  We present a new view of Gaussian belief propagation (GaBP) based on a
representation of the determinant as a product over orbits of a graph. We show
that the GaBP determinant estimate captures totally backtracking orbits of the
graph and consider how to correct this estimate. We show that the missing
orbits may be grouped into equivalence classes corresponding to backtrackless
orbits and the contribution of each equivalence class is easily determined from
the GaBP solution. Furthermore, we demonstrate that this multiplicative
correction factor can be interpreted as the determinant of a backtrackless
adjacency matrix of the graph with edge weights based on GaBP. Finally, an
efficient method is proposed to compute a truncated correction factor including
all backtrackless orbits up to a specified length.
