We present  sensitivity-based pruning algorithms for general Boltzmann
networks.  Central to our methods is the efficient calculation of a
second-order approximation to the true weight saliencies in a cross-entropy
error.  Building upon recent work which shows a formal correspondence between
linear {\em Boltzmann chains} and Hidden Markov Models (HMMs), we
argue that our method can be applied to HMMs as well.  We illustrate
pruning on {\em Boltzmann zippers}, which are equivalent to two HMMs with
cross-connection links.  We verify that our second-order approximation
preserves the rank ordering of weight saliencies and thus the proper
weight is pruned at each pruning step.  In all our experiments in small
problems, pruning reduces the generalization error; in most cases the pruned networks 
facilitate interpretation as well.