*Bayes factor being the weight of the evidence in favor of the God hypothesis over a particular alternative hypothesis.
Which isn't necessarily 33,000.
Again, for those keeping score at home, the 33,000 jt512 is using arises from Bayes' theorem:
p(cheat|royal) = p(royal|cheat) * p(cheat) / p(royal)
That is,
given that I have a royal flush, the probability that I cheated is equal to the probability that I have a royal flush if I cheat, times the probability that I cheated, divided by the probability of getting a royal flush fairly.
The odds that I get a royal flush, if I cheat, are one. The odds of getting a royal flush fairly are .003%. Given this, we can simplify to:
p(cheat|royal) = 1/.003% * p(cheat) = 33,333 * p(cheat)
But 33,333 is not necessarily the Bayes factor.
The Bayes factor is a comparative measure between two alternative hypotheses. It is denoted as
k, and is calculated as:
k = p(flush|cheating) / p(flush|other hypothesis)
Or,
"The Bayes factor for any two hypotheses about how I got this royal flush is equal to the probability of getting a flush in one hypothesis divided by the probability of getting a flush in the other."
We have already established that the probability that I get a flush if I cheat is one, so:
k = 1 / p(flush|other hypothesis)
Now, if we just say the other hypothesis is "drawing random cards", then yes, we end up with 1/.003%, and our original result.
But the thing is that you can put
literally anything else in there for "other hypothesis", and so long as it ends with saying "...so I get a royal flush",
p(flush|other hypothesis) is one. This is up to, and including, "the dealer happened to shuffle the deck in exactly this way, entirely unintentionally".
So the Bayes factor is not, in fact, 33,000. It can be literally anything, including 1/1,
even if the alternative hypothesis is that it happened entirely randomly. There is literally nothing preventing this sort of manipulation.
Which is why we go back to
what I said before. In order for any sort of Bayesian analysis to have any weight whatsoever, you must be able to provide evidence that indicates your hypothesis is
actually true.
Otherwise, you just end up with the
same old issues.