  We propose a new approach to the theoretical analysis of Loopy Belief
Propagation (LBP) and the Bethe free energy (BFE) by establishing a formula to
connect LBP and BFE with a graph zeta function. The proposed approach is
applicable to a wide class of models including multinomial and Gaussian types.
The connection derives a number of new theoretical results on LBP and BFE. This
paper focuses two of such topics. One is the analysis of the region where the
Hessian of the Bethe free energy is positive definite, which derives the
non-convexity of BFE for graphs with multiple cycles, and a condition of
convexity on a restricted set. This analysis also gives a new condition for the
uniqueness of the LBP fixed point. The other result is to clarify the relation
between the local stability of a fixed point of LBP and local minima of the
BFE, which implies, for example, that a locally stable fixed point of the
Gaussian LBP is a local minimum of the Gaussian Bethe free energy.
