De Finetti’s conjecture: First broken, then fixed; but nobody noticed, Part 1

Bruno de Finetti was an Italian statistician and probability theorist who contributed mightily to the Twentieth Century revival of Bayesian methods in statistics. One of his achievements was a gambling semantics for probability which explained Laplace’s observation that probability seemed to measure a person’s “degrees of belief” about uncertain prospects.

“Degrees of belief” are what, exactly? De Finetti’s answer would be as follows. Your “degree of belief” that the Boston Celtics will win the 2012 NBA Championship is the most you would pay to acquire a ticket that earns you $1 if they do win, and nothing otherwise.

Why is that price a probability? Because it has the properties:

The price is at least zero – no matter what you think of the Celtics, if somebody gave you the ticket for free, you’d take it.

The price is at most one – no matter what you think of the Celtics, you wouldn’t pay more for it than it will ever pay you.

Ticket prices for combinations of different teams are additive – Owning a ticket which pays $1 if either the Celtics or the Bulls win the Championship is the same as owning a Bulls ticket and also owning a Celtics ticket. Whether you acquire your “portfolio” one team at a time, or as a single ticket, you should pay the same price for the same portfolio either way.

These three properties of the prices for low-stakes interests in uncertain prospects are the only properties that simple probability has. That probability “prices” prospects was de Finetti’s signature insight.

However, before that, he had looked into another explanation for Laplace’s idea. Real people don’t always (or even usually?) think about uncertainty using numbers. Deciding what is the most they’d pay for any of those $1 tickets is a hard question for most people. On the other hand, many people find it fairly easy to answer “ordering” questions, for example, whether it is more likely that the Celtics will win the championship than the Lakers. And, of course, if you can’t answer that much, then putting number prices on the two tickets is hopeless.

So, for a time in the 1930’s, de Finetti explored conditions under which various constraints on the possible ordering of teams, and combinations of teams, would be “probability agreeing.” That is, that there would exist some probability distribution over the various teams so that

Whenever the person said “This combination is more likely than that combination,”

Then the probability of this combination is at least as great as the probability of that combination, according to at least one probability which sets a number upon every team.

There are some trivial properties which define what any “ordering” usually means, and for orderings of strength of belief, there should be privileged places at the top for the combination of all teams and at the bottom for no team at all. We’ll just call having those simple properties “being well-behaved.” To good behavior, de Finetti introduced something new that reminded him of an ordinal form of addition, an “as-if” addition, or quasi-additivity.

An ordering is “quasi-additive” if, whenever comparing two combinations, you can completely ignore whatever potential outcomes they have in common. You won’t change the order of two combinations by adding the same new outcome to each of the original combinations. You won’t change the order of two combinations by removing the same outcome from both.

Example: In a quasi-additive ordering, if the Bulls are more likely than the Knicks to win, then the combination of the Bulls or the Celtics is rated more likely than the Knicks or the Celtics. The common element, the Celtics, doesn’t change the order, and doesn’t change the order whether it is added to combinations (neither of which already has the Celtics), or removed from each of two combinations.

De Finetti didn’t prove that every well-behaved quasi-additive ordering was probability agreeing, nor did he prove otherwise. He was able to show that if there were an infinite store of alternative outcomes, then quasi-additivity and good behavior were enough for probability agreement.

Well, there are only 30 NBA teams. That set can be “padded out” indefinitely, say by coin tosses, to make an indefinitely big set. That was nice to know, but wasn’t a terribly attractive thing to do.

Worse, there are about 1 billion combinations of those 30 teams. If you express beliefs by numbers, then only thirty numbers are needed; the other billion numbers are just the various arithmetic sums. It might be easier to rank 30 teams than to assign 30 prices, but the individual rankings do not completely determine the ranking of the combinations, the way numbers do.

There were too many combinations of teams to compare for any practical application. Further, since the new property of quasi-additivity didn’t seem to be enough to ensure probability agreement, there wasn’t much immediate theoretical interest, either.

De Finetti turned his attention to the  gambling explanation of what Laplace was talking about. This move was very successful for him, and his reputation grew. He became a leading advocate for the subjective interpretation of probability, the Laplace interpretation, or as it most commonly called, the Bayesian interpretation.

