Proof of Immortality, the IV league

[Jedi]

OK, calculate the probability that a toaster helped them out of the desert.

same as the one your worked out for the God hypothesis:

The weight of the evidence in favor of H1 over H2, where H1 reflects our uncertainty about a toaster's helpfulness, is 850, about a third of that for our omnipotent toaster, but still strongly favoring H1 over H2.

 

[Nonpareil]

Really? Fill in the numbers and let's see.

That is what I just did, yes. Both p(cheating given a royal flush) and p(cheating) are unknown. Unknowns on both sides of the equation, neither established by your prior scenario. You cannot simply pull these out of thin air.

If you have a point to make, make it.

 

[jt512]

If you have a point to make, make it.

Write out your equation putting in all the numbers you know. After you have done that, we'll be able to see if the royal flush thing is evidence for cheating or not.

 

[jt512]

OK, calculate the probability that a toaster helped them out of the desert.

same as the one your worked out for the God hypothesis:

But I haven't calculated a probability for the God hypothesis.

The weight of the evidence in favor of H1 over H2, where H1 reflects our uncertainty about a toaster's helpfulness, is 850, about a third of that for our omnipotent toaster, but still strongly favoring H1 over H2.

So, you've worked out the weight of the evidence in favor of the toaster hypothesis, but not the probability of the toaster hypothesis. Come on. Take the next step.

 

[Jedi]

But I haven't calculated a probability for the God hypothesis.

So, you've worked out the weight of the evidence in favor of the toaster hypothesis, but not the probability of the toaster hypothesis. Come on. Take the next step.

come on jt512, stop being pedantic. my argument is that i can calculate the weight of the evidence in favor of a toaster (or bigfoot, or invisible aliens, etc.) in the same way as you calculated it for God. which is why it's trivial.

 

[Nonpareil]

Write out your equation putting in all the numbers you know. After you have done that, we'll be able to see if the royal flush thing is evidence for cheating or not.

I have already told you what we know. p(royal flush, independent of cheating) is very small - something like .003%, which I don't care enough to look up again - and p(royal flush given that I cheat) is one.

That leaves you to try and determine what p(cheating) and p(cheating given royal flush) are. It's a painfully simple equation. I just don't care to jump through hoops and type it out again for you.

If you think that you can calculate the value for p(cheating), it only takes ten seconds to demonstrate.

As a final note before I head off to bed, however: even if you could do this, it still wouldn't make anecdotes evidence of anything. In the above "calculations", again, "cheating" could be replaced with "probability wizard", "divine intervention", or even just "the stars aligned" with equal validity. It still has no explanatory power whatsoever.

 

[Jedi]

Write out your equation putting in all the numbers you know. After you have done that, we'll be able to see if the royal flush thing is evidence for cheating or not.

i think this was meant for Nonpareil.

 

[Slowvehicle]

come on jt512, stop being pedantic. my argument is that i can calculate the weight of the evidence in favor of a toaster (or bigfoot, or invisible aliens, etc.) in the same way as you calculated it for God. which is why it's trivial.

Or for ANY imaginary "savior"

 

[JoeMorgue]

You also can't prove the existence of immaterial souls by sticking your fingers in your ears and going "La la la everybody agrees with me la la la."

 

[Jedi]

Or for ANY imaginary "savior"

You also can't prove the existence of immaterial souls by sticking your fingers in your ears and going "La la la everybody agrees with me la la la."

exactly.

 

[JoeMorgue]

Mojo,
- Why do you think that's what I'm trying to do?

Oh Jesus bloody Christ.

BECAUSE THAT'S WHAT YOU ARE DOING.

 

[Nonpareil]

All right, I lied. One more for the road.

jt512's "gotcha", which he is trying desperately to lure me into thinking that I haven't seen it, is that the equation I gave above, with the numbers supplied, works out to something like:

p(cheating given a royal flush) = [a very large number] * p(cheating)

"Thusly," he says, "the probability of you having cheated is much greater if you got a royal flush, so you having a royal flush is evidence of you having cheated! Ah-ha!"

But no. That is still not an actual answer. Because, you see, there is a very, very simple and obvious solution to the above: zero.

If I don't cheat, then p(cheating) is zero. The odds that I cheated, given a royal flush, are zero. It doesn't matter how big that number gets. The odds are still zero.

jt512 must still establish a non-zero value for p(cheating) for his claims to have any weight. That is to say, he must show that I had the means, opportunity, and inclination to cheat. In terms of the survivors in the desert scenario, he must establish that God exists, God sometimes saves people lost in deserts, and so on.

