Why Malerin is Wrong About Bayes Theorem

[RandFan]

Yeah, but if you don't know whether it's biased towards heads or towards tails, do you have any more reason to bet on heads than on tails?

Let's assume that we don't know if the coin is biased. Are there any other mysteries? Our experience tells us that the coin is likely close to 50/50 even if it is biased but it is really beside the point. If we know statistics we know that there is never an absolute 50/50 chance of anything.

What reason do you have that there is a god as opposed to there is no god? Have you ever seen a god? Do you know what a god entails? Do you know what the requirements of a god are?

To hypothesize a known probability about something that is completly unknown is irrational. .5 is nonsense. It's human nature to assume that a corresponding conditional will render a 50/50 chance because we are so familiar with that thinking. The light is either on or off. The water is either cold or hot. So we see things that way when in truth that simply is not really the case. There is not black and white there are a lot of grays. There is not hot and cold there are degrees of warm.

For an example of the problem see The Monty Hall Problem.

 

[GreedyAlgorithm]

To hypothesize a known probability about something that is completly unknown is irrational. .5 is nonsense.

No. You are equivocating definitions of probability. To set a probability (Bayesian: represents degree of belief in something; other properties can be derived which usually end up the same as in other definitions) to 0.5 in the case of a binary question of which nothing else is known is simply definition. To extrapolate from that to believe that if you could ask that question many times, the person who set a probability at 0.5 is claiming you'd get approximately 50% yes and 50% no answers, is to equivocate the definition. To say it's not 0.5 because the probability God exists is not 0.5 is also ridiculous because in the case of the God question we have plenty of knowledge pointing to the nonexistence of God. It's just mathematics. Read up on Cox's theorems.

I believe you are making the common mistake of thinking that assigning a Bayesian probability of 0.5 means much more than it actually does. See here for the three most common objections and why they are wrong, or tell me your objection that's not one of those three, or tell me why one of the reasons the objections are wrong is wrong.

 

[RandFan]

To say it's not 0.5 because the probability God exists is not 0.5 is also ridiculous because in the case of the God question we have plenty of knowledge pointing to the nonexistence of God.

I can't parse this sentence. You seem to be contradicting yourself. We have evidence there is no god and no evidence that there is a god therefore it's ridiculous to say the probability is less than 0.5?

It's just mathematics. Read up on Cox's theorems.

It might just be mathematics but I can't make heads or tales of your statement. I'm currently reading Innumeracy by Paulos who I believe takes on your argument in Irreligion. I've ordered that book.

See here for the three most common objections...

Given zero information I couldn't assign one selection a greater value than another. I also wouldn't assume that it were likely a 50/50 (the world isn't filled with 50/50 propositions) only that my best bet given no information is 50/50. I know it's unintuitive, but then again, it's just mathematics.

In your above post you have taken my statement out of context but I concede that the problem is in part mine. If we have a binary with knowns (evidence that there is no god) and unknowns (no evidence for god) then to hypothesize about the unknown is irrational.

"Bayes's theorem provides a model of learning from experience,"

Given that I do know that the idea of god has been been looked at nine ways to Sunday I know that there is no information to suppose a god and there is plenty of information to suppose that there is no god.

 

[Ivor the Engineer]

Given H = 'God exists and created the universe' and E = 'Our knowledge of the universe', what initial estimates are reasonable for the application of Bayes rule?

IMO, P(H) = P(~H) = 0.5 is the only reasonable estimate for the upper two branches assuming something like a deistic god. We also know nothing about how a god, assuming one exists, decides to (or not to) create a universe, so P(E|H) = P(~E|H) = 0.5 also seem to be the only reasonable estimate. This leaves P(E|~H) to be estimated to allow us to be able to calculate P(H|E).

P(H|E) = P(E|H)*P(H) / [P(E|H)*P(H) + P(E|~H)*P(~H)]

P(H|E) = 0.5 * 0.5 / [0.5 * 0.5 + P(E|~H) * 0.5]

P(H|E) = 0.5 / [0.5 + P(E|~H)]

P(H|E) vs. P(E|~H) is plotted in one of my previous posts.

