Bayesian Blind Source Separation with Unknown Prior Covariance

Abstract

The task of blind source separation (BSS) is to recover original signal sources which are observed only via their superposition with unknown weights. Since we are interested in estimation of the number of relevant sources in noisy observation, we use the Bayesian formulation which automatically removes spurious sources. A tool for this behavior is joint estimation of the unknown prior covariance matrix of the sources in tandem with the sources. In this work, we study the effect of various choices of the covariance matrix structure. Specifically, we compare models using the automatic relevance determination (ARD) principle on the first and the second diagonal, as well as full covariance matrix with Wishart prior. We obtain five versions of the variational BSS algorithm. These are tested on synthetic data and on a selected dataset from dynamic renal scintigraphy. MATLAB implementation of the methods is available for download.

A Shaping Parameters of Posterior Distributions

Posterior distributions are $\tilde{f}(A|D)=t\mathcal {N}_{A}(\mu _{A},I_{p}\otimes {\varSigma }_{A}),$ $\tilde{f}(\xi _{k}|D)=\mathcal {G}_{\xi _{k}}\left( \phi _{k},\psi _{k}\right) ,$ $\tilde{f}(\mathbf {x}|D)=t\mathcal {N}_{\mathbf {x}}\left( \mu _{\mathbf {x}},{\varSigma }_{\mathbf {x}}\right) ,$ $\tilde{f}({\varUpsilon }|D)=\mathcal {W}_{{\varUpsilon },nr}\left( {\varSigma }_{{\varUpsilon }},\beta \right) ,$ $\tilde{f}(\omega |D)=\mathcal {G}_{\omega }\left( \vartheta ,\rho \right) ,$ with shaping parameters ${\varSigma }_{A}^{-1}=\left( \omega \widehat{X^{T}X}+\widehat{\Xi }\right) ,$ $\mu _{A}=\left( \omega D\widehat{X}\right) {\varSigma }_{A},$ $\phi =\phi _{0}+\frac{p}{2}\mathbf {1}_{r,1},$ $\psi =\psi _{0}+\frac{1}{2}\mathrm {diag}\left( {\widehat{A^{T}A}}\right) ,$ ${\varSigma }_{\mathbf {x}}^{-1}=\left( ({\widehat{\omega }}{\widehat{A^{T}A}})\otimes I_{n}+{\widehat{{\varUpsilon }}}\circ L\right)$ $\mu _{\mathbf {x}}={\varSigma }_{\mathbf {x}}\left( {\widehat{\omega }}\mathrm {vec}\left( D^{T}{\widehat{A}}\right) \right)$ ${\varSigma }_{{\varUpsilon }}^{-1}=\left( {\widehat{\mathbf {x}\mathbf {x}^{T}}}+\alpha _{0}^{-1}I_{nr}\right)$ $\beta =\beta _{0}+1$ $\vartheta =\vartheta _{0}+\frac{pn}{2},$ $\rho =\rho _{0}+\frac{1}{2}\mathrm {tr}\left( (D-{\widehat{A}}{\widehat{X^{T}}})(D-{\widehat{A}}{\widehat{X^{T}}})^{T}\right) .$