Results 1  10
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22
Prior Probabilities
 IEEE Transactions on Systems Science and Cybernetics
, 1968
"... e case of location and scale parameters, rate constants, and in Bernoulli trials with unknown probability of success. In realistic problems, both the transformation group analysis and the principle of maximum entropy are needed to determine the prior. The distributions thus found are uniquely determ ..."
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Cited by 166 (3 self)
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e case of location and scale parameters, rate constants, and in Bernoulli trials with unknown probability of success. In realistic problems, both the transformation group analysis and the principle of maximum entropy are needed to determine the prior. The distributions thus found are uniquely determined by the prior information, independently of the choice of parameters. In a certain class of problems, therefore, the prior distributions may now be claimed to be fully as "objective" as the sampling distributions. I. Background of the problem Since the time of Laplace, applications of probability theory have been hampered by difficulties in the treatment of prior information. In realistic problems of decision or inference, we often have prior information which is highly relevant to the question being asked; to fail to take it into account is to commit the most obvious inconsistency of reasoning and may lead to absurd or dangerously misleading results. As an extreme examp
A bayesian approach for blind separation of sparse sources
 IEEE Transactions on Speech and Audio Processing
, 2005
"... We present a Bayesian approach for blind separation of linear instantaneous mixtures of sources having a sparse representation in a given basis. The distributions of the coefficients of the sources in the basis are modeled by a Student t distribution, which can be expressed as a Scale Mixture of Gau ..."
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Cited by 49 (9 self)
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We present a Bayesian approach for blind separation of linear instantaneous mixtures of sources having a sparse representation in a given basis. The distributions of the coefficients of the sources in the basis are modeled by a Student t distribution, which can be expressed as a Scale Mixture of Gaussians, and a Gibbs sampler is derived to estimate the sources, the mixing matrix, the input noise variance and also the hyperparameters of the Student t distributions. The method allows for separation of underdetermined (more sources than sensors) noisy mixtures. Results are presented with audio signals using a Modified Discrete Cosine Transfrom basis and compared with a finite mixture of Gaussians prior approach. These results show the improved sound quality obtained with the Student t prior and the better robustness to mixing matrices close to singularity of the Markov Chains Monte Carlo approach.
Syntactic Measures of Complexity
, 1999
"... page 14 Declaration  page 15 Notes of copyright and the ownership of intellectual property rights  page 15 The Author  page 16 Acknowledgements  page 16 1  Introduction  page 17 1.1  Background  page 17 1.2  The Style of Approach  page 18 1.3  Motivation  page 19 1.4  Style of ..."
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Cited by 23 (2 self)
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page 14 Declaration  page 15 Notes of copyright and the ownership of intellectual property rights  page 15 The Author  page 16 Acknowledgements  page 16 1  Introduction  page 17 1.1  Background  page 17 1.2  The Style of Approach  page 18 1.3  Motivation  page 19 1.4  Style of Presentation  page 20 1.5  Outline of the Thesis  page 21 2  Models and Modelling  page 23 2.1  Some Types of Models  page 25 2.2  Combinations of Models  page 28 2.3  Parts of the Modelling Apparatus  page 33 2.4  Models in Machine Learning  page 38 2.5  The Philosophical Background to the Rest of this Thesis  page 41 Syntactic Measures of Complexity  page 3  3  Problems and Properties  page 44 3.1  Examples of Common Usage  page 44 3.1.1  A case of nails  page 44 3.1.2  Writing a thesis  page 44 3.1.3  Mathematics  page 44 3.1.4  A gas  page 44 3.1.5  An ant hill  page 45 3.1.6  A car engine  page 45 3.1.7  A cell as part of an organism ...
When did Bayesian inference become “Bayesian"?
 BAYESIAN ANALYSIS
, 2006
"... While Bayes’ theorem has a 250year history, and the method of inverse probability that flowed from it dominated statistical thinking into the twentieth century, the adjective “Bayesian” was not part of the statistical lexicon until relatively recently. This paper provides an overview of key Bayesi ..."
