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Old 2nd December 2016, 01:58 PM   #2121
JesseCuster
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Originally Posted by jt512 View Post
That's correct.




As I have explained, it tells us the amount by which the new evidence should increase our prior odds for the hypothesis. Multiply your prior odds* for wizards helping people out of the desert by magic by 850, and let us know how likely you now think that hypothesis is.

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*Use the simple odds: P(wizards) / [1 - P(wizards)]
I'm not sure that makes sense. If the prior odds of wizards are greater than 1/850 then the resulting likelihood of that hypothesis will now be greater than 1, which I'm pretty sure is a mathematical likelihood which means there's something wrong with the logic being used or it's somehow impossible for wizards (or any other scenario in this hypothesis) to have prior odds of greater than 1/850.

I've no idea what the prior odds for wizards rescuing stranded explorers are anyway or how such a thing could reasonably possibly be estimated. What are the odds of God rescuing stranded explorers and how was that number arrived at?

I'm also not sure what "it tells us the amount by which the new evidence should increase our prior odds for the hypothesis" means. What is "the new evidence" in this case? Does any new evidence multiply the likelihood of wizards or Gods or whatever rescuing travellers by a factor of 850? That doesn't sound right to me.

Sorry if this has been shown already in the thread, I've read dozens of pages in recent weeks of this thread and have no doubt missed some important points.
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Old 2nd December 2016, 02:03 PM   #2122
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Originally Posted by jt512 View Post
As I have explained, it tells us the amount by which the new evidence should increase our prior odds for the hypothesis. Multiply your prior odds* for wizards helping people out of the desert by magic by 850, and let us know how likely you now think that hypothesis is.
Which is to say, no. It doesn't tell us the odds that wizards actually rescued them. It only tells us that, if wizards were around to magic them up a drink, they would have had a greater chance of getting one.

Which is not helpful.
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Old 2nd December 2016, 02:57 PM   #2123
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Old 2nd December 2016, 03:02 PM   #2124
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Originally Posted by Mojo View Post
Jabba, you can't prove the existence of immaterial souls using an argument that relies on assuming the existence of immaterial souls.
Mojo,
- Why do you think that's what I'm trying to do?
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Old 2nd December 2016, 03:03 PM   #2125
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Originally Posted by Jabba View Post
Mojo,
- Why do you think that's what I'm trying to do?
Because in your OOFLam model, you include an immaterial soul.
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Old 2nd December 2016, 03:14 PM   #2126
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Originally Posted by Jabba View Post
Why do you think that's what I'm trying to do?
Because the enumerable "potential selves" concept in your P(E|H) construct is functionally identical to the notion of an immortal soul. Calling it something else doesn't affect the fact that no such concept -- by any name -- is required in the scientific model. Attempting to disguise it by insisting it is merely an abstract facet of the model does not enliven it; it is a merely a word-game basis for inventing the most crucial quantity in your mode: the infinity by which you can divide any other number away and make your model give the desired answer without any toehold in reality.
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Old 2nd December 2016, 03:38 PM   #2127
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Originally Posted by Jabba View Post
Mojo,
- Why do you think that's what I'm trying to do?

Because that's where you get the "infinity" in your "seven billion over infinity".
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Old 2nd December 2016, 03:40 PM   #2128
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Seriously, Jabba? Have you really forgotten your own argument?
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Old 2nd December 2016, 03:57 PM   #2129
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Originally Posted by Mojo View Post
Seriously, Jabba? Have you really forgotten your own argument?
I don't think it's a matter of having forgotten it. I think it's a matter of him believing his own hype. I don't think he's willing or capable of understanding that the scientific model has no concept of enumerable things that become selves, such that there can be an "infinite number" of them. He is unwilling or unable to conceive of the self as an emergent property of the brain, not some necessarily separate entity (even as an abstract concept). Yet this is what the self is in the scientific model. It's innumerable in the sense Jabba wants to use it. But the real concept of property seems to be philosophically beyond him. Whether he's just not adept enough to grasp the concept or whether he's ignoring it for his own purposes is anyone's guess.
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Old 2nd December 2016, 05:04 PM   #2130
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Originally Posted by JayUtah View Post
Whether he's just not adept enough to grasp the concept or whether he's ignoring it for his own purposes is anyone's guess.
The later is very high-end Dunning-Kruger: As you are a keen thinker, then some other people is not certifiable incompetent but they might be ignoring the reasonable steps for unknown reasons (they're intelligent with a plan instead of plain clumsy).

