The answer to that question must be 1.00 -- but that isn't what I'm asking.
Actually it is. You're trying hard to smudge the words and concepts to make it seem otherwise. You may not even realize it, because reasoning like a Texas sharpshooter seems to come so effortlessly to your fingertips. But in fact you're asking the likelihood of A given A.
There is a difference between our questions -- and the answer to my question is 1/140,000.
No there isn't. And your answer is based on a naive, first-year statistics simplification that we've already discussed. Want to prove me wrong? Describe "probability density function" in your own words and give an example of one you didn't just Google for. I'll give you one I didn't Google for. First, there aren't 140,000 centuries in all of elapsed time since the Big Bang. It's 14
billion years, not million. There are 140 million centuries in all of time. That's your mulligan for this round; even experienced people make order-of-magnitude errors.
As we discussed before, not all centuries are created equal. Especially when it comes to selecting one for some particular purpose. It's cosmologically obvious that the first century after the Big Bang -- when everything was still a hot mess -- is very much different in so many ways than the century 1942-2042. Those material differences are going to affect selection criteria for various problems we might contemplate. Different criteria give rise to different probability densities.
Oh, you thought 1942-2042 was just some arbitrary century among the 140 million you could have chosen for your proof. Or even among the 100 centuries that comprise all of recorded human history, or among the 2,000 centuries that comprise the sojourn of
H. sapiens in this hurly-burly universe. The last two are kind of important because -- and you may have forgotten this -- but we're talking about human immortality. It's hard to have that without humans. So reaching back 14 billion years is a little optimistic. You do seem to have a thing for oversized denominators.
A couple people have asked you why that 1942-2042 interval is so important. Because you don't know about probability distributions, you thought that the probability of that century for any given purpose would just be 1 divided by how many of those centuries were in the raffle barrel. As usual, your prior is way off. You're asking about the probability of a timepoint falling within a "random" increment in all of time. Oh it's random, to be sure, but not evenly distributed. Not very even at all.
You discretized the sample space for the conditioned event. That's okay, but...strange since you haven't discretized (or even defined) "now". You discretized it to a one-century interval. Why that, instead of fifty years? We're talking about cosmology, after all. Why not a ten-thousand-year eon? The starting point is a little on-the-nose, wouldn't you agree? I mean, it's a century but it's not on a Gregorian century boundary. Why? Hm, it's almost like this "arbitrary" century didn't arise at random, but is a specific human life span. That is, the life span of a
specific human. Which human? My money's on Jabba, whose birth year is known to many of us.
A probability density can occur for any reason pertinent to the problem. And for this particular problem, there's a gigantic spike in the density graph starting right about the time you were born and ending right about the time a normal human would pass away. That's why fifty years wouldn't have been a good increment: you've already lived longer than that. Bayes lets me take all the evidence I just outlined and express that evidence in terms of a probability distribution that isn't strictly frequentist. For all the finger-wagging you do about Bayes, you really don't know much about his work. Your conditioned event is not "some random century." Your conditioned event is "Jabba's lifetime." Because it was chosen according to those criteria and not an evenly distributed random variable, P(A) is most certainly
not 1 in 140 million.
That about does it for event A. Now let's look at event B.
Jesus said, "Now is the son of man betrayed into the hands of sinners." Taking that arguendo as history, that was still a long time ago. It's not happening today. Similarly ol' scoliotic Ricky -- third of his name -- rejoiced that "Now is the winter of our discontent made glorious summer..." Well, that was a thing for him. Dunno about you, but even though today is my birthday my winter out here in the mountain west is still pretty bleak and discontented. February is like second-string winter -- not as magical and bright as first-string winter, but it's sure trying. And moving closer to --- hehe, now -- we turn to Honest Abe. His day is tomorrow, so let me have my hurrah while it's still my day. He told his audience that "Now we are engaged in a great civil war..." which was actually over more than a century ago. Just because the word "now" persists in all those texts doesn't mean the concept of "now" persists in the way it was intended in each of those times.
Now always means the present instance from the perspective of who said it. When we read it in writings from long enough ago, we have to reset our mental frame of reference. In a way, the word "now" has two meanings. And your arguments seem to include a lot of words that have two meanings, but among which you never pick just one. I wonder why that is.
Now is subjective. Now, when spoken by a person, always means the time period of nebulous duration that surrounds when he said it. For Jesus' betrayal, the whole thing took less than 24 hours from the time he said the words. For Richard, it was the span of a spring. It's not like he's going to keep referring to that now as "now" long into the summer. And the "now" of the American civil war could be reckoned in years and still be valid. But the point remains that it's always the time when the speaker speaks, not some arbitrary time point in the 14 billion years of available time. So when Jesus speaks of "now" it refers to events happening in his lifetime. Same with Richard and Abe. Now always
has to fall in their lifetime.
Hm, that sounds familiar. Event A, we determined, was Jabba's lifetime. Event B, "now,"
must be in Jabba's lifetime because of the definition of now. Even B can't be outside Jabba's lifetime as long as he's the one writing the proof. It's like the easy example of the probability of being dealt a jack given that the card dealt was a face card. "The card is a jack" and "The card is a face card" are not independent events. Hence when we look at the likelihood ratio in Bayes, we see that its numerator in this case is 1, because the probability of a face card given a jack is 1.
Oh, and let's not forget that the denominator of the likelihood ratio works out to the probability of now being now. That is, the probability of the time point in question being the time point at which the phrase is uttered. It was true when Jesus said it. It was true when Richard III said it. It was true when Abe said it. And it's true every time you have said it over the past two days or so. But they are all different time points. Funny how equivocation and tautology amount to such things.
But now you can see that we have a degenerate likelihood ratio -- very close to, if not equal to, 1/1, or 1. That means the posterior really can't vary much from the prior, which is 1. The prior is the probability that now is now. The likelihood ratio is a number close to 1 (because both A and B stem from the same person's lifetime and are fundamentally the same event) over right around 1, the probability that you would have chosen your lifetime as the century in this example. I'm hurrying to finish this in time to go have birthday bourbon with the boys, so if there are mistakes I'll correct them tomorrow.
Besides that, what kind of question would ask for the likelihood of A -- given A?
Bayes' theorem describes the probabilistic relationship between two events, A and B. There is no requirement that A and B be independent events. In fact a lot of the practical uses of Bayesian inference (or likelihoods in general) involve events that are partially if not substantially dependent. You seem able to think of those parameters only as discrete either-or circumstances. That's your limitation, so please stop projecting it onto everyone else.
But to answer you directly: what kind of question would ask the likelihood of an event given the event? The kind of question that's trying very hard -- but failing -- to disguise a wanton commission of the Texas sharpshooter fallacy.
