Why Malerin is Wrong About Bayes Theorem

[rocketdodger]

Okay, second try, doing the math correctly this time. Let's start with a simple definition of conditional probability :

P(E) = P(E|H)P(H) + P(E|-H)P(H).

We now assume that I have access to an event generator that will generate events independent of E with probability 0.5. This is not a big assumption, since I have access to a whole jar-full of them; they're called coins. H is therefore just an event "I flipped heads," and P(H) = P (-H) = 0.5.

Furthermore, since this event is by construction independent of E, P(E|H) = P(E) and P(E|H)P(H) = P(E)/2 = P(E|H)/2.

Similarly, P(E|-H) = P(E) and P(E|-H)P(-H) = P(E)/2 = P(E|-H)/2.

And since P(E|H)/2 = P(E)/2 = P(E|-H)/2, we have that P(E) = P(E|H) = P(E|-H).

Your claim is that, in the absence of any other information about P(E), we should assume that P(E|H) + P(E|-H) = 1.

But, mathematically, this works out to be that P(E|H) + P(E|H) = 1.0, which in turn means that P(E|H) = 0.5, which in turn means that P(E) = 0.5

So your suggested constraint ends up being much stronger than the claim you rejected, namely that we cannot estimate P(E). We have. If I have no information on it whatsoever, then P(E) must be equal to 0.5.

So you end up either supporting Malerin's formulation --- the a priori probability of God existing is 0.5, or you end up needing to reject your own overconstraint.

Well, three things:

First, this the result I intended, and I think the math in my OP agrees with what you did -- if P(H) is 0.5, then P(E) must be 0.5 as well.

Second, this is not analagous to Malerin's example! He uses H = a universe creator and E = a life supporting universe. That means E is not independent of H -- specifically, he is assuming a value very close to unity for P(E|H) because the fine-tuning argument relies on a high probability of a life supporting universe given that a universe creator exists.

That is why I could not use the equivalences you did I.E. P(E) = P(E|H) and P(E) = P(E|~H). P(E) is definitely not equal to P(E|H) in the fine tuning argument.

Third, is arriving at a value for P(E) (assuming E is not independent of H) using the conditional probabilities P(E|H) and P(E|~H) still considered "a priori?"

Because my contention was that a valid use of Bayes Theorem requires a true a priori estimate for both priors, that in this case we can't get a good a priori estimate, that instead we must rely on the above relationship between the unconditional and the conditionals to get any value of P(E) in order to use Bayes at all nevermind properly, and since we rely on the conditionals all bets are off when it comes to interpreting the results -- illustrated by my claim that P(E) is dependent on P(H) when one expect it should not be for a proper analysis.

Do you still think this is wrong?

 

[fls]

I think that's reasonably good way to explain it. But then the conclusion one would draw is that the use of Bayes' theorem and its uninformative prior should be confined to situations where you have increasing knowledge - that is, absent information, its prior should be avoided (i.e. it can't serve as a guess for Randfan's scenario). And that when you find yourself in a situation where Bayes' theorem does not increase your knowledge (the likelihood ratio is derived intuitively (i.e. pulled out of your ass ), rather than empirically), such as Malerin's fine-tuning scenario, a maximally uninformative prior should also be avoided.

For those reasons, I don't think RandFan's scenario proves Malerin's case, because I think the application of p=0.5 is unsupported in either case.

Linda

Specifically, an approach similar to what rocketdodger outlined previously - a consideration of all states in which a million dollars is won vs. those in which a hundred dollars is lost, or states in which god is present vs. those in which god is absent - makes more sense.

Linda

 

[GreedyAlgorithm]

Specifically, an approach similar to what rocketdodger outlined previously - a consideration of all states in which a million dollars is won vs. those in which a hundred dollars is lost, or states in which god is present vs. those in which god is absent - makes more sense.