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Cited by 10 (1 self)
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While Bayes’ theorem has a 250year history, and the method of inverse probability that flowed from it dominated statistical thinking into the twentieth century, the adjective “Bayesian” was not part of the statistical lexicon until relatively recently. This paper provides an overview of key Bayesian developments, beginning with Bayes’ posthumously published 1763 paper and continuing up through approximately 1970, including the period of time when “Bayesian” emerged as the label of choice for those who advocated Bayesian methods.
Robust Bayesianism: Imprecise and Paradoxical Reasoning
, 2004
"... We are interested in understanding the relationship between Bayesian inference and evidence theory, in particular imprecise and paradoxical reasoning. The concept of a set of probability distributions is central both in robust Bayesian analysis and in some versions of DempsterShafer theory. Most of ..."
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Cited by 6 (1 self)
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We are interested in understanding the relationship between Bayesian inference and evidence theory, in particular imprecise and paradoxical reasoning. The concept of a set of probability distributions is central both in robust Bayesian analysis and in some versions of DempsterShafer theory. Most of the literature regards these two theories as incomparable. We interpret imprecise probabilities as imprecise posteriors obtainable from imprecise likelihoods and priors, both of which can be considered as evidence and represented with, e.g., DSstructures. The natural and simple robust combination operator makes all pairwise combinations of elements from the two sets. The DSstructures can represent one particular family of imprecise distributions, Choquet capacities. These are not closed under our combination rule, but can be made so by rounding. The proposed combination operator is unique, and has interesting normative and factual properties. We compare its behavior on Zadeh's example with other proposed fusion rules. We also show how the paradoxical reasoning method appears in the robust framework.
Robust Bayesianism: Relation to evidence theory
 J. Advances in Information Fusion
"... We are interested in understanding the relationship between Bayesian inference and evidence theory. The concept of a set of probability distributions is central both in robust Bayesian analysis and in some versions of DempsterShafer’s evidence theory. We interpret imprecise probabilities as impreci ..."
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Cited by 4 (0 self)
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We are interested in understanding the relationship between Bayesian inference and evidence theory. The concept of a set of probability distributions is central both in robust Bayesian analysis and in some versions of DempsterShafer’s evidence theory. We interpret imprecise probabilities as imprecise posteriors obtainable from imprecise likelihoods and priors, both of which are convex sets that can be considered as evidence and represented with, e.g., DSstructures. Likelihoods and prior are in Bayesian analysis combined with Laplace’s parallel composition. The natural and simple robust combination operator makes all pairwise combinations of elements from the two sets representing prior and likelihood. Our proposed combination operator is unique, and it has interesting normative and factual properties. We compare its behavior with other proposed fusion rules, and earlier efforts to reconcile Bayesian analysis and evidence theory. The behavior of the robust rule is consistent with the behavior of Fixsen/Mahler’s modified Dempster’s (MDS) rule, but not with Dempster’s rule. The Bayesian framework is liberal in allowing all significant uncertainty concepts to be modeled and taken care of and is therefore a viable, but probably not the only, unifying structure that can be economically taught and in which alternative solutions can be modeled, compared and explained. Manuscript received April 20, 2006; released for publication April
What is the Problem of Simplicity?
"... Abstract: The problem of simplicity involves three questions: How is the simplicity of a hypothesis to be measured? How is the use of simplicity as a guide to hypothesis choice to be justified? And how is simplicity related to other desirable features of hypotheses that is, how is simplicity to be ..."
