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...

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An algebra is presented for a simple probabilistic data model that may be regarded as an extension of the standard relational model. The probabilistic algebra is developed in such a way that (restricted to α-acyclic database schemes) the relational algebra is a homomorphic image of it. Strictly probabilistic results are emphasized. Variations on the basic probabilistic data model are discussed. The algebra is used to explicate a commonly used statistical smoothing procedure and is shown to be potentially very useful for decision support with uncertain information.

We develop a flexible model-based procedure for clustering functional data. The technique can be applied to all types of curve data but is particularly useful when individuals are observed at a sparse set of time points. In addition to producing final cluster assignments, the procedure generates predictions and confidence intervals for missing portions of curves. Our approach also provides many useful tools for evaluating the resulting models. Clustering can be assessed visually via low dimensional representations of the curves, and the regions of greatest separation between clusters can be determined using a discriminant function. Finally, we extend the model to handle multiple functional and finite dimensional covariates and show how it can be applied to standard finite dimensional clustering problems involving missing data.

We introduce a technique for extending the classical method of Linear Discriminant Analysis to data sets where the predictor variables are curves or functions. This procedure, which we call functionallinear discriminant analysis (FLDA), is particularly useful when only fragments of the curves are observed. All the techniques associated with LDA can be extended for use with FLDA. In particular FLDA can be used to produce classifications on new (test) curves, give an estimate of the discriminant function between classes, and provide a one or two dimensional pictorial representation of a set of curves. We also extend this procedure to provide generalizations of quadratic and regularized discriminant analysis.

A new method of pointwise adaptation has been proposed and studied in Spokoiny (1998) in context of estimation of piecewise smooth univariate functions. The present paper extends that method to estimation of bivariate grey-scale images composed of large homogeneous regions with smooth edges and observed with noise on a gridded design. The proposed estimator # f(x) at a point x is simply the average of observations over a window # U(x) selected in a data-driven way. The theoretical properties of the procedure are studied for the case of piecewise constant images. We present a nonasymptotic bound for the accuracy of estimation at a specific grid point x as a function of the number of pixel n, of the distance from the point of estimation to the closest boundary and of smoothness properties and orientation of this boundary. It is also shown that the proposed method provides a near optimal rate of estimation near edges and inside homogeneous regions. We briefly discuss algorithmic aspects and the complexity of the procedure. The numerical examples demonstrate a reasonable performance of the method and they are in agreement with the theoretical issues. An example from satellite (SAR) imaging illustrates the applicability of the method. # The authors thank A.Juditski, O. Lepski, A.Tsybakov and Yu.Golubev for important remarks and discussion. polzehl, j. and spokoiny, v. 1 1

Methods are developed for design of linear tomographic reconstruction algorithms with specified properties. Assuming a starting model with constant slowness, an algorithm with the following properties is found: (1) The optimum constant for the starting model is determined automatically. (2) The weighted least-squares error between the predicted and measured traveltime data is as small as possible. (3) The variance of the reconstructed slowness from the starting model is minimized. (4) Rays with the greatest length have the least influence on the reconstructed slowness. (5) Cells with most ray coverage tend to deviate least from the background value. The resulting algorithm maps the reconstruction problem into a vector space where the contribution to the inversion from the background slowness remains invariant, while the optimum contributions in orthogonal directions are found. For a starting model with nonconstant slowness, the reconstruction algorithm has analogous properties. -- 2 ...

Quantitative science requires the assessment of uncertainty, and this means that measurements and inferences should be described as probability distributions. This is done by building data into a probabilistic likelihood function which produces a posterior “answer ” by modulating a prior “question”. Probability calculus is the only way of doing this consistently, so that data can be included gradually or all at once while the answer remains the same. But probability calculus is only a language: it does not restrict the questions one can ask by setting one’s prior. We discuss how to set sensible priors, in particular for a large problem like image reconstruction. We also introduce practical modern algorithms (Gibbs sampling, Metropolis algorithm, genetic algorithms, and simulated annealing) for computing probabilistic inference.

Ensemble forecasting involves the use of several integrations of a numerical model. Even if this model is assumed to be known, ensembles are needed due to uncertainty in initial conditions. The ideas discussed in this paper incorporate aspects of both analytic model approximations and Monte Carlo arguments to gain some efficiency in the generation and use of ensembles. Efficiency is gained through the use of importance sampling Monte Carlo. Once ensemble members are generated, suggestions for their use, involving both approximation and statistical notions such as kernel density estimation and mixture modeling are discussed. Fully deterministic procedures derived from the Monte Carlo analysis are also described. Examples using the three-dimensional Lorenz system are described. Address: Mark Berliner Department of Statistics Ohio State University 1958 Neil Ave. Columbus, OH 43210-1247 USA e-mail: mb@stat.ohio-state.edu Keywords and Phrases: Chaos, Importance sampling, Kernel density es...

In a number of engineering topics we are faced with the inverse problem of recovering the spatial distribution of some scalar or vector quantity from measurements of the interaction of an investigated medium with an incident wave. The common feature of such image reconstruction problems is that they are often ill-posed or ill-conditioned. We review first the basic aspects of standard regularization theory. Then, using an information-based approach, we show that existing regularization criteria, which were introduced in the literature using very different approaches, can be interpreted as special cases of an entropy, in spite of their apparent variety. Finally, we discuss its limitations and present the Bayesian statistical approach which allows local properties to be introduced in the estimated image through Markov random fields and associated local energy functions.