Linda

This is called prior knowledge. If I asked you a binary question Q, and all you knew about it was that it is a binary question, you should set p=0.5 for "yes" and for "no". If you actually have some knowledge about states in which Q is yes vs not, then you should incorporate that. It may be very hard to do so. In Malerin's case we have lots of prior knowledge. By saying "I choose an uninformative prior p=0.5", Malerin is explicitly throwing away that prior knowledge. As such, his results apply to pretty much no one that exists, but given the premises it's fine -- unless he claims his results mean anything at all other than "given this hypothetical knowledge set which no one actually has..."

 

[Ivor the Engineer]

H = God Exists.
E = Universe Exists.

The graph below shows how P(H|E) varies with P(E|~H) if we assume P(H) = P(~H) = 0.5 and P(E|H) = P(~E|H) = 0.5.

Notice that even when P(E|~H) = 1, P(H|E) is still 0.33. I.e., this debate would still be going on even if we had the technology to create universes with intelligent life in them.

 

[MattusMaximus]

No, because I have other information based on other closets and other people.

In a sense, you're actually proving Malerin's case for him. If you knew nothing about money and closets, then it would be a good bet for you. You know it's a bad bet because you have an informed prior. Really, it's no different than your offering to play craps with me using a pair of dice that you know are weighted and I don't.

Ah, I see the point of your criticism now.

Nuts - well, it was funny while it lasted.

Dr. Kitten, a question. Would you agree that if Malerin makes the assumption that the "agnostic" probability of God's existence would be 0.5 then the "agnostic" probability of the existence of the FSM must also be 0.5?

 

[Malerin]

Ah, I see the point of your criticism now.

Nuts - well, it was funny while it lasted.

Dr. Kitten, a question. Would you agree that if Malerin makes the assumption that the "agnostic" probability of God's existence would be 0.5 then the "agnostic" probability of the existence of the FSM must also be 0.5?

There is no evidence for the FSM, and it would seem to violate basic biological truths and evolutionary truths. As CJ has pointed out, there is evidence for God (subjective spiritual experiences, NDE accounts), though not the kind of evidence everyone finds convincing, and the existence of God does not violate any basic scientific truths. Therefore, one can be agnostic about God (maybe the experiences are genuine, maybe the NDE accounts are accurate) while rejecting the FSM (no evidence for it, how would it be possible biologically).

 

[Malerin]

This is called prior knowledge. If I asked you a binary question Q, and all you knew about it was that it is a binary question, you should set p=0.5 for "yes" and for "no". If you actually have some knowledge about states in which Q is yes vs not, then you should incorporate that. It may be very hard to do so. In Malerin's case we have lots of prior knowledge. By saying "I choose an uninformative prior p=0.5", Malerin is explicitly throwing away that prior knowledge. As such, his results apply to pretty much no one that exists, but given the premises it's fine -- unless he claims his results mean anything at all other than "given this hypothetical knowledge set which no one actually has..."

All you would need is an agnostic who's never heard of the FT argument.

 

[MattusMaximus]

There is no evidence for the FSM, and it would seem to violate basic biological truths and evolutionary truths. As CJ has pointed out, there is evidence for God (subjective spiritual experiences, NDE accounts), though not the kind of evidence everyone finds convincing, and the existence of God does not violate any basic scientific truths. Therefore, one can be agnostic about God (maybe the experiences are genuine, maybe the NDE accounts are accurate) while rejecting the FSM (no evidence for it, how would it be possible biologically).

I beg to differ Malerin. There is, according to your own definition, evidence for the FSM. In fact, I had a subjective spiritual experience with the FSM just yesterday. Thus - again, by your own definition & arguments - you must accept the agnostic probability of the FSM's existence to be 0.5.

Prove me wrong. Go on, I dare you.

In the meantime, may you be touched by His Noodly Appendage. RA-men!

 

[fls]

This is called prior knowledge. If I asked you a binary question Q, and all you knew about it was that it is a binary question, you should set p=0.5 for "yes" and for "no". If you actually have some knowledge about states in which Q is yes vs not, then you should incorporate that. It may be very hard to do so. In Malerin's case we have lots of prior knowledge. By saying "I choose an uninformative prior p=0.5", Malerin is explicitly throwing away that prior knowledge. As such, his results apply to pretty much no one that exists, but given the premises it's fine -- unless he claims his results mean anything at all other than "given this hypothetical knowledge set which no one actually has..."

