5000-1 odds on Leicester, what does it mean?
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.