Compatible Measures and Merging

Abstract

Two measures, μ and μ̃, are updated as more information arrives. If with μ-probability 1, the predictions of future events according to both measures become close, as time passes, we say that μ̃ merges to μ. Blackwell and Dubins (Blackwell, D., L. Dubins. 1962. Merging of opinions with increasing information. Ann. Math. Statist. 38 882–886.) showed that if μ is absolutely continuous with respect to μ̃ then μ̃ merges to μ. Restricting the definition to prediction of near future events and to a full sequence of times yields the new notion of almost weak merging (AWM), presented here. We introduce a necessary and sufficient condition and show many cases with no absolute continuity that exhibit AWM. We show, for instance, that the fact that μ̃ is diffused around μ implies AWM.