That is to say, he must provide evidence. Otherwise, that entire formula boils down to exactly what all these claims of anecdotes as evidence are worth - a big, fat zero.

 

[jt512]

I have already told you what we know. p(royal flush, independent of cheating) is very small - something like .003%, which I don't care enough to look up again - and p(royal flush given that I cheat) is one.

Okay, let's assume that in the version of poker we're playing the probability of getting a royal flush is .003% if the player is not cheating. Let's also assume, that if the player is cheating then he'll get a royal flush with probability of 1. So if the player gets a royal flush, we have the following:

(Bayes factor) = P(R.F.|cheating) / P(R.F.|not cheating) = 1/.003% = 33,333.

By Bayes' Theorem, we have:

(Posterior odds of cheating) = (Bayes factor) × (Prior odds of cheating)
(Posterior odds of cheating) = 33,333 × (Prior odds of cheating)

So, whatever the prior odds of cheating are, once we've observed the player getting the royal flush, the odds that he is cheating increase by a factor of over 33,000. Are you seriously trying to tell us that an observation that increases the odds of X by more than a factor of 33,000 is not evidence for X? Clearly, it is. Furthermore, it is evidence for X regardless of what the prior odds of X were. It's the change in the odds of X brought about by an observation that determines whether the observation is evidence for X, not the value of the odds themselves.

That leaves you to try and determine what p(cheating) and p(cheating given royal flush) are. It's a painfully simple equation. I just don't care to jump through hoops and type it out again for you.

My point is that whether an observation is evidence for a proposition does not depend on the odds themselves, only on the change in odds, and we don't need to know the odds to know the how the evidence changes them—the change in odds is the same regardless of the odds themselves.

 

[jt512]

come on jt512, stop being pedantic. my argument is that i can calculate the weight of the evidence in favor of a toaster (or bigfoot, or invisible aliens, etc.) in the same way as you calculated it for God. which is why it's trivial.

You were talking about calculating the probability of the god/toaster hypothesis, not the weight of the evidence. Your conflation of the two concepts suggests you don't understand the difference.

 

[marplots]

Okay, let's assume that in the version of poker we're playing the probability of getting a royal flush is .003% if the player is not cheating. Let's also assume, that if the player is cheating then he'll get a royal flush with probability of 1. So if the player gets a royal flush, we have the following:

(Bayes factor) = P(R.F.|cheating) / P(R.F.|not cheating) = 1/.003% = 33,333.

By Bayes' Theorem, we have:

(Posterior odds of cheating) = (Bayes factor) × (Prior odds of cheating)
(Posterior odds of cheating) = 33,333 × (Prior odds of cheating)

So, whatever the prior odds of cheating are, once we've observed the player getting the royal flush, the odds that he is cheating increase by a factor of over 33,000. Are you seriously trying to tell us that an observation that increases the odds of X by more than a factor of 33,000 is not evidence for X? Clearly, it is. Furthermore, it is evidence for X regardless of what the prior odds of X were. It's the change in the odds of X brought about by an observation that determines whether the observation is evidence for X, not the value of the odds themselves.

My point is that whether an observation is evidence for a proposition does not depend on the odds themselves, only on the change in odds, and we don't need to know the odds to know the how the evidence changes them—the change in odds is the same regardless of the odds themselves.

Still no good because of the framing problem and the dependence on my ability to create models.

Suppose we grant everything you posted. Player A gets a royal flush. We conclude that there is now evidence Player A is cheating and the odds Player A is cheating have increased.

Now, consider a few other things. First, there are five players in the game. I know Player A to be generally honest (which is why I set the prior low for cheating for Player A). But the four other players, B,C,D and E, are either unknown to me or known to actually be dishonest. In fact, I actually suspect player D has studied the fine art of card cheating.

Now, player A gets a royal flush. This is evidence of cheating (based on the rarity of a royal flush). However, it is stronger evidence that one or more of the other players (not player A) are cheating, since they have higher priors than A, and when summed, it's higher still.

We now have a situation where the royal flush dealt to player A is both evidence of player A cheating, and evidence that player A has not cheated, depending on what we decide to model - either player A alone or the game as a whole. The same fact is both evidence for and against player A cheating.

