Bayesian Inference for Nonnegative Matrix Factorisation Models

Algorithm 1

Variational nonnegative matrix factorisation.

(1) Initialise:

      $L_t^{(0)}=E_t^{(0)}~𝒢(·;A_t\mathrm{,}{B}_t{\mathrm{.}/A}_t)\hspace{1em}\hspace{1em}\hspace{1em}{L}_v^{(0)}=E_v^{(0)}~𝒢(·;A_v\mathrm{,}{B}_v{\mathrm{.}/A}_v)$

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

(3)  Source sufficient statistics

        ${\mathrm{\Sigma }}_t^{(n)}:=L_t^{(n-1)}\mathrm{.}*(((X\mathrm{.}*M)\mathrm{.}/(L_t^{(n-1)}{L}_v^{(n-1)})){L_v^{(n-1)}}^{\top })$

        ${\mathrm{\Sigma }}_v^{(n)}:=L_v^{(n-1)}\mathrm{.}*({L_t^{(n-1)}}^{\top }((X\mathrm{.}*M)\mathrm{.}/(L_t^{(n-1)}{L}_v^{(n-1)})))$

(4)  Means

     $E_t^{(n)}:={\alpha }_t^{(n)}\mathrm{.}*{\beta }_t^{(n)}\hspace{1em}\hspace{1em}{\alpha }_t^{(n)}=A_t+{\mathrm{\Sigma }}_t^{(n)}\hspace{1em}\hspace{1em}{\beta }_t^{(n)}=1./(A_t\mathrm{.}/B_t+M{E_v^{(n-1)}}^{\top })$

     $E_v^{(n)}:={\alpha }_v^{(n)}\mathrm{.}*{\beta }_v^{(n)}\hspace{1em}\hspace{1em}{\alpha }_v^{(n)}=A_v+{\mathrm{\Sigma }}_v^{(n)}\hspace{1em}\hspace{1em}{\beta }_v^{(n)}=1./(A_v\mathrm{.}/B_v+{E_t^{(n)}}^{\top }M)$

(5)  Optional: Compute Bound (See Appendix, (A.13)

(6)  Means of Logs

      $L_t^{(n)}=\text{exp}\left(\mathrm{\Psi }\right({\alpha }_t^{(n)}\left)\right)\mathrm{.}*{\beta }_t^{(n)}\hspace{1em}\hspace{1em}{L}_v^{(n)}=\text{exp}\left(\mathrm{\Psi }\right({\alpha }_v^{(n)}\left)\right)\mathrm{.}*{\beta }_v^{(n)}$

(7)  Optional: Update Hyperparameters (See Appendix, Section )

(8) end for