Uncertainty in KRACH probability estmates (thread drift)

[jtwcornell91]

Now, of course, this is within the context of the Bradley-Terry model, and you could say the probability relationships assumed by that model (odds ratio for A:C = odds ratio for A:B times odds ratio for B:C) are not right.  But you see similar effects in a binomial experiment, where you're making the same comparison over and over again to try to estimate a single probability.  The posterior on the Cornell-Clarkson odds ratio (using the results from when I made the original post, not the current ones) is pretty close to what you'd get if, instead of the whole season, the only results you had were nine games between the two teams, of which Cornell won 6 and Clarkson won 3.  Then, with basically only the assumption that each game is an independent comparison with the same probability that Cornell will win (which might not be literally true if the two teams really played 9 times, but is more or less the case in the real season), you get a posterior for the win probability of a very specific form (the beta distribution), which in terms of odds ratio, comes out to the red dashed curve on the plot below.  (In this simple model, we do have the exact posterior available to us.)  We see that it's pretty close to the Gaussian approximation (blue dash-dot curve), but again skews slightly away from 50:50, just as in the Bradley-Terry case.  (The Gaussian approximation to the marginal Bradley-Terry posterior is shown in light grey for comparison.)

[jtwcornell91]

For completeness, here are the same posteriors plotted in terms of win probability rather than odds ratio.

One other objection is that all of these posteriors use the Haldane prior (no fictitious games), and you might put on a prior that reflects the reasonable supposition that all log-odds-ratios are not equally likely a priori.  But that doesn't change the fundamental nature of the underlying beta distribution.  For instance, if you assumed a priori that any win probability was equally likely (the Bayes-Laplace prior for the binomial experiment) then the posterior would be a beta(7,4) distribution rather than the beta(6,3) plotted here, and it would still be skewed slightly to the right on a log-odds-ratio plot, although the peak itself would be at a lower value (7:4=1.75:1 rather than 6:3=2:1).

[ugarte]

John, stop fucking with me. The world is ending and I want to feel smart at the end.

[scoop85]

John, stop fucking with me. The world is ending and I want to feel smart at the end.

::banana::::banana::::banana::

[ugarte]

John, stop fucking with me. The world is ending and I want to feel smart at the end.

::banana::::banana::::banana::

hahaha the bananas are dancing

[scoop85]

John, stop fucking with me. The world is ending and I want to feel smart at the end.

::banana::::banana::::banana::

hahaha the bananas are dancing

Been a looooong time since I did me some dancing bananas and it seemed just right!

[jtwcornell91]

John, stop fucking with me. The world is ending and I want to feel smart at the end.

We're all just seeking enlightenment. ::innocent::

[Trotsky]

John, stop fucking with me. The world is ending and I want to feel smart at the end.

We're all just seeking enlightenment. ::innocent::

Speak for yourself.

[ugarte]

John, stop fucking with me. The world is ending and I want to feel smart at the end.

We're all just seeking enlightenment. ::innocent::

Speak for yourself.

i don't know who that is but ... uh ... do you know the provenance of that phrase

[Trotsky]

i don't know who that is but ... uh ... do you know the provenance of that phrase

Of course.

[adamw]

I'd just like to say - I should've accepted the bet on whether North Dakota, Minnesota State, Cornell would really finish 1-2-3 in the Pairwise.

It stayed that way for six weeks, and right to the bitter end. Never budged.

(and it wouldn't have budged, either)