Merging of Forecasts in Markov Models

Blackwell and Dubins (1962) and Kalai and Lehrer (1994) showed that absolute continuity is necessary and sufficient for merging of opinions. This paper suggests the concept of merging of forecasts which is a modification of merging of opinions in Markov models where the underlying state of nature may change over time. We define the merging of forecasts as the conditional probabilities of the future state given the past observations of signals drawn conditional on the state get close to each other for different agents; it allows for the event that agents agree on the future evolution of the states even if they have not agreed in the distant past. For an ergodic Markov chain, any forecasts merge. In particular, we can dispense with the absolute continuity for merging of forecasts.