George sandbags his old buddy Bruno

There matters might have remained had not the journal Dialectica produced a special issue in 1949 on the foundations of probability theory, with a stellar cast of contributors. One of them was Bruno de Finetti. He wrote a masterly exposition of the subjective perspective of probability theory. There were few surprises in de Finetti’s paper; it was his “stump speech.”

Another contributor was George Polya. Polya and de Finetti were personal friends. Polya had made important contributions to mathematical analysis. He also had taken an interest in the foundations of mathematical conjectures, the “guesses” that typically preceded new theorems. This led him to contemplate principles for formally describing the “plausibility” of conjectures, and sure enough, Polya’s plausibility resembled probability. One crucial feature of Polya’s theory is that plausibility is inherently ordinal in character, there were no numbers, and there should be no numbers.

Key to Polya’s thinking was what he called the “heuristic syllogism,” in its crudest form,

If A implies B, and B is observed, then A should become more credible.

There could easily be a separate article on why this principle, as stated, is unsatisfactory as a rule of inductive, or “heuristic” inference. Polya knew that it needed tightening and elaboration in order to work. The more he refined the thought, the more closely his system resembled an ordinal probabilistic scheme, in which B would function as “evidence” for A, much as an experimental outcome functions as evidence about a hypothesis for statisticians who use numbers.

Polya also came to think that his plausibility was a more realistic basis for most real-life dealings with uncertainty than numerical probability. So, in a way that is almost painful to read, Polya’s contribution to the Dialectica special issue seemed like a rebuttal to de Finetti’s paper. If there were few surprises in the latter, Polya’s thrust seems to have thoroughly surprised his old friend. Real beliefs typically resemble probability only qualitatively. Numbers, such as maximum buying prices, are often unrealistic or even undesirable.

De Finetti was energized, and in the few months between the appearance of the special issue and the end of the year, he cranked out a paper for the meeting of the Italian Society for the Progress of Science. An English translation has been added to the Unlinks section of this blog, or click here.

In his paper, de Finetti revived his pre-gambling work, determined to show that probability was not married to numbers, that an ordinal system with properties Polya admired was within probability’s grasp. De Finetti boldly conjectured that any well-behaved quasi-additive ordering, even with only a finite set of alternatives, is probability-agreeing.

In his paper, de Finetti showed that if there are only two alternatives, then this was so. He  showed that it was also true for three alternatives. With some effort, he managed to show that it was also true for four alternatives. Five alternatives stumped him. Time probably ran out as the conference loomed. He left the matter as a conjecture.

Ironically, de Finetti’s approach to the problem was more or less as Polya had prescribed as how the development of a mathematical conjecture rightly ought to proceed. In the crude from of the “heuristic syllogism,” de Finetti was writing

If well-behaved quasi-additivity works to ensure probability agreement for any number of alternatives, then it must work for two alternatives. Behold, it does work for two alternatives. Therefore, the conjecture is more credible than before.

By successively proving special cases of two, three and four alternatives, De Finetti was justified, according to Polya’s theory, in entertaining ever greater confidence in his conjecture, that it would turn out to be true for all cases, for every finite domain whatsoever.

However, one of the objections that Polya had about numerical probability models of belief was that “ever greater confidence” was indistinguishable from ever more closely approach certainty. This was unacceptable to Polya for mathematical hypotheses. For those, something is proven, which is certainty, or else it is unproven, which is uncertainty. There is no “close to” certain, nor “closer to” either.  “Greater confidence” in Polya’s mind would mean something like “more worthy of further attention and work,” despite an unchanging status as something uncertain.

Fair enough. That de Finetti’s conjecture held true for two, three and four alternatives says nothing about whether the conjecture also holds for five alternatives. The successful cases only provide justification for someone to devote resources to investigating the question further. And that is what de Finetti’s paper invited the community to do, to investigate his conjecture further.

Next time: Come on, you just know it isn’t going to work for five. So how do you fix it? Why did so few people accept the fix when it turned up, not even the person who found it?

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