New England won the SuperBowl XVI coin toss snapping a 14-game streak by NFC teams, which lead ~2 to 1 in tosses.
The graphic below shows the coin toss process realization (red) to SB XVI, superimposed on 100 pseudorandom i.i.d and unbiased binomial random walks (white) which displays the concentration of measure relative to the cone of possible outcomes (boundary: green dashed lines).
From a physics standpoint, there is no coin bias or NFC "kabbala" or telekinesis: each year there are distinct coins, different people toss the coins, and most important, it's not heads vs. tails that determines the outcome but how the tosses are called. In other words the toss process is fair and any determination to the contrary is "Gambler's Fallacy".
Nevertheless, we can assume unknown bias in the form of a uniform (ie, uniformative) prior distribution of bias in [0,1]. The sequence of Bayesian posteriors is computed via the binomial distribution because of the Cox consistency axiom as described in Sivia & Skilling Data Analysis.
Normalizing to probability distributions at each step, the sequence of posterior distributions is:
Where each curve is one of the posterior distributions and ones based on more data (ie, up to more recent SB's) are rendered in darker gray. The straight line corresponds to the posterior after SB I (ie, with one data point available).
For example the Maximum a posteriori estimator (MAP) after 46 tosses is ~.65 in favor of NFC. Ok that's driven by the data and the a-priori physical information "fades". This introduces a bias in this case.
My question however concerns the narrowing of the posterior distribution over time. This limitation of Bayesian analysis has been noted in Guyonnet and Ferson's "Bayesian methods in risk assessment".
In general, since the posterior distribution is obtained by multiplying the prior by the likelihood function, whenever these overlap only a little, the posterior will be very narrow, see G&F, p.12.
Is there a general methodology that can mitigate this? I don't have a particular solution in mind, but some way to complement the tendency to narrow down, some enveloping method that gives a conservative estimate to counter the bias? Or would any amendement to how the posterior is computed introduce inconsistencies with respect to the Laplace/Cox/Polya axioms?
If we can't address the limitations of the methods in the simplest possible stochastic process - coin tossing - how can we apply Bayesian methods to complex systems like climate forecasting, biomedical risk assessment etc where these tools are being currently used?
Since the posterior is derived from Bayes theorem, which in turn is based on the product rule: P(A|B)P(B)=P(AB)=P(B|A)P(B). I wonder if Bayes theorem can be balanced by looking at a (suitably normalized) sum of distributions? That may lead to an envelope or conservative posterior estimate?