As for the logical contradictions the definition of God often raises, aren't most if not all of these removed if one assumes our universe is a simulation and God to be a programmer who can rearrange anything in the simulation in (as far as we are concerned) zero time?

 

[RandFan]

What is more likely?

1. A exists.
2. A does not exist.

Given that there are a finite number of things that exist and an infinite number of things that don't exist (like a blind purple people eater living in my basement) then statement #2 is statistically more likely (FTR: I don't have a basement).

The mistake being made is that, from a statistical standpoint, a proposition can be viewed as 50/50 only if you know nothing of the premises.


What is more likely?

1. A is B
2. A is not B.

A polar bear is green with yellow polka dots is false.

Since there are an infinite number of things that A is not then it is more likely that A is not B.

 

[fls]

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.

Right. As Malerin has amply demonstrated, the mathematics can be applied to any damned thing. I was speaking as to the choice of an appropriate approach in order to provide a useful answer to a question.

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."

Yes, something needs to be precise - the prior, the posterior, or the likelihood ratio. If none of them are, it's the wrong tool. Which is why it's the wrong answer for both Randfan's question and Malerin's argument.

Linda

 

[fls]

The idea of agnosticism is related to the idea of indifference - that two things are different in name only. In order to determine whether you truly think the choice is binary or whether you don't know which is which consider the following.

I have formed a bunch of paired sets:

One is the set of all closets with a million dollars in them, one is the set of all closets without a million dollars in them.
One is the set of all universes with God, one is the set of all universes without God
One is the set of all coin tosses that came up "heads" from a fair, two-sided coin, one is the set of all coin tosses that came up "tails" from a fair, two-sided coin.

Unfortunately, I forgot to label these sets. Your task is to put labels on them for me. If you think you can figure out which label goes on which set, or if you care which label goes on which set, you are not agnostic.

Anyone who chooses the set which is much, much smaller as the one to label "closets with a million dollars" is not agnostic about closets and money.

Anyone who thinks or cares that whether or not the set of universes with a creative or controlling force acting (somewhat) on their behalf is the one labeled God, cannot claim to be agnostic about God.

Linda

 

[Ichneumonwasp]

Do we really need a formal mathematical way of saying that "God did it" solves problems but explains nothing? I thought we already knew that.

 

[Ivor the Engineer]

<snip>

Unfortunately, I forgot to label these sets.

<snip>

I don't believe you.

 

[Ivor the Engineer]

Given that there are...an infinite number of things that don't exist...

Now that's a hard statement to parse.

 

[Ichneumonwasp]

Anyone who thinks or cares that whether or not the set of universes with a creative or controlling force acting (somewhat) on their behalf is the one labeled God, cannot claim to be agnostic about God.

Linda

Good luck with that.

I tried that approach initially -- that if we want to arrive at any real estimate of the probability of God, instead of whether or not life adds anything to the picture, by pointing out that "God" either means substance dualism or is true by definition. I'm not sure how one could assign a value for the probability of substance dualism, but it's got to be nearly zero. Sure, it's logically possible, but it can only work by means of magic.

I don't think we even need to go there, though, since the chance of life existing sometime and somewhere given the existence of one universe that came into existence without obvious cause and with infinite time and space to work with is 1. Without constraints on the number of possible universes, there will necessarily be an infinite number of them. So, the existence of life provides no useful information in deciding the existence of a designer. If we want to assign 0.5 (maximum ignorance), then we are still stuck with maximum ignorance.

 

[Robin]

Here is how I see it.

I find a device with a blank LCD screen and one button marked "Random Number". I guess that it will display a random number on the screen, but I have no idea what set of values they will come from. It might be a sort of electronic coin tosser and give only 0 or 1, or it might be an electronic dice and have a range from 1 to 6. Or it might be for generating random numbers for cryptographic purposes, or selecting from a range of numbers for a remote password.

