ABSTRACT
Convenient expressions for conditional distributions and forecast distributions are available for HMMs. This makes it easy, for example, to check for outliers or to make interval forecasts. In this chapter, we first show (in Section 5.1) how to compute the conditional distribution of an observation under an HMM, i.e. the distribution of the observation at time t given the observations at all other times. In Section 5.2 we derive the forecast distribution of an HMM. Then, in Section 5.3, we demonstrate how, given the HMM and the observations, one can deduce information about the states occupied by the underlying Markov chain. Such inference is known as decoding. We continue to use the earthquakes series as our illustrative example. Our results are stated for the case of discrete observations Xt; if the observations are continuous, probabilities will need to be replaced by densities.
