Did you read the very article you linked to just a couple of posts above?
The problem is that it's probably
all he's read on the subject, at least lately. There's no use his pretending he has any sort of real expertise in this sort of reasoning, but he's already cast himself in the role of teacher, so now he's stuck to some extent having to display knowledge he doesn't have. Now it's obvious he's frantically Googling around for tutorial web sites or anything he can regurgitate here to make it seem like he knows what he's talking about. But he can't actually have a discussion; all he can do is make vague references to those sources, which he doesn't really understand, and hope that having made the reference seems enough like an answer. It's the same as the cargo cults hoping their bamboo "control towers" are accurate enough to attract the sky gods.
You really should. Therein you might even find this:
He quoted that line above. That's his whole point here. Jay says Bayes involves two events, but Jabba's web site
seems to say something different. Therefore Jay must be wrong on this point, and Jabba thinks he has his little "gotcha!" Jay says "event" but in Jabba's mind an event can only be data. His author says "hypothesis," which is a different thing than data.
See, the problem is that Bayes' theorem is used in ways other than Jabba's proposal, so more generalized presentations of the concepts don't use the "data/hypothesis" terminology. I can go to my shelf and find any number of textbooks that present Bayes' theorem as I did: a relationship between two events. When we use it the narrow way Jabba is proposing all usage should look like, event B is the observed data. Event A is a model describing how all such events as B might arise. Event A is a hypothesis. It's an event in the sense of the hypothesis being true or false. Those are unrealized outcomes while we reason, but surely an event as Bayesians conceive the term. It's a likelihood precisely because it has not been realized as an outcome, but remains -- prosaically enough -- a hypothetical outcome.
When we apply Bayes to practical problems -- which is what I and my employees do every day at work -- the probability P(B) is the probability that specific datum B would be collected out of all the possible data. That is, the probability that out of all the possible values we could measure, B is the answer that came up on the gauge. And normally that's not a number that bears a lot of scrutiny. It's really only there to normalize P(B|A). What's amusing is that in Jabba's models, this is the number he too often fudges. He's "normalizing" the likelihood out of existence. He further tends to conflate P(B|A) and P(B), but that's a tangent for today's post.
In his present distraction, P(B) is the probability that, out of all the centuries available in all of time, the period 1942-2042 would be the one "measured" or selected. If it were chosen at random, the value would indeed be 1 in 14 million. In experimental science, P(B) can be thought of as representing the null hypothesis, and the ratio P(B|A)/P(B) is the normalized likelihood that hypothetical event A accounts better or worse for event B than the null. But B was not chosen at random. That is, B did not arise as the one-time outcome of some random variable, uniformly distributed or otherwise. Jabba explicitly tells us he chose it
according to criteria -- specifically according to the criterion that it include now. Therefore P(B) is 1.
If you take pains to make sure B is the measured data, it's not random and the algebra of random selection does not apply. The null hypothesis, in science, means that the outcome is effectively a random variable with respect to any hypothesis, independent of the contemplated model. In a successful proof, A
predicts that B will be the data measured. But the measurement method itself is blind to A. Jabba screws up here in his proof, because he proposes different measurements for P(B) depending upon whether the hypothetical A is materialism or immortality.
If X, for example, is a model of heating based on physical science and Y is another model of heating based on faeries farting, then B might be some measured outcome such as temperature for some change in another variable Q, which could be mean molecular motion. The null says that the temperature is unchanged as a function of Q. Another independent variable might be R, introduction of faerie-fart gases. The null for testing that hypothesis would be that temperature B is unchanged as a factor of R. The random variable associated with either case would establish a high density around the prevailing temperature prior to applying the independent variable. We measure B as temperature the same way, regardless of what X or Y might propose as the mechanism of heating. What Jabba does is insist that since he believes Y works by means of faeries, P(X|B) must measure not only just the fact of temperature, but must also look for dissolved faerie-fart gases. None found, so in his mind X is a poor explanation. P(X|B) predicts that the temperature will vary more according to changes in Q than according to the PDF P(B). The ratio P(B|A)/P(B) expresses that. P(Y|C) does the same for changes in R, dissolved faerie farts. If the test for X is successful, the likelihood ratio evaluates to greater than 1, amplifying the prior P(X). If while varying R, the temperature does not change, P(Y|B) is driven low, because it predicts the temperature will rise. P(B)'s random variable then dominates and the likelihood ratio attenuates P(Y).
Returning to Jabba's century question, if the denominator P(B) is 1, as it is here, then the likelihood P(B|A)/P(B) is determined entirely by P(B|A). A is "present time is included in some given century." Jabba wants to pretend that the determination of now P(A) and the determination of the candidate century P(B) are independent random events. They aren't. P(B) is not a random event. Nor is it an independent event. Thus the likelihood P(B|A), "the likelihood that Jabba's lifetime includes a moment of time that Jabba can call 'now'" is 1. Every time Jabba says "now" it's within his lifetime, and cannot be otherwise. The likelihood ratio degenerates to 1/1. Thus the posterior is the prior. The posterior P(A|B), "the likelihood that a time point Jabba calls 'now' will occur in Jabba's lifetime" is both algebraically and intuitively 1.
Normally I would have to invoke Bayesian black magic to justify that P(B) is not a random variable but is instead a probability density function clustered around the century of Jabba's life. That's what I had to do yesterday, before dashing off to drink Pappy Van Winkle's at my local speakeasy. Today Jabba has helpfully admitted that P(B) is not random in any way, but was in fact selected according to the same criteria that gives rise to "now" in his model.
Gee, it's almost like he drew the target where the bullet landed.