The meaning and value of evidence is context dependent. To make a clean calculation requires abstraction. Abstraction can erase important contextual relationships. Contextual relationships are not always obvious to us in the first place and are generally the thing we want to find out.

 

[jt512]

Still no good because of the framing problem and the dependence on my ability to create models.

Flash: Guy on internet calls Bayes' Theorem "No Good"!

Suppose we grant everything you posted. Player A gets a royal flush. We conclude that there is now evidence Player A is cheating and the odds Player A is cheating have increased.

Now, consider a few other things. First, there are five players in the game. I know Player A to be generally honest (which is why I set the prior low for cheating for Player A). But the four other players, B,C,D and E, are either unknown to me or known to actually be dishonest. In fact, I actually suspect player D has studied the fine art of card cheating.

Now, player A gets a royal flush. This is evidence of cheating (based on the rarity of a royal flush).

Right.

However, it is stronger evidence that one or more of the other players (not player A) are cheating, since they have higher priors than A, and when summed, it's higher still.

Wrong. How many times do I have to write that the weight of the evidence (the Bayes factor) is independent of the prior odds?

We now have a situation where the royal flush dealt to player A is both evidence of player A cheating, and evidence that player A has not cheated, depending on what we decide to model - either player A alone or the game as a whole. The same fact is both evidence for and against player A cheating.

That doesn't even follow from your previous statement, which was itself incorrect.

 

[Jedi]

You were talking about calculating the probability of the god/toaster hypothesis, not the weight of the evidence. Your conflation of the two concepts suggests you don't understand the difference.

i'm not interested in discussing priors or calculating nonsensical probabilities so you can avoid explaining why your argument isn't trivial. calculating these probabilities doesn't change my argument, as i can still interject 'toaster' wherever you have the word 'God.' it adds nothing to the discussion we're having except obfuscation. and if it makes you feel better, i retract my question about probabilities and replace it with odds for H1 relative to H2. my argument remains the same either way.

you know exactly what i'm getting at, as i've been making this same argument for a while. and you still haven't answered it. what does that suggest?

 

[jt512]

ii retract my question about probabilities and replace it with odds for H1 relative to H2.

Nope. Still missing the point. Review the odds form of Bayes theorem, and get straight what terms you are talking about.

you know exactly what i'm getting at, as i've been making this same argument for a while. and you still haven't answered it. what does that suggest?

I don't even think you know what you're getting at. I think you still don't understand the difference between the Bayes factor (or weight of the evidence) and the probability (or odds) of a hypothesis.

 

[marplots]

Flash: Guy on internet calls Bayes' Theorem "No Good"!

Oh, the theorem is fine, it's the application I'm disputing.

If you want general criticism of the method itself, here's a paper from 2008 which lists the most common: http://www.stat.columbia.edu/~gelman/research/published/badbayesmain.pdf

My favorite has always been along the lines of, "If I'm going to guess the priors anyhow, why not save time and just guess the posteriors too?" After all, any posterior is just the prior for a different calculation.

This actually works quite well for the "desert survivors" problem. There is no God, so any evidence which implies there is is wrong. Easy peasy.

Wrong. How many times do I have to write that the weight of the evidence (the Bayes factor) is independent of the prior odds?

Write it as often as you like. Heck, eventually it might actually apply to the point someone else is trying to make. It's actually another illustration of what I've been hammering on. You have a statement which is true in a certain context and to preserve that truth value, the context has to be managed to fit. When I try to bring up something else, to demonstrate context dependence, just hand wave it away and repeat the statement.

Here's how it looks from my end:

10 True statement in context 1
20 Receive suggestion that context 1 is artificial and doesn't apply
30 Reply with: No, that's wrong because then the statement wouldn't be true
40 Goto 10

 

[aleCcowaN]

As I wrote in the post you're replying to, the odds of X = P(X) / [1–P(X)], where P(X) means the "probability of X." So, if the prior odds of the wizard hypothesis were, say 2/850, the odds after the new data, ie, the posterior odds, would by 850 × 2/850 = 2, or "2-to-1" in favor of the hypothesis. Odds of 2:1 is equivalent to a probability of 2/3, or about 0.67, which would be the posterior probability of the hypothesis.

...

Is this your peculiar version of Euler's "Sir, [a+bⁿ]/n=x, hence God exists—reply!"? Do you expect everyone to flee Russia, like Diderot?

Your ridiculous un-statistical anti-scientific glossolalia has gone too far!