To me is as simple as this one an similar threads can go on for ever just because knowledge has limitations, but ignorance hasn't.
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Old 2nd December 2016, 05:15 PM   #2131
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Originally Posted by Jabba View Post
Mojo,
- Why do you think that's what I'm trying to do?
Because, by your own rules, by your continued silence you have admitted that you do not have any evidence of any kind for the existence of a "soul", much less its "immortality"...
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Old 2nd December 2016, 06:23 PM   #2132
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Originally Posted by Jabba View Post
Mojo,
- Why do you think that's what I'm trying to do?

Currently, I think you're trying to shift attention away from the part of your argument that has most recently come under fire. It would be great for you if we all went back to trying to talk about how you arrived at different givens and what numbers you're multiplying by what.

But you personally asked that we pick one issue and drill down to its very core. We did. We came to a point where you needed to show immaterial souls were at least possible. We discussed what would or would not be evidence of that. You provided no evidence nor any real understanding of what evidence might look like, instead begging the question of "believing" some story about a film about a story by a doctor who gave no specifics whatsoever.

The issue is the lack of evidence that immaterial souls are even possible. Anything you do to deviate from that issue should be, by your own rules, disallowed.
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Old 2nd December 2016, 06:25 PM   #2133
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Originally Posted by JesseCuster View Post
I'm not sure that makes sense. If the prior odds of wizards are greater than 1/850 then the resulting likelihood of that hypothesis will now be greater than 1, which I'm pretty sure is a mathematical likelihood which means there's something wrong with the logic being used or it's somehow impossible for wizards (or any other scenario in this hypothesis) to have prior odds of greater than 1/850.

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.

Quote:
I've no idea what the prior odds for wizards rescuing stranded explorers are anyway or how such a thing could reasonably possibly be estimated.

Seriously? You have no idea about the probability that wizards exist and can use magic to save explorers lost in the desert? Seriously? No idea at all?

Quote:
I'm also not sure what "it tells us the amount by which the new evidence should increase our prior odds for the hypothesis" means.

It means that you multiply the prior odds by 850 to get the posterior odds.

Quote:
What is "the new evidence" in this case?

The very evidence you suggested: that 2 people came back from the desert and reported they were saved by wizards.

Quote:
Does any new evidence multiply the likelihood of wizards or Gods or whatever rescuing travellers by a factor of 850? That doesn't sound right to me.

Whether it sounds right to you or not, it is mathematically correct (by Bayes' Theorem). I think if you come up with a reasonable common-sense estimate of the probability that wizards helped our lost explorers using magic, and multiply it by 850, you'll be satisfied with resulting posterior odds.
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Old 2nd December 2016, 06:29 PM   #2134
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Originally Posted by Nonpareil View Post
Which is to say, no. It doesn't tell us the odds that wizards actually rescued them. It only tells us that, if wizards were around to magic them up a drink, they would have had a greater chance of getting one.

Not only is it helpful in and of itself, there is a whole school of statistics that thinks all we should do is calculate ratios of likelihoods. It is also a necessary step toward determining the posterior probability of the hypothesis.
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Old 2nd December 2016, 06:30 PM   #2135
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Originally Posted by Jedi View Post
which is completely subjective and can change from person to person based on the assumptions you're making. which is why it's a trivial argument. you can show that a toaster helped people out of the desert using the exact same assumptions.

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

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Old 2nd December 2016, 06:59 PM   #2136
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Originally Posted by jt512 View Post
Not only is it helpful in and of itself, there is a whole school of statistics that thinks all we should do is calculate ratios of likelihoods. It is also a necessary step toward determining the posterior probability of the hypothesis.
I'm sure there is.

That still doesn't make a given royal flush evidence of cheating.
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Old 2nd December 2016, 07:03 PM   #2137
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Originally Posted by Nonpareil View Post

That still doesn't make a given royal flush evidence of cheating.

No matter how many times you write that nonsensical statement, it will still be nonsensical.
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Old 2nd December 2016, 07:43 PM   #2138
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Originally Posted by jt512 View Post
No matter how many times you write that nonsensical statement, it will still be nonsensical.
It's hardly nonsensical. Very straightforward, really.

Your posts thus far have consisted largely of some very basic observations on probability. If God leads you to water in the desert, you are guaranteed to survive and tell others you saw God; on the other hand, if he doesn't, you aren't. The probability of observing survivors claiming to have seen God is therefore very high if they actually did see God, but lower if they did not.

That is, if God saved these people, the probability of them saying that he did is much higher than if he didn't. This is true, and is not in dispute; in the same way, the fact that your chances of drawing a royal flush are much higher if you cheated are higher than if you didn't is not in dispute.

But drawing a royal flush is not evidence that you cheated to do so. The fact that the survivors claimed to be saved by God is not evidence that they actually were. The fact that a given observation is consistent with a hypothetical explanation is not evidence that said explanation is actually the correct one.

