Ever since Kim and Pearl provided an exact message-passing algorithm for updating probability in singly connected Bayesian networks (BN), attempts at extending it to general BN have been made. Brute variable instantiation – or global conditioning (GC) – implies unnecessary computations which more refined local conditioning (LC) methods try to avoid. By using the concept of subnetwork of a BN (BSN), and identifying each message in the BN with messages in a set of singly connected BSN, we are able to identify the parameters actually required by each message, to give the expression of local computations, and thus fully justify a general LC method applying to any BN.