To know that something is a binary question gives you a lot of information (specifically excluding the fake binary question raised in this thread - I have a million dollars in my closet or not)

The problem has been that Malerin does not restrict his claim as to what the results mean.

Linda

 

[GreedyAlgorithm]

There is no evidence for the FSM, and it would seem to violate basic biological truths and evolutionary truths. As CJ has pointed out, there is evidence for God (subjective spiritual experiences, NDE accounts), though not the kind of evidence everyone finds convincing, and the existence of God does not violate any basic scientific truths. Therefore, one can be agnostic about God (maybe the experiences are genuine, maybe the NDE accounts are accurate) while rejecting the FSM (no evidence for it, how would it be possible biologically).

...wait a tick. You're saying that because you have some evidence about H, you can assign an uninformative prior to H?

Come on!

 

[drkitten]

I think that's reasonably good way to explain it.

Thanks.

But then the conclusion one would draw is that the use of Bayes' theorem and its uninformative prior should be confined to situations where you have increasing knowledge - that is, absent information, its prior should be avoided (i.e. it can't serve as a guess for Randfan's scenario).

I think you're confusing two separate potential errors. As far as I can tell, there is nothing wrong, mathematically, with answering any question with "Damfino" and representing that with the maximally uninformative distribution.

Including using that representation as the prior before you start doing your statistics. If you know nothing, you know nothing, and this is appropriately represented as the real number 0.5.

The other error creeps in when you and your proctologist start pulling out conditional probabilities using nothing but your intuition. Because these numbers will lend a spurious air of precision to what amounts to wild-assed guessing.

As an example of what I'm talking about, imagine finding a dead body floating in an (unmarked) lifeboat in the high seas, and as the investigator you need to determine what its nationality is (so that you can check records appropriately). Based on what I've told you so far, I think it's quite reasonable to use Bayesian statistics and represent your initial knowledge with "damfino." But you need to use real data and real numbers to start your hypothesis testing. You can't just say "well, I think that more Latvians are blonde than brunette."

 

[drkitten]

First, this the result I intended, and I think the math in my OP agrees with what you did -- if P(H) is 0.5, then P(E) must be 0.5 as well.

Which means that the probability of any event about which I know nothing must be 0.5? The chances that my aunt's college roomate's eldest grandchild being a senator is 0.5, simply because I chose to flip a coin?

Nonsense.

Second, this is not analagous to Malerin's example! He uses H = a universe creator and E = a life supporting universe. That means E is not independent of H -- specifically, he is assuming a value very close to unity for P(E|H) because the fine-tuning argument relies on a high probability of a life supporting universe given that a universe creator exists.

But your argument hinges on P(E|H) + P(E|-H) = 1 in the general case; I chose the simplest case (a coin flip for H) to generate a counterexample.

Third, is arriving at a value for P(E) (assuming E is not independent of H) using the conditional probabilities P(E|H) and P(E|~H) still considered "a priori?"

Yes(-ish), because we're just doing arithmetic manipulation. You can solve for the prior using the posterior, you can solve for the posterior using the prior -- or for that matter, you can solve for the prior just using the constraints.

P(H) is the prior probability of the hypothesis being true. P(H|E) is the posterior probability of the hypothesis being true, given the evidence E. P(E) is the (prior) probability of the evidence happening, while P(E|H) is the (posterior) probability of the evidence happening given the truth of the hypothesis. If you have any three, you can plug in and solve for the fourth.

Because my contention was that a valid use of Bayes Theorem requires a true a priori estimate for both priors,

And you're wrong. If you have a good a prior estimate of the probability of the evidence, then you can use a maximally uninformative prior for the hypothesis and get a reasonably good guess at the likelihood of the hypothesis. If you have a good prior estimate of the hypothesis, then you can use that to get a reasonable hypothesis about the evidence-space.

What you can't do is pull numbers out of thin air for both, which is what Malerin is trying to do.

Do you still think this is wrong?

