# Nonnegative Matrix Factorization as Maximum Likelihood

Elements of Statistical Learning has this on such NMF loss function (section 14.6 Non-negative Matrix Factorization):

The matrices $\mathbf{W}$ and $\mathbf{H}$ are found by maximizing $$L(\mathbf{W}, \mathbf{H}) = \sum_{i=1}^N\sum_{j=1}^p[x_{ij} \log(\mathbf{WH})_{ij} − (\mathbf{WH})_{ij} ]. \; \; \; \; \; \; \; \; \; (14.73)$$This is the log-likelihood from a model in which $x_{ij}$ has a Poisson distribution with mean $(\mathbf{WH})_{ij}$— quite reasonable for positive data.

What if $x_{ij}$ is not a natural number? Is there a class of distributions that generalizes Poisson distribution that has such log-likelihood (up to a term that doesn't depend on distribution mean)?