Bayesian Inference for Nonnegative Matrix Factorisation Models

Algorithm 2

Gibbs sampler for nonnegative matrix factorisation.

(1) Initialize:

      $T^{(0)}=~𝒢(·;A_t\mathrm{,}{B}_t)\hspace{1em}\hspace{1em}{V}^{(0)}~𝒢(·;A_v\mathrm{,}{B}_v)$

(2) for $n=1\dots$ MAXITER do

(3)   Sample Sources

(4)   for $\tau =1\cdots K$,$\nu =1\cdots W$ do

(5)    $p_{\nu \mathrm{,1}:I\mathrm{,}\tau }^{(n)}=T^{(n-1)}(\nu \mathrm{,1}:I)\mathrm{.}*V^{(n-1)}{(1:I\mathrm{,}\tau )}^{\top }\mathrm{.}/(T^{(n-1)}(\nu \mathrm{,1}:I)V^{(n-1)}(1:I\mathrm{,}\tau ))$

(6)    $S^{(n)}(\nu \mathrm{,1}:I\mathrm{,}\tau )~ℳ(s_{\nu \mathrm{,1}:I\mathrm{,}\tau };x_{\nu \mathrm{,}\tau }\mathrm{,}{p}_{\nu \mathrm{,1}:I\mathrm{,}\tau }^{(n)})$

(7)   end for

        ${\mathrm{\Sigma }}_t^{(n)}={\displaystyle \sum _{\tau }}{S}_{\nu \mathrm{,}i\mathrm{,}\tau }^{(n)}\hspace{1em}\hspace{1em}{\mathrm{\Sigma }}_v^{(n)}={\displaystyle \sum _{\nu }}{S}_{\nu \mathrm{,}i\mathrm{,}\tau }^{(n)}$

(8)   Sample Templates

      $\begin{array}{c}{\alpha }_t^{(n)}=A_t+{\mathrm{\Sigma }}_t^{(n)}\hspace{1em}\hspace{1em}{\beta }_t^{(n)}=1./\left(A_t\mathrm{.}/B_t+1_W{\left(V^{(n-1)}{1}_K\right)}^{\top }\right)\end{array}$

      $T^{(n)}~𝒢(T;{\alpha }_t^{(n)}\mathrm{,}{\beta }_t^{(n)})$

(9)   Sample Excitations

      $\begin{array}{c}{\alpha }_v^{(n)}=A_v+{\mathrm{\Sigma }}_v^{(n)}\hspace{1em}\hspace{1em}{\beta }_v^{(n)}=1./\left(A_v\mathrm{.}/B_v+{\left(1_W^{\top }{T}^{(n-1)}\right)}^{\top }{1}_K^{\top }\right)\end{array}$

      $\begin{array}{c}{V}^{(n)}~𝒢(V;{\alpha }_v^{(n)}\mathrm{,}{\beta }_v^{(n)})\end{array}$

(10) end for