I don't even know what size the numbers will be so that I can get an idea of the sort of range they will be in.

But it would be silly then to ask "is the number going to be a one?" and on the basis that the response will be a binary "yes/no" assign a probability of 0.5 to each.

I would simply have no basis on which to assign any sort of probability.

 

[fls]

Here is how I see it.

I find a device with a blank LCD screen and one button marked "Random Number". I guess that it will display a random number on the screen, but I have no idea what set of values they will come from. It might be a sort of electronic coin tosser and give only 0 or 1, or it might be an electronic dice and have a range from 1 to 6. Or it might be for generating random numbers for cryptographic purposes, or selecting from a range of numbers for a remote password.

I don't even know what size the numbers will be so that I can get an idea of the sort of range they will be in.

But it would be silly then to ask "is the number going to be a one?" and on the basis that the response will be a binary "yes/no" assign a probability of 0.5 to each.

I would simply have no basis on which to assign any sort of probability.

Could you answer the question, "should I assign a probability of 0.5?"

Linda

 

[drkitten]

But don't you have information about the world in general? Don't you know that most binary choices are not really 50/50? Don't you know about statistics and deviation from the mean?

Better than you, apparently.

The 50/50 is, as has been amply demonstrated, the agnostic probability. The one you hold if you have no information whatsoever.

It's a starting point, not an ending point.

If you have information, you feed it into the starting (prior) probability to get a revised (posterior) probability and you end up with an informed probability that is almost certainly not 50/50.

You sound like the sort of person who insists on eating the cake mix, because you don't understand the difference between the cake mix (a starting point) and the cake (an ending point).

 

[Ichneumonwasp]

Here is how I see it.

I find a device with a blank LCD screen and one button marked "Random Number". I guess that it will display a random number on the screen, but I have no idea what set of values they will come from. It might be a sort of electronic coin tosser and give only 0 or 1, or it might be an electronic dice and have a range from 1 to 6. Or it might be for generating random numbers for cryptographic purposes, or selecting from a range of numbers for a remote password.

I don't even know what size the numbers will be so that I can get an idea of the sort of range they will be in.

But it would be silly then to ask "is the number going to be a one?" and on the basis that the response will be a binary "yes/no" assign a probability of 0.5 to each.

I would simply have no basis on which to assign any sort of probability.

But isn't the point of all this that if you don't know, then you assign a probability of 0.5 (it either is or is not a 1) and then perform some further modification so that you end up with a better idea of it being a 1 or not? If there is no information at all, then can you even use Bayes' theorem?

 

[drkitten]

But isn't the point of all this that if you don't know, then you assign a probability of 0.5 (it either is or is not a 1) and then perform some further modification so that you end up with a better idea of it being a 1 or not? If there is no information at all, then can you even use Bayes' theorem?

Yup. It's astonishing how many people in this thread seem to think that you simply say "yes, it's a 50/50 chance of being a 1" and then walk away.

 

[fls]

Better than you, apparently.

The 50/50 is, as has been amply demonstrated, the agnostic probability. The one you hold if you have no information whatsoever.

It's a starting point, not an ending point.

If you have information, you feed it into the starting (prior) probability to get a revised (posterior) probability and you end up with an informed probability that is almost certainly not 50/50.

You sound like the sort of person who insists on eating the cake mix, because you don't understand the difference between the cake mix (a starting point) and the cake (an ending point).

I think the problem is that the issue of agnosticism as p=0.5 is almost never raised under situations where one is truly agnostic. For one thing, we are almost never really agnostic - we bring some knowledge about the way the world works to our presuppositions. Even Robin's example of an LCD and a 'random' button wasn't really agnostic, since she/he brings knowledge to the situation about how many numbers there are that are not '1'. We also recognize that there are many options other than 'binary', which inhibits us from agreeing that a scenario with no information should be assumed to be binary. And most importantly, it's almost always brought up here when we're talking about God, and it's pretty clear that even people who claim to be agnostic about God are actually acting upon prior knowledge.

