We propose a method to bound the syndrome entropy of linear block codes from their factor graph representation. It is specially suited for sparse graphs such as those of low density parity check codes. It is based on the chain rule decomposition of the entropy and the confinement of dependencies within code subgraphs. After forcing or assuming the subgraphs to have a tree structure, the computation is done by means of density evolution as for a belief propagation analysis. We employ this method to compute upper bounds of the LDPC code syndrome entropy, which allows us to obtain asymptotic MAP upper bounds that match the ones obtained by the generalized area theorem.