  We present a new probabilistic model to address semi-nonnegative matrix
factorization (SNMF), called Skellam-SNMF. It is a hierarchical generative
model consisting of prior components, Skellam-distributed hidden variables and
observed data. Two inference algorithms are derived: Expectation-Maximization
(EM) algorithm for maximum \emph{a posteriori} estimation and Variational Bayes
EM (VBEM) for full Bayesian inference, including the estimation of parameters
prior distribution. From this Skellam-based model, we also introduce a new
divergence $\mathcal{D}$ between a real-valued target data $x$ and two
nonnegative parameters $\lambda_{0}$ and $\lambda_{1}$ such that
$\mathcal{D}\left(x\mid\lambda_{0},\lambda_{1}\right)=0\Leftrightarrow
x=\lambda_{0}-\lambda_{1}$, which is a generalization of the Kullback-Leibler
(KL) divergence. Finally, we conduct experimental studies on those new
algorithms in order to understand their behavior and prove that they can
outperform the classic SNMF approach on real data in a task of automatic
clustering.
