Suppose we have a large number of symbol sequences emitted from an HMM that has a particular transition probability ai!j! = 0 for some single value of i# and j#. We use such sequences to train a new HMM, one that happens also to start with its ai!j! = 0. Prove that this parameter will remain 0 throughout training by the Forward-backward algorithm. In other words, if the topology of the trained model (pattern of non-zero connections) matches that of the generating HMM, it will remain so after training.