The Well-Calibrated Bayesian

Abstract

Suppose that a forecaster sequentially assigns probabilities to events. He is well cdihrated if, for example, of those events to which he assigns a probability 30 percent, the long-run proportion that actually occurs turns out to be 30 percent. We prove a theorem to the effect that a coherent Bayesian expects to be well calibrated, and consider its destructive implications for the theory of coherence.

Belief Affirming in Learning Processes

“…We say that the learning process is belief affirming (with respect to average payoff) if lim t Ä (E(t)&A(t))=0, for each player. 2 The results of this paper indicate that belief affirming in fictitious play and even more general learning processes is the rule rather than the exception. We consider first, in Section 2, the classical fictitious play in which time is discrete.…”

“…That is, (H, H ) is played in [0, 1), (H, T) in [1,2) and so on. Both players use the same averaging function f, according to which they average history uniformly, starting with the beginning of the previous round.…”

“…Let N= [1,2] be the set of players. By Theorem A, E 1 (t) A 1 (t) and E 2 (t) A 2 (t)=&A 1 (t) for each t. By Robinson's Theorem, lim t Ä E 1 (t)=v and lim t Ä E 2 (t)=&v. Thus lim sup t Ä A 1 (t) v and lim sup t Ä A 2 (t) &v. As lim inf t Ä A 1 (t)=&lim sup t Ä A 2 (t), it follows that lim t Ä A 1 (t)=v.…”

“…The oldest consistency condition is the convergence of the process, namely the convergence of players' beliefs concerning their rivals' strategies. Foster and Vohra [4] suggested calibration of the forecasts, in a learning process, as a consistency condition (see Dawid [2]) and showed that in calibrated processes the joint distribution of players' actions converge to the set of correlated equilibria. Another consistency condition was suggested by Milgrom and Roberts [9].…”

Conditional Universal Consistency

“…Kalai, Lehrer and Smorodinsky [1996] use Dawid's [1982] stronger notion of calibration with respect to checking rules, which can be viewed as a generalization of our conditioning scheme. They show that calibration with respect to all checking rules is equivalent to the merging of opinions, and since the merging of opinions with respect to all models is impossible, so is calibration with respect to all checking rules.…”