Factorial Hidden Markov

One of the basic probabilistic tools used for time series modeling is the hidden Markov model (HMM). In an HMM, information about the past of the time series is conveyed through a single discrete variable|the hidden state. We present a generalization of HMMs in which this state is factored into multiple state variables and is therefore represented in a distributed manner. Both inference and learning in this model depend critically on computing the posterior probabilities of the hidden state variables given the observations. We present an exact algorithm for inference in this model, and relate it to the Forward{Backward algorithm for HMMs and to algorithms for more general belief networks. Due to the combinatorial nature of the hidden state representation , this exact algorithm is intractable. As in other intractable systems, approximate inference can be carried out using Gibbs sampling or mean eld theory. We also present a structured approximation in which the the state variables are decoupled, based on which we derive a tractable learning algorithm. Empirical comparisons suggest that these approximations are eecient and accurate alternatives to the exact methods. Finally, we use the structured approximation to model Bach's chorales and show that it outperforms HMMs in capturing the complex temporal patterns in this dataset.