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Cited by 2 (0 self)
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Abstract: The problem of simplicity involves three questions: How is the simplicity of a hypothesis to be measured? How is the use of simplicity as a guide to hypothesis choice to be justified? And how is simplicity related to other desirable features of hypotheses that is, how is simplicity to be tradedoff? The present paper explores these three questions, from a variety of viewpoints, including Bayesianism, likelihoodism, and the framework of predictive accuracy formulated by Akaike (1973). It may turn out that simplicity has no global justification that its justification varies from problem to problem. Scientists sometimes choose between rival hypotheses on the basis of their simplicity. Nonscientists do the same thing; this is no surprise, given that the methods used in science often reflect patterns of reasoning that are at work in everyday life. When people choose the simpler of two theories, this “choosing ” can mean different things. The simpler theory may be chosen because it is aesthetically more pleasing, because it is easier to understand or remember, or because it is easier to test. However, when philosophers talk about the “problem of simplicity,” they usually are thinking about another sort of choosing. The idea is that choosing the simpler 1 theory means regarding it as more plausible than its more complex rival. Philosophers often describe the role of simplicity in hypothesis choice by talking about the problem of curvefitting. Consider the following experiment. You put a sealed pot on a stove. The pot has a thermometer attached to it as well as a device that measures how much pressure the gas inside exerts on the walls of the pot. You then heat the pot to various temperatures and observe how much pressure there is in the pot. Each temperature reading with its associated pressure reading can be represented as a point in the coordinate system depicted below. The problem is to decide what the general relationship is between temperature and pressure for this system, given the data. Each hypothesis about this general relationship takes the form of a line. Which line is most plausible, given the observations you have made?
Probabilistic Reasoning and Inference for Systems Biology
, 2007
"... One of the important challenges in Systems Biology is reasoning and performing hypotheses testing in uncertain conditions, when available knowledge may be incomplete and the experimental data may contain substantial noise. In this thesis we develop methods of probabilistic reasoning and inference th ..."
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Cited by 1 (0 self)
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One of the important challenges in Systems Biology is reasoning and performing hypotheses testing in uncertain conditions, when available knowledge may be incomplete and the experimental data may contain substantial noise. In this thesis we develop methods of probabilistic reasoning and inference that operate consistently within an environment of uncertain knowledge and data. Mechanistic mathematical models are used to describe hypotheses about biological systems. We consider both deductive model based reasoning and model inference from data. The main contributions are a novel modelling approach using continuous time Markov chains that enables deductive derivation of model behaviours and their properties, and the application of Bayesian inferential methods to solve the inverse problem of model inference and comparison, given uncertain knowledge and noisy data. In the first part of the thesis, we consider both individual and population
The Marginalization Paradox and the Formal Bayes ’ Law
, 708
"... Abstract. It has recently been shown that the marginalization paradox (MP) can be resolved by interpreting improper inferences as probability limits. The key to the resolution is that probability limits need not satisfy the formal Bayes ’ law, which is used in the MP to deduce an inconsistency. In t ..."
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Abstract. It has recently been shown that the marginalization paradox (MP) can be resolved by interpreting improper inferences as probability limits. The key to the resolution is that probability limits need not satisfy the formal Bayes ’ law, which is used in the MP to deduce an inconsistency. In this paper, I explore the differences between probability limits and the more familiar pointwise limits, which do imply the formal Bayes ’ law, and show how these differences underlie some key differences in the interpretation of the MP.
Simplicity and Prediction
, 1994
"... In this paper I investigate the relation between simplicity and prediction. I firstly discuss several classical ideas about the concept of simplicity and how it is related to the acceptability of scientific theories. I further investigate a formal simplicity definition that stems from research in ..."
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In this paper I investigate the relation between simplicity and prediction. I firstly discuss several classical ideas about the concept of simplicity and how it is related to the acceptability of scientific theories. I further investigate a formal simplicity definition that stems from research in machine learning. In that approach simplicity plays an important role in the probability of predictions, as explicated by Ray Solomonoff. According to Solomonoff we should trust the theory which implications can be generated by the shortest computerprogramme that can generate a description of our known observational data. A shorter computerprogramme uses more patterns from that data, and hence provides more probable predictions. It is proved that this simplicity measure is reasonably independent of the computerlanguage that is used. I demonstrate that the approach of Solomonoff subsumes most of the other approaches and ideas, and solves most of their problems. My general conclu...