It is possible to derive the fixed point of the bound propagation
algorithm analytically for a Boltzmann ring if we take a single value
for all weights () and one for all thresholds (). Due to this
symmetry all single node marginals are equal in such a network. Moreover
all upper and lower bounds on the single nodes should be identical.
This implies that for the fixed point the following holds for any :

and similarly for the lower bound. The conditional probability in Equation is given by

(13) |

In Figure a the difference between the upper and lower bound on the means is shown for various choices of the weight and threshold. As we saw before tight bounds can be obtained for small weights and somewhat larger, but positive weights, whereas negative weights result in rather poor, but still non-trivial bounds. It should be mentioned, however, that these results correspond to the choice of the smallest clusters ( and ) for the bound propagation algorithm. The bounds can easily be improved by enlarging the clusters as we saw in Figure .

Close to the fixed point the bound propagation algorithm converges exponentially. The distance to the asymptotic value can be written as , where is the number of iterations and is a number between zero and one indicating the convergence speed. The closer to one is, the slower the algorithm converges. In Figure b is shown for the same weights and thresholds. It is clear that larger weights induce a slower convergence. That is what we see in general: probabilities that are close to deterministic slow down the convergence.