  The Bethe approximation, discovered in statistical physics, gives an
efficient algorithm called belief propagation (BP) for approximating a
partition function. BP empirically gives an accurate approximation for many
problems, e.g., low-density parity-check codes, compressed sensing, etc.
Recently, Vontobel gives a novel characterization of the Bethe approximation
using graph cover. In this paper, a new approximation based on the Bethe
approximation is proposed. The new approximation is derived from Vontobel's
characterization using graph cover, and expressed by using the edge zeta
function, which is related with the Hessian of the Bethe free energy as shown
by Watanabe and Fukumizu. On some conditions, it is proved that the new
approximation is asymptotically better than the Bethe approximation.
