. The final result is a fully automatic algorithm for clustering that works in the presence of noise and outliers. 1 Introduction Detection of independently moving objects is an essential but often neglected precursor to problems in computer vision e.g. e#cient video compression [3], video editing, surveillance, smart tracking of objects etc. The work in this paper stems from the desire to develop a g...

Introduction To motivate the methods described in this chapter, consider the following inference problem in astronomy (Soubiran, 1993). Until fairly recently, it has been believed that the Galaxy consists of two stellar populations, the disk and the halo. More recently, it has been hypothesized that there are in fact three stellar populations, the old (or thin) disk, the thick disk, and the halo, distinguished by their spatial distributions, their velocities, and their metallicities. These hypotheses have different implications for theories of the formation of the Galaxy. Some of the evidence for deciding whether there are two or three populations is shown in Figure 1, which shows radial and rotational velocities for n = 2; 370 stars. A natural model for this situation is a mixture model with J components, namely y i = J X j=1 ae j

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Radial wavelet networks have recently been proposed as a method for nonparametric regression. In this paper we analyse their performance within a Bayesian framework. We derive probability distributions over both the dimension of the networks and the network coefficients by placing a prior on the degrees of freedom of the model. This process bypasses the need to test or select a finite number of networks during the modelling process. Predictions are formed by mixing over many models of varying dimension and parameterization. We show that the complexity of the models adapts to the complexity of the data and produces good results on a number of benchmark test series. Keywords: Wavelets, radial basis functions, model choice, Bayesian neural networks, reversible jump Markov chain Monte Carlo, nonparametric regression, splines. 1 Introduction Wavelet networks have previously been studied in relation to nonparametric regression by Zhang (1997), Kugarajah and Zhang (1995), Zhang and Benveni...

Covariate and confounder selection in case-control studies is most commonly carried out using either a two-step method or a stepwise variable selection method in logistic regression. Inference is then carried out conditionally on the selected model, but this ignores the model uncertainty implicit in the variable selection process, and so underestimates uncertainty about relative risks. We report on a simulation study designed to be similar to actual case-control studies. This shows that p-values computed after variable selection can greatly overstate the strength of conclusions. For example, for our simulated case-control studies with 1,000 subjects, of variables declared to be "significant" with p-values between.01 and.05, only 49 % actually were risk factors when stepwise variable selection was used. We propose Bayesian model averaging as a formal way of taking account of model uncertainty in case-control studies. This yields an easily interpreted summary, the posterior probability that a variable is a risk factor, and our simulation study indicates this to be reasonably well calibrated in the situations simulated. The methods are applied and compared

Deterministic simulation models are used in many areas of science, engineering and policy-making. Typically, they are complex models that attempt to capture underlying mechanisms in considerable detail, and they have many user-specified inputs. The inputs are often specified by some form of trial-and-error approach in which plausible values are postulated, the corresponding outputs inspected, and the inputs modified until plausible outputs are obtained. Here we address the issue of more formal inference for such models. Raftery et al. (1995a) proposed the Bayesian synthesis approach in which the available information about both inputs and outputs was encoded in a probability distribution and inference was made by restricting this distribution to the submanifold specifid by the model. Wolpert (1995) showed that this is subject to the Borel paradox, according to which the results can depend on the parameterization of the model. We show that this dependence is due to the presence of a prior on the outputs. We propose a modified approach, called Bayesian melding, which takes full account of information and uncertainty about both inputs and outputs to the model, while avoiding the Borel paradox. This is done by recognizing the existence of two priors, one implicit and one explicit, on each input and output � these are combined via logarithmic pooling. Bayesian melding is then