  We introduce a probabilistic model with implicit norm regularization for
learning nonnegative matrix factorization (NMF) that is commonly used for
predicting missing values and finding hidden patterns in the data, in which the
matrix factors are latent variables associated with each data dimension. The
nonnegativity constraint for the latent factors is handled by choosing priors
with support on the nonnegative subspace, e.g., exponential density or
distribution based on exponential function. Bayesian inference procedure based
on Gibbs sampling is employed. We evaluate the model on several real-world
datasets including Genomics of Drug Sensitivity in Cancer (GDSC $IC_{50}$) and
Gene body methylation with different sizes and dimensions, and show that the
proposed Bayesian NMF GL$_2^2$ and GL$_\infty$ models lead to robust
predictions for different data values and avoid overfitting compared with
competitive Bayesian NMF approaches.
