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81
Bayesian Interpolation
 Neural Computation
, 1991
"... Although Bayesian analysis has been in use since Laplace, the Bayesian method of modelcomparison has only recently been developed in depth. In this paper, the Bayesian approach to regularisation and modelcomparison is demonstrated by studying the inference problem of interpolating noisy data. T ..."
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Cited by 533 (18 self)
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Although Bayesian analysis has been in use since Laplace, the Bayesian method of modelcomparison has only recently been developed in depth. In this paper, the Bayesian approach to regularisation and modelcomparison 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...
Removing camera shake from a single photograph
 ACM Trans. Graph
, 2006
"... Camera shake during exposure leads to objectionable image blur and ruins many photographs. Conventional blind deconvolution methods typically assume frequencydomain constraints on images, or overly simplified parametric forms for the motion path during camera shake. Real camera motions can follow c ..."
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Cited by 195 (13 self)
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Camera shake during exposure leads to objectionable image blur and ruins many photographs. Conventional blind deconvolution methods typically assume frequencydomain constraints on images, or overly simplified parametric forms for the motion path during camera shake. Real camera motions can follow convoluted paths, and a spatial domain prior can better maintain visually salient image characteristics. We introduce a method to remove the effects of camera shake from seriously blurred images. The method assumes a uniform camera blur over the image and negligible inplane camera rotation. In order to estimate the blur from the camera shake, the user must specify an image region without saturation effects. We show results for a variety of digital photographs taken from personal photo collections.
Bayesian color constancy
 Journal of the Optical Society of America A
, 1997
"... The problem of color constancy may be solved if we can recover the physical properties of illuminants and surfaces from photosensor responses. We consider this problem within the framework of Bayesian decision theory. First, we model the relation among illuminants, surfaces, and photosensor response ..."
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Cited by 143 (21 self)
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The problem of color constancy may be solved if we can recover the physical properties of illuminants and surfaces from photosensor responses. We consider this problem within the framework of Bayesian decision theory. First, we model the relation among illuminants, surfaces, and photosensor responses. Second, we construct prior distributions that describe the probability that particular illuminants and surfaces exist in the world. Given a set of photosensor responses, we can then use Bayes’s rule to compute the posterior distribution for the illuminants and the surfaces in the scene. There are two widely used methods for obtaining a single best estimate from a posterior distribution. These are maximum a posteriori (MAP) and minimum meansquarederror (MMSE) estimation. We argue that neither is appropriate for perception problems. We describe a new estimator, which we call the maximum local mass (MLM) estimate, that integrates local probability density. The new method uses an optimality criterion that is appropriate for perception tasks: It finds the most probable approximately correct answer. For the case of low observation noise, we provide an efficient approximation. We develop the MLM estimator for the colorconstancy problem in which flat matte surfaces are uniformly illuminated. In simulations we show that the MLM method performs better than the MAP estimator and better than a number of standard colorconstancy algorithms. We note conditions under which even the optimal estimator produces poor estimates: when the spectral properties of the surfaces in the scene are biased. © 1997 Optical Society of America [S07403232(97)016074] 1.
Inducing probabilistic grammars by bayesian model merging
 In: Int. Conf. Grammatical Inference. URL: citeseer.nj.nec.com/stolcke94inducing.html
, 1994
"... We describe a framework for inducing probabilistic grammars from corpora of positive samples. First, samples are incorporated by adding adhoc rules to a working grammar; subsequently, elements of the model (such as states or nonterminals) are merged to achieve generalization and a more compact repr ..."
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Cited by 131 (0 self)
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We describe a framework for inducing probabilistic grammars from corpora of positive samples. First, samples are incorporated by adding adhoc rules to a working grammar; subsequently, elements of the model (such as states or nonterminals) are merged to achieve generalization and a more compact representation. The choice of what to merge and when to stop is governed by the Bayesian posterior probability of the grammar given the data, which formalizes a tradeoff between a close fit to the data and a default preference for simpler models (‘Occam’s Razor’). The general scheme is illustrated using three types of probabilistic grammars: Hidden Markov models, classbasedgrams, and stochastic contextfree grammars. 1
Bestfirst Model Merging for Hidden Markov Model Induction
, 1994
"... This report describes a new technique for inducing the structure of Hidden Markov Models from data which is based on the general `model merging' strategy (Omohundro 1992). The process begins with a maximum likelihood HMM that directly encodes the training data. Successively more general models ..."
