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Bayesian Inference via Approximation of Log-likelihood for Priors in Exponential Family
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara, Turkey.
Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper, a Bayesian inference technique based on Taylor series approximation of the logarithm of the likelihood function is presented. The proposed approximation is devised for the case where the prior distribution belongs to the exponential family of distributions. The logarithm of the likelihood function is linearized with respect to the sufficient statistic of the prior distribution in exponential family such that the posterior obtains the same exponential family form as the prior. Similarities between the proposed method and the extended Kalman filter for nonlinear filtering are illustrated. Further, an extended target measurement update for target models where the target extent is represented by a random matrix having an inverse Wishart distribution is derived. The approximate update covers the important case where the spread of measurement is due to the target extent as well as the measurement noise in the sensor.

National Category
Signal Processing
Identifiers
URN: urn:nbn:se:liu:diva-121616OAI: oai:DiVA.org:liu-121616DiVA: diva2:857311
Available from: 2015-09-28 Created: 2015-09-28 Last updated: 2015-10-05Bibliographically approved
In thesis
1. Analytical Approximations for Bayesian Inference
Open this publication in new window or tab >>Analytical Approximations for Bayesian Inference
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Bayesian inference is a statistical inference technique in which Bayes’ theorem is used to update the probability distribution of a random variable using observations. Except for few simple cases, expression of such probability distributions using compact analytical expressions is infeasible. Approximation methods are required to express the a priori knowledge about a random variable in form of prior distributions. Further approximations are needed to compute posterior distributions of the random variables using the observations. When the computational complexity of representation of such posteriors increases over time as in mixture models, approximations are required to reduce the complexity of such representations.

This thesis further extends existing approximation methods for Bayesian inference, and generalizes the existing approximation methods in three aspects namely; prior selection, posterior evaluation given the observations and maintenance of computation complexity.

Particularly, the maximum entropy properties of the first-order stable spline kernel for identification of linear time-invariant stable and causal systems are shown. Analytical approximations are used to express the prior knowledge about the properties of the impulse response of a linear time-invariant stable and causal system.

Variational Bayes (VB) method is used to compute an approximate posterior in two inference problems. In the first problem, an approximate posterior for the state smoothing problem for linear statespace models with unknown and time-varying noise covariances is proposed. In the second problem, the VB method is used for approximate inference in state-space models with skewed measurement noise.

Moreover, a novel approximation method for Bayesian inference is proposed. The proposed Bayesian inference technique is based on Taylor series approximation of the logarithm of the likelihood function. The proposed approximation is devised for the case where the prior distribution belongs to the exponential family of distributions.

Finally, two contributions are dedicated to the mixture reduction (MR) problem. The first contribution, generalize the existing MR algorithms for Gaussian mixtures to the exponential family of distributions and compares them in an extended target tracking scenario. The second contribution, proposes a new Gaussian mixture reduction algorithm which minimizes the reverse Kullback-Leibler divergence and has specific peak preserving properties.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. 79 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1710
National Category
Signal Processing
Identifiers
urn:nbn:se:liu:diva-121619 (URN)10.3384/diss.diva-121619 (DOI)978-91-7685-930-8 (print) (ISBN)
Public defence
2015-11-06, Visionen, B-huset, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2015-10-05 Created: 2015-09-28 Last updated: 2015-10-07Bibliographically approved

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