# Bayesian Modeling of Directional Data with Acoustic and Other Applications

Sammanfattning: A direction is defined here as a multi-dimensional unit vector. Such unitvectors form directional data. Closely related to directional data are axialdata for which each direction is equivalent to the opposite direction.Directional data and axial data arise in various fields of science. In probabilisticmodeling of such data, probability distributions are needed whichcount for the structure of the space from which data samples are collected.Such distributions are known as directional distributions and axial distributions.This thesis studies the von Mises-Fisher (vMF) distribution and the(complex) Watson distribution as representatives of directional and axialdistributions.Probabilistic models of the data are defined through a set of parameters.In the Bayesian view to uncertainty, these parameters are regarded as randomvariables in the learning inference. The primary goal of this thesis is todevelop Bayesian inference for directional and axial models, more precisely,vMF and (complex) Watson distributions, and parametric mixture modelsof such distributions. The Bayesian inference is realized using a family ofoptimization methods known as variational inference. With the proposedvariational methods, the intractable Bayesian inference problem is cast asan optimization problem.The variational inference for vMF andWatson models shall open up newapplications and advance existing application domains by reducing restrictiveassumptions made by current modelling techniques. This is the centraltheme of the thesis in all studied applications. Unsupervised clustering ofgene-expression and gene-microarray data is an existing application domain,which has been further advanced in this thesis. This thesis also advancesapplication of the complex Watson models in the problem of blind sourceseparation (BSS) with acoustic applications. Specifically, it is shown thatthe restrictive assumption of prior knowledge on the true number of sourcescan be relaxed by the desirable pruning property in Bayesian learning, resultingin BSS methods which can estimate the number of sources.Furthermore, this thesis introduces a fully Bayesian recursive frameworkfor the BSS task. This is an attempt toward realization of an online BSSmethod. In order to reduce the well-known problem of permutation ambiguityin the frequency domain, the complete BSS problem is solved in one unified modeling step, combining the frequency bin-wise source estimationwith the permutation problem. To realize this, all time frames and frequencybins are connected using a first order Markov chain. The model cancapture dependencies across both time frames and frequency bins, simultaneously,using a feed-forward two-dimensional hidden Markov model (2-DHMM).

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