In this comment at Gelman's Blog "Ney" asks "Does the 5000:1 mean that a team like Leicester would be expected to win the English championship only once within 5000 years?"

My answer to that is no. At least if a Bayesian gives a 5000:1 odds on something like a team winning a championship or a particular earthquake occurring or a particular financial event, it need not have any frequency of occurrence in the long run interpretation. But what is the interpretation?

Bayesian probabilities under Cox/Jaynes probability just says that different things have different degrees of credibility or plausibility or believability or whatever. In this system, Probability is like Energy, if we write energy in units where all the energy in the universe is 1 unit, then since energy is conserved, we can account for what fraction of the universes energy there is in any one object. Same idea for Bayesian probability, what fraction of our credibility is associated with a particular value or small range of values?

So, we could imagine a whole series of events, say each week, there are some soccer games played, and then someone wins, and that means that different matches happen next week, and then someone wins, and etc etc etc. By the end of the season there is some enormous combinatorial explosion of different possible "paths" through the season. A 5000:1 odds for a Bayesian roughly means that of these N possible paths through the season 5000/5001 N of them have Leicester losing and 1/5001 N of them have Leicester winning. Now, it's not quite that simple, because there's no reason why each of the N possible paths have to have equal plausibility, so some paths might "count more" than others, but it's clear that we're not talking about what will happen in 5000 future seasons, we're talking about the weight of plausibility among all the different detailed ways in which the outcome LEICESTER WINS THIS SEASON could occur.

4 Responses leave one →
June 5, 2016

If the bookmaker sets odds at 5000:1, it has a very solid and clear meaning in terms of a payout in case this happens. That'd be de Finettian Bayes, not Jaynesian.
Also, the bookmaker will be very interested in frequencies of payouts, and therefore it's actually related to a prediction of the frequency distribution of bets, which is more important to the bookmaker than predicting results. I'm pretty sure that for the sake of odds setting the bookmaker would be much happier about having a good estimator of bet frequencies than about the prospect of reading Jaynes.

• June 5, 2016

As I pointed out elsewhere on that comment thread, the bookmaker is really in the business of being a go-between for a market containing thousands of people. Their odds don't reflect what they "really" think they reflect what they believe will maximize their income given what thousands of bettors "really" think. For example, suppose the bookmaker knows exactly who is going to win the championship because they've bought off all the referees etc. What odds should they offer? It certainly shouldn't be 0:1 (ie. pay them a dollar now and they will give you your dollar back at the end of the season) even though that represents their certainty, they should offer odds that cause the maximum profit from betting, which involves how much demand they can induce on both sides in such a way that the demand for the bet is imbalanced in their favor. So, bookmaker odds are not some sort of consistent Bayesian probabilities they're some kind of mixture of imbalance between multiple people's probabilities.

On the other hand an individual bettor who doesn't hold a portfolio of bets on both sides of many different outcomes should bet based on their Bayesian probabilities. So, rather than thinking about what the Bookmaker setting odds of 5000-1 means, think about what it means for one sports aficionado to accept the bet at 5000-1. There can be no sense in which accepting that bet reflects the frequency of how often that bet will pay out, that bet will happen only one time, but it can reflect the plausibility that the bettor assigns to various different ways you might get from the first game of the season all the way through the championship to the last game and wind up winning (according to the bettor not the bookmaker). Actually it reflects the idea that the bettor believes that the odds of them winning are BETTER than 5000-1 so it's only a lower bound on their probability, and it only represents the probability in the context where the bet is small, because as the bet gets big the risk aversion and utility of money issues come into play. In reality there's a sense in which the bettor is also buying entertainment. Even if they virtually know they're going to lose \$5, they can like the feeling of having some support for their team, a think to talk about with their friends, a reason to watch all the games and go drink beers at the sports bar, etc etc. None of that actually involves probability at all.

Jaynes and DeFinetti both justify the same math, it's just that DeFinetti specifically justifies it in contexts where it makes sense to discuss betting whereas Cox/Jaynes provide a more general context for the justification.

The point of the post was that there is an interpretation for 5000-1 odds that makes logical sense and has nothing to do with frequency of occurrence of winning this bet under repeated playing of this bet in hypothetical alternative worlds or hypothetical future years.

Where B_i is their winnings on the i'th bet, O_i is the odds offered on the i-th bet, N_i is the number of bets people make with them when the odds across all bets are $\vec O$ (note the odds on other bets can change the number of people willing to take bet i).
By altering $\vec O$ they can alter $N_i$ but they also alter the payout B_i(O_i). If they have their own Bayesian probabilities for the outcomes of each of the $B_i$, $p(B_i|Knowledge)$ then then they can derive probabilities over their winnings $p(W|Knowledge)$ and they can set $\vec O$ to maximize the expected payout
by adjusting $\vec O$ according to whatever their model is for supply and demand, and payout together. That's why odds offered are not their Bayesian probabilities. But, odds taken by an individual bettor does indicate something about that individual's Bayesian probabilities.