This chapter shows how to compute the conditional distribution of the observation at time t given the observations at all other times. It explores the forecast distribution of a hidden Markov models (HMM). The chapter demonstrates that how, given the HMM and the observations, one deduce information about the states occupied by the underlying Markov chain. Convenient expressions for conditional distributions and forecast distributions are available for HMMs. Global decoding is the main objective in many applications, especially when there are substantive interpretations for the states. It is therefore of interest to investigate the performance of global decoding in identifying the correct states. In contrast, primarily in the engineering literature, HMMs are often applied solely for the purpose of classifying observations on the state-dependent process into 'real' states. In such applications one usually begins by taking a training sample in which both the states and the state-dependent observations are recorded.