Absotively.

 

[drkitten]

Dr. Kitten, a question. Would you agree that if Malerin makes the assumption that the "agnostic" probability of God's existence would be 0.5 then the "agnostic" probability of the existence of the FSM must also be 0.5?

Absolutely. Of course, neither of us are really "agnostic" in this sense about either; while I can do the math pretending that I don't have an opinion about whether or not God exists, that's not the same as actually not having an opinion.

However, if I do the math properly, I can codify the information that I used to get my opinion into a numeric form and essentially duplicate my process of opinion forming. For example, what is the probability that X exists given that X has a well-defined creator who has described X as a parody?
(Well, the creator could be lying, I suppose....)

The problem here is that to do the math "properly" requires a degree of statistical accuracy that I don't really have --- which is where Malerin's tail-chasing comes in. The problem isn't with assuming agnostically that God either exists or He doesn't. That's entirely valid. The problem is with the assumption that if He exists, then there is a X% chance of any particular thing happening, since we have no idea how many universes He might make in his spare time or what they look like.

 

[rocketdodger]

To know that something is a binary question gives you a lot of information (specifically excluding the fake binary question raised in this thread - I have a million dollars in my closet or not)

The problem has been that Malerin does not restrict his claim as to what the results mean.

Linda

Exactly.

If a decision is truly binary, the resultant states differ only in the outcome of the decision. Given state S, if the decision is "yes" the resultant state will be Sy and if the decision is "no" it will be Sn and Sy differs from Sn in only one way. That is why it makes sense to assign 0.5 to an uniformed prior.

But a completely unknown event is not truly binary. In particular, it is not clear how, supposing we begin with a pre-universe-creator state of S, the post-universe-creator states Sh would differ from the post-not-universe-creator states Se. If the universe creator event didn't occur, what would happen? How many other possible states might occur? We can't say because of Malerin's intentionally vague description of the event.

So the only logical thing to do is treat the universe creator like any other possible event we have no prior knowledge of and assign all the resultant states equal probability, leading to a value of 1 / |{all possible states that could occur as a result of an unknown event}|.

 

[drkitten]

There is no evidence for the FSM, and it would seem to violate basic biological truths and evolutionary truths.

If you're acquainted with this level of detail about the FSM hypothesis, then you are by definition not using the "agnostic" probability.

As CJ has pointed out, there is evidence for God (subjective spiritual experiences, NDE accounts), though not the kind of evidence everyone finds convincing, and the existence of God does not violate any basic scientific truths.

Yes, because there's nothing logically wrong about an omnipotent, omnibenevolent, omniscient being. [Insert laughing dog].

I could easily use those three facts in conjunction to prove that the probability of God existing is zero.

But I wouldn't be being "agnostic" about it.

Therefore, one can be agnostic about God (maybe the experiences are genuine, maybe the NDE accounts are accurate) while rejecting the FSM (no evidence for it, how would it be possible biologically).

You're not playing fair. If you expect us to suspend disbelief on all the things that we know about God that render Him logically incoherent and therefore impossible, you must also suspend disbelief in all the things you know about the FSM.

 

[Ivor the Engineer]

<snip>

You're not playing fair. If you expect us to suspend disbelief on all the things that we know about God that render Him logically incoherent and therefore impossible, you must also suspend disbelief in all the things you know about the FSM.

But that's one of the main reasons we invented gods in the first place: so we don't have to play fair!

 

[rocketdodger]

Which means that the probability of any event about which I know nothing must be 0.5? The chances that my aunt's college roomate's eldest grandchild being a senator is 0.5, simply because I chose to flip a coin?

Nonsense.

No no no I am talking about the case where E is not independent of H and we have only a single trial to gather evidence from. That is very different from your coin example because with a coin you know the flip is independent of whatever else and you know from experience that coins land on heads/tails approximately 50% of the time.

But your argument hinges on P(E|H) + P(E|-H) = 1 in the general case; I chose the simplest case (a coin flip for H) to generate a counterexample.

No! Again, I am talking about the specific case when E is not independent of H and there is only a single trial (ever) to use as evidence.

