@TB_Helmet @primetime ,
I do not claim to have the same expertise in Bayesian analysis as a social scientist, but I don't agree that the reference to Bayes in the original post would cause Bayes's or anyone else's head to explode. However, I will be more explicit here:
From a subjectivist point of view, we all have a prior belief that a 40 year old quarterback would benefit from extra rest compared to the same quarterback when he was younger. This loosely speaking is the prior.
We also have a body of specific evidence pertaining to whether Brady himself would benefit from increased rest.
A. Some of this evidence suggests that Brady would benefit from increased rest (by which I specifically will mean skipping games). This evidence includes the analysis in Fahey's report, which argued that Brady, late in the season, showed signs of weakening arm strength. Other evidence includes the increased interception rate in the last few games and weaker play in the most recent Dolphins game and possibly the Bills game.
B. Other evidence suggests that Brady would
not benefit from increased rest. This evidence includes Brady's performance in Q4 SBLI (although of course Brady had two byes not long prior so even that could be argued to be partly in the
A camp), in his body of work late in the season, and in arguments that Brady's apparent decline in ability in the last few games either (a) was illusory or (b) was due to injury. The (a) argument would be that for example the interception in the Bills game was not actually Brady's fault; that he actually played very well in the Miami game and was handicapped by injuries to other Patriots or by poor game planning. The (b) argument, even if true, seems somewhat equivocal from the standpoint of arguing against increased rest, since increased rest might help prevent and recover from injuries anyway, but it's been raised. It is also possible to argue, as primetime did above, that Brady's long record both this season and in prior seasons supports
B.
Let us divide these two kinds of Brady-specific evidence relevant to the question of whether he would benefit from increased rest into the pro evidence,
A, and the con evidence,
B.
We let the posterior
P(R|A,B) to be the chance the Brady benefits from increased rest. Now, in a subjectivist Bayesian formulation, we all have a prior belief
C that a randomly chosen 40 year old QB benefits from increased rest.
C is based on our personal experiences, on medical experiences, on the history of older athletes both in the NFL (and perhaps in other sports). It is something that different people certainly quantify differently but nearly anyone would agree that
C is close to 1.
The subjectivist Bayesian approach is therefore to evaluate
A and
B with reference to
C:
P(R|A,B) is implicitly
P(R|A,B,C).
An objectivist Bayesian approach would view
C as a base rate among all possible 40 year old quarterbacks that they benefit from increased rest. When the base rate is very high, we need very strong evidence
B to overcome the supposition that most likely a particular QB is part of the set of QBs measured by
C. The distinction between the subjectivist and objectivist approaches is always a bit delicate and not that important to this discussion. What is important to the discussion is that the amount of evidence
C is very high is enormously high compared to
B (or to
A for that matter). Therefore, any argument that
B outweighs
A must account for
C.
To be fair it is also possible that the base rate
C is d-separated by other Brady-specific evidence, which is to say, that using the base rate
C is inherently misleading because the typical 40-year old QB would also demonstrate other signs of declining play that Brady has not demonstrated. We might say that it is more accurate to limit
C to the rate amongst quarterbacks who continue to play at a very high level. Since that hasn't been done, to my knowledge, it is difficult to quantify.
This discussion here isn't coming to any particular conclusion. It's just a way of framing the main issues, and in particular, whatever the mathematical formalism suggested, in noting that our prior belief in decreased stamina amongst older athletes compared to younger athletes should be accounted for in assessing the likely impact of playing or not playing Brady every game. It is this explicit accounting for a prior which I called Bayesian, contrasting with frequentist approaches which are typically prior-free. (In fact, it's a very difficult philosophical question, one which to the best of my knowledge has not been solved, how to rigorously apply any statistical or probabilistic tools to unique, one-off, individualistic questions of this kind. People try anyway but I've never been all that persuaded of the well-foundedness of the approaches. The issue is that any frequentist interpretation of probability assumes replicability that one-off events do not have. And the subjectivist interpretations are very difficult to apply in practice because of the vast and ill-defined nature of the evidence - here, nearly all sports and medical data would technically be relevant. But that's a side issue.)
