A few days ago our reading group ran through Phil Dawid'sProbability, Causality and the Empirical World: A Bayes-de Finetti-Popper-Borel Synthesis and learned that not only does probability not exist for him, like for de Finetti, nor does causality. A common characterisation of positions concerning probabilities split them into four groups:

1) Frequentist: probabilities are limiting frequencies of outcomes in sequences of events.
2) Propensity theorist: an individual event has a propensity to display a certain outcome.
3) Subjective Bayesian: a subjective degree of belief in an outcome.
4) Objective Bayesian: relative to specific background knowledge, there is an objective value an agent should accord to their degree of belief in an outcome.

Dawid (pronounced 'David') holds a Bayesian position, made evident in his involvement in theSally Clark case, in which a mother was unjustly jailed after her two children died. But for Dawid, while the position that probabilities are consistent assignments of degrees of belief is all well and good, and it avoids the problems of taking probabilities to exist out there in the world, at some point you should want tocalibrate the probabilities that an individual is spewing out. I could lock myself in a dark room keeping my degrees of belief consistent, but if I have England as 99% likely to win the World Cup, and other oddities, you will want to have a framework in which you can criticise me. This idea of calibration takes place in weather forecasting. You might score the forecaster's rain predicitions by forming the sum of (x_i - y_i)², where x_i = 0 if it is dry on the i^th day, and 1 if it rains, and y_i is the forecasted percentage chance of rain - the Brier score.

Intuitively, if a forecaster believes in their forecasts for rain they ought to be happy to accept bets made either for or against it raining at the odds they give. You'd think then that we could winkle out the bad from the good forecaster by noting which ones would go broke within a short space of time. The trouble is in telling when a 'good' forecaster's being very unlucky, and when a 'bad' forecaster's being very lucky. You are forced to appeal to an infinitely long series of predictions. And this is Dawid's position. Probabilities don't exist, they're theoretical tools that mediate between our theories and the world. The only way they hook onto the world is by what they rule out as impossible (this is the Borel part of the synthesis). For example, among other things, a forecaster who says 30% chance of rain every day is ruling out the possibility that in the long run it will rain on 40% of the days. This has the paradoxical consequence that if, say, two weather forecasters make predictions for rain over the next century and agree on every day except tomorrow when one says 10% and the other say 90%, there is no way you can say whether one was right. They've both ruled out various sets of sequences of outcomes, like those for which the average differs from the limit of the average of their probabilities. But neither rules out anything that the other doesn't.

Leaving aside the problem that runs aren't infinite, this game theoretic interpretation of probability theory is certainly very interesting. Shafer and Vovk have written a book-length treatment of the idea in theirProbability and Finance: It's Only a Game!. Game theoretic ideas are also used to understand maximum entropy distributions. They correspond to stable points in zero-sum games played between a decision maker and nature. FlemmingTopsoe has many good papers on this, and there's also one byDawid with Peter Grunwald. Different entropies match up with different loss functions, such as the Brier score above. Something to add to the pot is Dawid'sThe geometry of proper scoring rules, a longer version of apaper written with Steffen Lauritzen. Now we have game theory blending with the differential geometric approach of Amari known as Information Geometry, discussed inearlierposts. I wish I could understand all this properly.