Yes(-ish), because we're just doing arithmetic manipulation. You can solve for the prior using the posterior, you can solve for the posterior using the prior -- or for that matter, you can solve for the prior just using the constraints.

P(H) is the prior probability of the hypothesis being true. P(H|E) is the posterior probability of the hypothesis being true, given the evidence E. P(E) is the (prior) probability of the evidence happening, while P(E|H) is the (posterior) probability of the evidence happening given the truth of the hypothesis. If you have any three, you can plug in and solve for the fourth.

But typically -- and in this case -- we don't know P(H|E). So we need values for the other three, including P(E), right?

And P(E) can be written as P(E|H)P(H) + P(E|~H)P(~H), right? So intuitively, we need to somehow extract three degrees of freedom from those two terms. We are already using P(E|H) and P(H) in Bayes, and we can calculate P(~H) from P(H). So the only place a third degree of freedom could be hiding is in P(E|~H).

But if P(E|~H) is dependent on P(E|H), then it isn't free. That is my argument. If, as I contend, P(E|H) and P(E|~H) must sum to a given value, there are only two degrees of freedom whereas Malerin contends there are three.

I guess this all hinges on whether there is a relationship between P(E|H) and P(E|~H).

If P(E|~H) was dependent on P(E|H) like I propose, would I be correct on the rest of this stuff? Would there indeed only be two degrees of freedom available?

 

[Malerin]

If you're acquainted with this level of detail about the FSM hypothesis, then you are by definition not using the "agnostic" probability.



Yes, because there's nothing logically wrong about an omnipotent, omnibenevolent, omniscient being. [Insert laughing dog].

I could easily use those three facts in conjunction to prove that the probability of God existing is zero.

The Christian God is a logical contradiction? That would be an interesting proof.

But I wouldn't be being "agnostic" about it.

I wasn't talking about the Christian God. I was talking about be agnostic about God (the existence of some powerful supernatural deity).

You're not playing fair. If you expect us to suspend disbelief on all the things that we know about God that render Him logically incoherent and therefore impossible, you must also suspend disbelief in all the things you know about the FSM.

Again, you're thinking of the Christian God. Plenty of people believe there might be a God that started the whole thing and doesn't really have much to do with it now.

 

[Malerin]

...wait a tick. You're saying that because you have some evidence about H, you can assign an uninformative prior to H?

Come on!

No one has a totally uninformative prior. Everything we believe in is influenced by our background knowledge. However, you could find an agnostic whose background knowledge doesn't include any evidence about physical constants or fine tuning. Trust me, it's not that hard. Most people have never heard of cosmological fine-tuning. If such an agnostic believes that Pr(E/~H) is pretty low (putting aside the problem of old evidence for the moment), they're probably going to revise their agnosticism.

For someone like me, though, the FT argument won't have much impact. It's already worked its way into my prior knowledge and is just one of those things that increases my belief in theism.

 

[drkitten]

But typically -- and in this case -- we don't know P(H|E). So we need values for the other three, including P(E), right?

And P(E) can be written as P(E|H)P(H) + P(E|~H)P(~H), right? So intuitively, we need to somehow extract three degrees of freedom from those two terms. We are already using P(E|H) and P(H) in Bayes, and we can calculate P(~H) from P(H). So the only place a third degree of freedom could be hiding is in P(E|~H).

But if P(E|~H) is dependent on P(E|H), then it isn't free.

And P(E|-H) is not dependent on P(E|H) in the general case.

If, as I contend, P(E|H) and P(E|~H) must sum to a given value,

And we know from upthread that they do not; we've got the examples both of a possibly unfair coin and of a child of indeterminate sex who may or may not be named Sue.

Since we know that it's possible that they don't add up to the necessary number, it makes no sense whatsoever to assert that, in all cases where we don't have information enough to confirm that they don't, they must.

If P(E|~H) was dependent on P(E|H) like I propose, would I be correct on the rest of this stuff? Would there indeed only be two degrees of freedom available?

Yes, but if that were the case, I think Bayes' theorem itself would fall down.