Conjugate Gamma Markov random fields for modelling nonstationary sources A. Taylan Cemgil 1 and Onur Dikmen 2 ⋆⋆ 1 Engineering Dept., University of Cambridge, CB2 1PZ, Cambridge, UK, atc27@eng.cam.ac.uk, WWW home page: http://www-sigproc.eng.cam.ac.uk/∼atc27 2 Dept. of Computer Eng., Bogazici University, 80815 Bebek, Istanbul, Turkey Abstract. In modelling nonstationary sources, one possible strategy is to define a latent process of strictly positive variables to model varia- tions in second order statistics of the underlying process. This can be achieved, for example, by passing a Gaussian process through a positive nonlinearity or defining a discrete state Markov chain where each state encodes a certain regime. However, models with such constructs turn out to be either not very flexible or non-conjugate, making inference some- what harder. In this paper, we introduce a conjugate (inverse-) gamma Markov Random field model that allows random fluctuations on vari- ances which are useful as priors for nonstationary time-frequency energy distributions. The main idea is to introduce auxiliary variables such that full conditional distributions and sufficient statistics are readily available as closed form expressions. This allows straightforward implementation of a Gibbs sampler or a variational algorithm. We illustrate our approach on denoising and single channel source separation. 1 Introduction In the Bayesian framework, various signal estimation problems can be cast into posterior inference problems. For example, source separation [6, 5, 9, 11, 3, 2], can be stated as p(s|x)= 1 Z x  dΘ o dΘ s p(x|s,Θ o )p(s|Θ s )p(Θ o )p(Θ s ) (1) where s ≡ s 1:K,1:N and x ≡ x 1:K,1:M . Here, the task is to infer N source signals s k,n given M observed signals x k,m where n =1 ...N , m =1 ...M at each index k where k =1 ...K. Here, k typically denotes time or a time-frequency atom in a linear transform domain. In Eq.(1), the (possibly degenerate, deterministic) con- ditional distribution p(x|s,Θ o ) specifies the observation model where Θ o denotes ⋆⋆ This research is funded by EPSRC (Engineering and Physical Sciences Research Council of UK) under the grant EP/D03261X/1 entitled “Probabilistic Modelling of Musical Audio for Machine Listening” and by TUBITAK (Scientific and Technolog- ical Research Council of Turkey) under the grant entitled “Time Series Modelling for Bayesian Source Separation”.