To put this in terms of Bayesian calculations:

p(cheating|royal flush) = p(royal flush|cheating) * p(cheating) / p(royal flush)

For those following along at home, the above reads: the probability that I cheated, given that I do have a royal flush, is equal to the odds of me getting a royal flush if I did cheat, times the odds that I cheated, divided by the odds of getting a royal flush fairly.

The probability of me having a royal flush given that I cheat is, for the sake of argument, one hundred percent. And I the odds of me drawing a royal flush without cheating are very low. That fills in two of the three terms on the right-hand side.

But you don't know the probability of me cheating. Or, in the terms of your original argument, you don't know how likely it is that God exists and would help these two. And you do not have enough information to actually establish it.

So yes. For the sake of argument, the probability of surviving and telling that story is very high if God exists and helps you. And it is very low if he doesn't.

But that doesn't mean that the fact that someone survived and told this story is evidence of divine intervention. The probability of that happening is not dependent on or affected by whether or not this story was told. It is entirely independent, and you do not have sufficient information to establish it.

A given royal flush is not evidence of cheating.
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Old 2nd December 2016, 07:56 PM   #2139
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Originally Posted by Nonpareil View Post
A given royal flush is not evidence of cheating.

But what if a book of anecdotes cited a film in which a doctor tells a story about having once seen a royal flush and believed it was evidence of cheating?

Surely, that makes the cheating theory unassailable.
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Old 2nd December 2016, 08:02 PM   #2140
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Originally Posted by Nonpareil View Post

But drawing a royal flush is not evidence that you cheated to do so.

p(cheating|royal flush) = p(royal flush|cheating) * p(cheating) / p(royal flush)

Really? Fill in the numbers and let's see.
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Old 2nd December 2016, 08:35 PM   #2141
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Originally Posted by jt512 View Post
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.
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Old 2nd December 2016, 08:59 PM   #2142
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Originally Posted by jt512 View Post
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.
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Old 2nd December 2016, 09:07 PM   #2143
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Originally Posted by Jedi View Post
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.
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Old 2nd December 2016, 09:10 PM   #2144
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Originally Posted by jt512 View Post
OK, calculate the probability that a toaster helped them out of the desert.
Originally Posted by Jedi View Post
same as the one your worked out for the God hypothesis:

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


Quote:
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.
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Old 2nd December 2016, 09:24 PM   #2145
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Originally Posted by jt512 View Post
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.
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Old 2nd December 2016, 09:24 PM   #2146
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Originally Posted by jt512 View Post
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.
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Old 2nd December 2016, 09:25 PM   #2147
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Originally Posted by jt512 View Post
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.
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Old 2nd December 2016, 09:32 PM   #2148
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Originally Posted by Jedi View Post
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"
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Old 2nd December 2016, 09:34 PM   #2149
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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."
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Old 2nd December 2016, 09:35 PM   #2150
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Originally Posted by Slowvehicle View Post
Or for ANY imaginary "savior"
Originally Posted by JoeBentley View Post
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.

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Old 2nd December 2016, 09:41 PM   #2151
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Originally Posted by Jabba View Post
Mojo,
- Why do you think that's what I'm trying to do?
Oh Jesus bloody Christ.

BECAUSE THAT'S WHAT YOU ARE DOING.
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Old 2nd December 2016, 09:44 PM   #2152
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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.
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Old 2nd December 2016, 10:02 PM   #2153
jt512
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Originally Posted by Nonpareil View Post
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.

Quote:
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.

Last edited by jt512; 2nd December 2016 at 10:07 PM.
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Old 2nd December 2016, 10:11 PM   #2154
jt512
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Originally Posted by Jedi View Post
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.
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Old 2nd December 2016, 10:47 PM   #2155
marplots
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Originally Posted by jt512 View Post
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.
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Old 2nd December 2016, 10:58 PM   #2156
jt512
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Originally Posted by marplots View Post
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"!

Quote:
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.

Quote:
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?

Quote:
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.
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Old 2nd December 2016, 11:03 PM   #2157
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Originally Posted by jt512 View Post
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?
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Old 2nd December 2016, 11:08 PM   #2158
jt512
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Originally Posted by Jedi View Post
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.

Quote:
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.
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Old 2nd December 2016, 11:09 PM   #2159
marplots
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Originally Posted by jt512 View Post
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...dbayesmain.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.

Originally Posted by jt512 View Post
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

Last edited by marplots; 2nd December 2016 at 11:25 PM.
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Old 2nd December 2016, 11:23 PM   #2160
aleCcowaN
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Originally Posted by jt512 View Post
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!
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