Blackwell ea’s Merging of Opinions

David Blackwell and Lester Dubins Merging of Opinions with Increasing Information The Annals of Mathematical Statistics, Vol. 33, No. 3 (Sep., 1962), pp. 882-886.

This is often quoted as the source for the view that, with enough data, subjective probabilities tend to merge. The treatment is entirely mathematical.

Interpretation

Usually, there is essentially only one conditional distribution Qn of the future given the past. Therefore, our theorem may be interpreted to imply that if the opinions of two individuals, as summarized by P and Q, agree only in that P(D) > 0 ↔  > 0, then they are certain that after a sufficiently large finite number of observations x1, … , xn, their opinions will become and remain close to each other, where close means that for every event E the probability that one man assigns to E differs by at most e from the probability that the other man assigns to it, where e does not depend on E. Leonard J. Savage observed that our theorem applies to the particularly interesting case in which P and Q are symmetric (or exchangeable). …

Caution

Though the conditional distributions of the future Pn and Qn merge as n becomes large, this need not happen to the unconditional distributions of the future. …

Comments

It is a key assumption that one has probability distributions in the mathematical sense. Thus it might be better to say that if the two individuals believe that a process is stably probabilistic and agree on which events have positive probability, then they should believe that their views, conditioned on a commonly observed past, will converge. Similarly, only if we share their view that they are observing a stable, random, probabilistic, process should we too expect their views to converge.

It should also be noted that the theory relates to probability distributions, not hypotheses. If two individuals associate different likelihoods with hypotheses, their views may diverge with greater evidence. So it is not always the case that all reasonable people will agree, given enough shared evidence.

Example

Suppose that A, B & C are playing roulette, and all doing quite well.

  • A, only, is aware that there is a correlation between where the ball ends up and where it was released relative to the wheel, which gives a slightly positive expected return.
  • B is playing normally, but has been lucky.
  • C has noticed that A and B have been doing well, speculates that it is the location of the number on the wheel that matters, and bets on numbers that are between those of A and B.

A misreading of the above theory would suggest that if they play for long enough the opinions of A, B & C about the game should converge. But it seems more reasonable that:

  • A will do much better than B and at least slightly better than as C, and will carry on.
  • B  will do worse. They may carry on or may swap to a C-like strategy, betting ‘near’ A and C.
  • C may carry on, or may notice that A is now doing worse and may just go close to A.

It does not seem certain that B or C will notice the correlation between the ball release and its resting place, and nor is it certain that B will realise that it is the position on the wheel that matters.

It is not enough that the players ‘observe’ the same events: they need to appreciate what may be significant and perform enough correlations to identify even slight dependencies. 

Dave Marsay

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