Bayesian Inference for Nonnegative Matrix Factorisation Models

Figure 1

(a) A schematic description of the NMF model with data augmentation. (b) Graphical model with hyperparameters. Each source element $s_{\nu \mathrm{,}i\mathrm{,}\tau }$ is Poisson distributed with intensity $t_{\nu \mathrm{,}i}{v}_{i\mathrm{,}\tau }$. The observations are given by $x_{\nu \mathrm{,}\tau }={\sum }_i{s}_{\nu \mathrm{,}i\mathrm{,}\tau }$. In matrix notation, we write $X=\sum S_i$. We can analytically integrate out over $S$. Due to superposition property of Poisson distribution, intensities add up, and we obtain $\langle X\rangle =TV$. Given $X$, the NMF algorithm is shown to seek the maximum likelihood estimates of the templates $T$ and excitations $V$. In our Bayesian treatment, we further assume that elements of $T$ and $V$ are Gamma distributed with hyperparameters $\mathrm{\Theta }$.

(a)

(b)