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Cited by 95 (7 self)
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This report describes a new technique for inducing the structure of Hidden Markov Models from data which is based on the general `model merging' strategy (Omohundro 1992). The process begins with a maximum likelihood HMM that directly encodes the training data. Successively more general models are produced by merging HMM states. A Bayesian posterior probability criterion is used to determine which states to merge and when to stop generalizing. The procedure may be considered a heuristic search for the HMM structure with the highest posterior probability. We discuss a variety of possible priors for HMMs, as well as a number of approximations which improve the computational efficiency of the algorithm. We studied three applications to evaluate the procedure. The first compares the merging algorithm with the standard BaumWelch approach in inducing simple finitestate languages from small, positiveonly training samples. We found that the merging procedure is more robust and accurate, part...
Comparison of Approximate Methods for Handling Hyperparameters
 NEURAL COMPUTATION
"... I examine two approximate methods for computational implementation of Bayesian hierarchical models, that is, models which include unknown hyperparameters such as regularization constants and noise levels. In the 'evidence framework' the model parameters are integrated over, and the resu ..."
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Cited by 69 (1 self)
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I examine two approximate methods for computational implementation of Bayesian hierarchical models, that is, models which include unknown hyperparameters such as regularization constants and noise levels. In the 'evidence framework' the model parameters are integrated over, and the resulting evidence is maximized over the hyperparameters. The optimized
From Laplace To Supernova Sn 1987a: Bayesian Inference In Astrophysics
, 1990
"... . The Bayesian approach to probability theory is presented as an alternative to the currently used longrun relative frequency approach, which does not offer clear, compelling criteria for the design of statistical methods. Bayesian probability theory offers unique and demonstrably optimal solutions ..."
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Cited by 52 (2 self)
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. The Bayesian approach to probability theory is presented as an alternative to the currently used longrun relative frequency approach, which does not offer clear, compelling criteria for the design of statistical methods. Bayesian probability theory offers unique and demonstrably optimal solutions to wellposed statistical problems, and is historically the original approach to statistics. The reasons for earlier rejection of Bayesian methods are discussed, and it is noted that the work of Cox, Jaynes, and others answers earlier objections, giving Bayesian inference a firm logical and mathematical foundation as the correct mathematical language for quantifying uncertainty. The Bayesian approaches to parameter estimation and model comparison are outlined and illustrated by application to a simple problem based on the gaussian distribution. As further illustrations of the Bayesian paradigm, Bayesian solutions to two interesting astrophysical problems are outlined: the measurement of wea...
L_0  The First Five Years of an Automated Language Acquisition Project
, 1996
"... The L0 project at ICSI and UC Berkeley attempts to combine not only vision and natural language modelling, but also learning. The original task was put forward in #Feldman et al. 1990a# as a touchstone task for AI and cognitive science. The task is to build a system that can learn the appropriate ..."
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Cited by 26 (7 self)
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The L0 project at ICSI and UC Berkeley attempts to combine not only vision and natural language modelling, but also learning. The original task was put forward in #Feldman et al. 1990a# as a touchstone task for AI and cognitive science. The task is to build a system that can learn the appropriate fragmentofany natural language from sentencepicture pairs. Wehave not succeeded in building such a system, but wehave made considerable progress on component subtasks and this has led in a number of productive and surprising directions. 1 Introduction The L 0 project at ICSI and UC Berkeley attempts to combine not only vision and natural language modelling, but also learning. The original task was put forward in #Feldman et al. 1990a# as a touchstone task for AI and cognitive science in a very simple form: The system is given examples of pictures paired with true statements about those pictures in an arbitrary natural language. #See Figure 1.# The system is to learn the relevant porti...
Bayesian Model Comparison and Backprop Nets
 ADVANCES IN NEURAL INFORMATION PROCESSING SYSTEMS 4
, 1992
"... The Bayesian model comparison framework is reviewed, and the Bayesian Occam's razor is explained. This framework can be applied to feedforward networks, making possible (1) objective comparisons between solutions using alternative network architectures; (2) objective choice of magnitude and ..."
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Cited by 21 (0 self)
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The Bayesian model comparison framework is reviewed, and the Bayesian Occam's razor is explained. This framework can be applied to feedforward networks, making possible (1) objective comparisons between solutions using alternative network architectures; (2) objective choice of magnitude and type of weight decay terms; (3) quantified estimates of the error bars on network parameters and on network output. The framework also generates a measure of the effective number of parameters determined by the data. The relationship