If we want to multiply a weight to a product of unequal matrices, may someone explain why we have to write the notation in all three dimensions? For example, I read the example in Jensen's inequality weight ($\pi_{ijk}$) of NMF. Despite it sums over $k$, the $\pi_{ijk}$ has all $ijk$ notation. Is this important?
Example
To apply Jensen's inequality, we introduce $\sum_k \pi_{ijk} = 1$ (on product $W_{ik}H_{kj}$)
\begin{equation*} d_{\mathrm{KL}} (\mathbf{V}\ \vert\vert \mathbf{WH}) = \sum_{ij}-V_{ij}\log\sum_{k}\pi_{ijk}\frac{W_{ik}H_{kj}}{\pi_{ijk}} + \sum_{ij}\sum_{k}{W_{ik}H_{kj}} \end{equation*}