I can't even remember a situation here where this advice would have been relevant. Not that that ever seems to hold people back.

Linda

 

[rocketdodger]

Here is how I see it.

I find a device with a blank LCD screen and one button marked "Random Number". I guess that it will display a random number on the screen, but I have no idea what set of values they will come from. It might be a sort of electronic coin tosser and give only 0 or 1, or it might be an electronic dice and have a range from 1 to 6. Or it might be for generating random numbers for cryptographic purposes, or selecting from a range of numbers for a remote password.

I don't even know what size the numbers will be so that I can get an idea of the sort of range they will be in.

But it would be silly then to ask "is the number going to be a one?" and on the basis that the response will be a binary "yes/no" assign a probability of 0.5 to each.

I would simply have no basis on which to assign any sort of probability.

This is equivalent to my number example.

My contention is that, were you to execute a single trial and receive a 1, it would be valid to estimate the posterior probability of getting a 1 to be 1.0. If we executed more trials and observed a different frequency of 1 occuring then it would not be valid -- but we didn't, we only had a single trial.

Is your contention that we shouldn't even do that? This seems to be what drkitten is saying as well. But it doesn't seem graceful to me, to have special cases where probability analysis is invalid. If nothing else, statistics should still be valid albeit useless in such cases.

 

[rocketdodger]

What is more likely?

1. A exists.
2. A does not exist.

Given that there are a finite number of things that exist and an infinite number of things that don't exist (like a blind purple people eater living in my basement) then statement #2 is statistically more likely (FTR: I don't have a basement).

The mistake being made is that, from a statistical standpoint, a proposition can be viewed as 50/50 only if you know nothing of the premises.


What is more likely?

1. A is B
2. A is not B.

A polar bear is green with yellow polka dots is false.

Since there are an infinite number of things that A is not then it is more likely that A is not B.

Yep, this is my sentiment as well. This is also what fls is talking about.

The only question is, are we right?

drkitten and greedyalgorithm are pretty darn smart so if they don't agree then I am not going to be confident in this lol.

EDIT -- upon more thought, maybe this is what they mean by "agnostic?" As in, our knowledge of the existence of things cannot be brought to bare if we are truly agnostic? And in that case 0.5 is the optimal choice since we don't even know what the difference between a binary and "existential" proposition is?

 

[Ichneumonwasp]

I'm no mathematician and I have exactly zero training in using Bayes' theorem, as will be immediately evident from the question I will now ask:


Bur first the preamble --
Let's say that we actually wanted to use Bayes' theorem to decide if there is or is not an intelligent designer or God. I imagine we would have to run numbers, using various different conditions each time, through the equations numerous times in order to "pare down to" a reasonable guesstimate.

It's silly (certainly not rational) for anyone to use a single condition/example to arrive at the actual likelihood of something as complex as an intelligent designer or God. Even if we wanted to restrict ourselves, as in the original example, to a single universe (as you can tell I didn't want to play that game because I think it unusually restrictive), there are many other considerations when it comes to God or an intelligent designer. How could anyone possibly decide that we are better off believing in God or an intelligent designer on the basis of a single characteristic anyway?

So, let's say we begin with a prior of 0.5 -- true agnosticism -- and use whatever estimate of life's existence with and without an intelligent designer and arrive at a posterior probability of whatever -- say for the purpose of the example that it turns out 0.9 or something on that order.

Here's the question now:

I can see two different ways of asking the next question about this intelligent designer -- we could do a series of runs through Bayes all with a prior probability of 0.5 and average the outputs. Or we could take the new posterior probility and insert it into the equation as a prior probability (with a little more info) and run it through again -- say in this second run we take into account the probability of this intelligent designer existing as an immaterial entity.

My gut tells me we just keep using the posterior probabilities as new prior probabilities and refine our knowledge each time we apply Bayes. Is that correct?