As I recall Bayes' theorem, it doesn't quite align with how you're presenting it. Here's what I remember: P(A) is the prior probability, P(A∣DX) is the posterior probability, and P(DX) is the llkelihood. But the terminology in Bayes' theorem has always puzzled me—there’s nothing inherently 'prior' about the prior nor is it necessary for posterior to come after. As you mentioned, prior(and posterior) is not tied to time or sequence in the way the term might suggest. Anyway, that's how I recall it. And what’s this about deep kimchi? Kimchi is a staple in Korea.
Good questions. First, recall the most crucial lesson of this entire class: there are no such things as "P(A)" and "P(DX)" and the like. There are only probabilities dependent on information, like Pr(A|X) and Pr(D|X).
I didn't actually give the proof of Bayes here, but it could be done in the usual way. Pace:
Pr(A|DX) = Pr(D|A)Pr(A|X) / Pr(D|X).
And by breaking out Pr(D|X) it's usual total probability.
I wanted to emphasize that point, which is key, that all that matters is Pr(A|DX), the probability of the proposition of interest with respect to ALL the information we assume.
We do not strictly need Bayes, though it's nice to have. We just want Pr(A|DX), and however we get it is good enough.
Okay, thanks! Bayes’ theorem is simple enough to prove, but that homework problem? That’s a different story—gets a bit messy with all the combinations and whatnot. I’ll work on it and send you what I come up with. I got a kick out of you pointing out that the math turns into kimchi when applied to the real world. So true! The theory is beautiful, but applying it? Practically impossible. Just think about random variables—so many theorems start with that assumption, but in the real world? Random variables don’t really exist. How could they? Everything's subject to conditions. Even rolling a die isn’t truly random. And clinical trials? Perfect example. You’re dealing with supposedly 'random' volunteers, but wait—did I say random? Not so random if they volunteered!
As I recall Bayes' theorem, it doesn't quite align with how you're presenting it. Here's what I remember: P(A) is the prior probability, P(A∣DX) is the posterior probability, and P(DX) is the llkelihood. But the terminology in Bayes' theorem has always puzzled me—there’s nothing inherently 'prior' about the prior nor is it necessary for posterior to come after. As you mentioned, prior(and posterior) is not tied to time or sequence in the way the term might suggest. Anyway, that's how I recall it. And what’s this about deep kimchi? Kimchi is a staple in Korea.
Paul,
Good questions. First, recall the most crucial lesson of this entire class: there are no such things as "P(A)" and "P(DX)" and the like. There are only probabilities dependent on information, like Pr(A|X) and Pr(D|X).
I didn't actually give the proof of Bayes here, but it could be done in the usual way. Pace:
Pr(A|DX) = Pr(D|A)Pr(A|X) / Pr(D|X).
And by breaking out Pr(D|X) it's usual total probability.
I wanted to emphasize that point, which is key, that all that matters is Pr(A|DX), the probability of the proposition of interest with respect to ALL the information we assume.
We do not strictly need Bayes, though it's nice to have. We just want Pr(A|DX), and however we get it is good enough.
Okay, thanks! Bayes’ theorem is simple enough to prove, but that homework problem? That’s a different story—gets a bit messy with all the combinations and whatnot. I’ll work on it and send you what I come up with. I got a kick out of you pointing out that the math turns into kimchi when applied to the real world. So true! The theory is beautiful, but applying it? Practically impossible. Just think about random variables—so many theorems start with that assumption, but in the real world? Random variables don’t really exist. How could they? Everything's subject to conditions. Even rolling a die isn’t truly random. And clinical trials? Perfect example. You’re dealing with supposedly 'random' volunteers, but wait—did I say random? Not so random if they volunteered!