  Graphical models represent multivariate and generally not normalized
probability distributions. Computing the normalization factor, called the
partition function, is the main inference challenge relevant to multiple
statistical and optimization applications. The problem is of an exponential
complexity with respect to the number of variables. In this manuscript, aimed
at approximating the PF, we consider Multi-Graph Models where binary variables
and multivariable factors are associated with edges and nodes, respectively, of
an undirected multi-graph. We suggest a new methodology for analysis and
computations that combines the Gauge Function technique with the technique from
the field of real stable polynomials. We show that the Gauge Function has a
natural polynomial representation in terms of gauges/variables associated with
edges of the multi-graph. Moreover, it can be used to recover the Partition
Function through a sequence of transformations allowing appealing algebraic and
graphical interpretations. Algebraically, one step in the sequence consists in
application of a differential operator over gauges associated with an edge.
Graphically, the sequence is interpreted as a repetitive elimination of edges
resulting in a sequence of models on decreasing in size graphs with the same
Partition Function. Even though complexity of computing factors in the sequence
models grow exponentially with the number of eliminated edges, polynomials
associated with the new factors remain bi-stable if the original factors have
this property. Moreover, we show that Belief Propagation estimations in the
sequence do not decrease, each low-bounding the Partition Function.
