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15
Choice of Basis for Laplace Approximation
 Machine Learning
, 1998
"... Maximum a posterJori optimization of parameters and the Laplace approximation for the marginal likelihood are both basisdependent methods. This note compares two choices of basis for models parameterized by probabilities, showing that it is possible to improve on the traditional choice, the prob ..."
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Cited by 35 (1 self)
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Maximum a posterJori optimization of parameters and the Laplace approximation for the marginal likelihood are both basisdependent methods. This note compares two choices of basis for models parameterized by probabilities, showing that it is possible to improve on the traditional choice, the probability simplex, by transforming to the softmax' basis.
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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Cited by 1 (0 self)
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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.
University of Cambridge.
"... Bayesian inference offers us a powerful tool with which to tackle the problem of data modelling. However, the performance of Bayesian methods is crucially dependent on being able to find good models for our data. The principal focus of this thesis is the development of models based on Gaussian proce ..."
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Bayesian inference offers us a powerful tool with which to tackle the problem of data modelling. However, the performance of Bayesian methods is crucially dependent on being able to find good models for our data. The principal focus of this thesis is the development of models based on Gaussian process priors. Such models, which can be thought of as the infinite extension of several existing finite models, have the flexibility to model complex phenomena while being mathematically simple. In this thesis, I present a review of the theory of Gaussian processes and their covariance functions and demonstrate how they fit into the Bayesian framework. The efficient implementation of a Gaussian process is discussed with particular reference to approximate methods for matrix inversion based on the work of Skilling (1993). Several regression problems are examined. Nonstationary covariance functions are developed for the regression of neuron spike data and the use of Gaussian processes to model the potential energy surfaces of weakly bound molecules is discussed. Classification methods based on Gaussian processes are implemented using variational methods. Existing bounds (Jaakkola and Jordan 1996) for the sigmoid function are used to tackle binary problems and multidimensional bounds on the softmax function are presented for the multiple class case. The performance of the variational classifier is compared with that of other methods using the CRABS and PIMA datasets (Ripley 1996) and the problem of predicting the cracking of welds based on their chemical composition is also investigated. The theoretical calculation of the density of states of crystal structures is discussed in detail. Three possible approaches to the problem are described based on free energy minimization, Gaussian processes and the theory of random matrices. Results from these approaches are compared with the stateoftheart techniques (Pickard 1997)
zur Erklärung des akademischen Grades
"... of high complexity data for the inversion of metric InSAR in urban environments Vom Fachbereich Elektrotechnik und Informatik der ..."
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of high complexity data for the inversion of metric InSAR in urban environments Vom Fachbereich Elektrotechnik und Informatik der
unknown title
, 807
"... The complementarity of astrometric and radial velocity exoplanet observations Determining exoplanet mass with astrometric snapshots ..."
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The complementarity of astrometric and radial velocity exoplanet observations Determining exoplanet mass with astrometric snapshots
Letter to the Editor
, 902
"... Bayesian analysis of the radial velocities of HD 11506 reveals another planetary companion ..."
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Bayesian analysis of the radial velocities of HD 11506 reveals another planetary companion
unknown title
, 807
"... On the complementarity of astrometric and radial velocity exoplanet observations Determining exoplanet mass with astrometric snapshots ..."
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On the complementarity of astrometric and radial velocity exoplanet observations Determining exoplanet mass with astrometric snapshots
University of Cambridge.
"... Bayesian inference offers us a powerful tool with which to tackle the problem of data modelling. However, the performance of Bayesian methods is crucially dependent on being able to find good models for our data. The principal focus of this thesis is the development of models based on Gaussian proce ..."
Abstract
 Add to MetaCart
Bayesian inference offers us a powerful tool with which to tackle the problem of data modelling. However, the performance of Bayesian methods is crucially dependent on being able to find good models for our data. The principal focus of this thesis is the development of models based on Gaussian process priors. Such models, which can be thought of as the infinite extension of several existing finite models, have the flexibility to model complex phenomena while being mathematically simple. In thesis, I present a review of the theory of Gaussian processes and their covariance functions and demonstrate how they fit into the Bayesian framework. The efficient implementation of a Gaussian process is discussed with particular reference to approximate methods for matrix inversion based on the work of Skilling (1993). Several regression problems are examined. Nonstationary covariance functions are developed for the regression of neuron spike data and the use of Gaussian processes to model the potential energy surfaces of weakly bound molecules is discussed. Classification methods based on Gaussian processes are implemented using variational methods. Existing bounds (Jaakkola and Jordan 1996) for the sigmoid function are used to tackle binary problems and multidimensional bounds on the softmax function are presented for the multiple class case. The performance of the variational classifier is compared with that of other methods using the CRABS and PIMA datasets (Ripley 1996) and the problem of predicting the cracking of welds based on their chemical composition is also investigated. The theoretical calculation of the density of states of crystal structures is discussed in detail. Three possible approaches to the problem are described based on free energy minimization, Gaussian processes and the theory of random matrices. Results from these approaches are compared with the stateoftheart techniques (Pickard 1997)