Thursday, January 06, 2011

Richard Carrier on Bayes' Theorem

I don't know if Tim McGrew is going to call this crappy. But he has told me he has numerous objections to it.

129 comments:

Tim said...

Everyone interested in the question of Richard's competence with Bayesian mathematics should download a copy of his pdf file, linked from the title of Vic's post. Come back here and indicate with a quick comment once you've got your copy.

Paul said...

Got it.

mattghg said...

OK Tim I've got it. Not that I'm in a position to assess Carrier's credentials but I'm interested in what other commentators have to say.

Victor Reppert said...

I have it downloaded.

Mr Veale said...

Got it Tim.

Mr Veale said...

Tim

I've posted a copy to David at University of Ulster. We doubt he'll be able join in the discussion here, but we're planning on putting something accessible on the web on the ex-apologists and Bayes Theorem. (and other issues). So we'll enjoy this thread.

Graham

Jason Pratt said...

Got it (though I don't know when I'll get to read it).

I should point out that I'm on record around here as being one of those people who thinks it's entirely wrong to treat Bayes' Theorem as a mathematical formula, plugging in percentages and trying to come up with a numerical result (such as by dividing various elements by each other).

I don't recall offhand whether Tim and/or Richard are the sort who treat it as a math problem with elements to be solved by plugging numbers into variables. (I'm sure I ought to remember, but it's been too many years since I discussed things with either of you on this topic. Sorry. {g}) I just thought I should state my position up front.

JRP

Jason Pratt said...

Crap... adding another comment because I forgot to check the comment tracking box the first time... {wry g}

Mr Veale said...

Not to jump the gun, but often numbers are put in for sake of illustration when the author is talking about a low epistemic probability.

Swinburne does this,for example, and is often misunderstood on internet discussions. I find that a bit frustrating.

Graham

Victor Reppert said...

Yes, you get this nonsense about "You think you can prove that the probability of the resurrection is 94%, isn't that ridiculous?", taking the claim out of context.

Duke of Earl said...

Got it.

Mr Veale said...

I'm half way through the piece.
O my goodness!

Jason Pratt said...

Graham,

Swineburne has only himself to blame for that, since he promotes the results as numerical percentages in a popular fashion. Yes, I know he knows it's wrong to do so (I have his original book on the topic). But he does it anyway.


For what it's worth, I'm pretty sure I overstated my rejection earlier; mainly because I keep forgetting that the theorem isn't exclusive to describing how human inductive reasoning collates evidence. When used in regard to actual probabilistic systems (like randomly rolling a six-sided dice), it works well enough as a math operation. (Maybe... it's been a few years since I went through all this in depth. But that fits with what Bayes was originally doing, using a ball falling onto a point on a sheet and rolling to rest to analogically describe human induction vs. whatever it was that Hume had proposed.)


I'm not adamantly married to being right on this point, btw. {g} If I learn better this time the discussion goes round, no problem. One of the reasons I like to pay attention when Victor brings up the topic, is out of continuing curiosity to figure out whether and where I'm making a huge mistake that would then allow me to line up with the majority of practicing scholars on proper Bayesian usage for estimating "probability" of historical hypotheses.

JRP

Mr Veale said...

Jason
As much as I love Swinburne's work, I don't think that he has a gift for popularisation. So maybe that does account for some misunderstanding.

I don't get the impression that many philosophers of history see the advantages in Bayesian reasoning. I think that's a shame.

Graham

Mr Veale said...

Just reading Carrier on how the criteria of authenticity are used..oh dear, oh dear, oh dear! I'm stunned. Absolutely stunned. The hubris is unspeakable.

Greg Welty said...

"Swineburne has only himself to blame for that, since he promotes the results as numerical percentages in a popular fashion. Yes, I know he knows it's wrong to do so (I have his original book on the topic). But he does it anyway."

You have evidence for this claim? He explains what he is doing in nearly every published treatment of the matter. He doesn't 'promote the numerical percentages'; rather, he stresses the opposite.

Swinburne, _The Resurrection of God Incarnate_, p. 215:

"Now I stress again that we cannot really give exact values to these probabilities, nor to analogous probabilities in science or history. You cannot give an exact value to the probability that quantum theory is true or to the probability that King Arthur lived at Glastonbury. But we can conclude that these things are probable, or not very probable on the basis of other things being very probable, or not very probable or most unlikely; and that is all I am doing here. But I am giving artificial numbers to capture the 'not very probable' etc. characters of these other things to bring out what is at stake."

Blue Devil Knight said...

Jason wrote:
"I'm on record around here as being one of those people who thinks it's entirely wrong to treat Bayes' Theorem as a mathematical formula"

Ummm, sorry, that's what Bayes' Theorem is, it's a theorem from probability theory. Are you saying that it's a mistake to use it for historical analysis?

Seems Carrier was unclear on this distinction too.

Anonymous said...

I've got it, Tim.

Tim said...

Okay, several people have indicated that they have downloaded their own copies.

Let's have a little fun.

First, flip to p. 26. (I know, we could start at the beginning. I'll get there.) Check out the calculation about "Mark" under numeral 4 -- the one that begins "... for example, if you picked 10 people ..." and ends, "This is the kind of statistical fallacy you need to be aware of if you decide to employ statistical logic in your historical method."

There's a cookie for the first person who can explain why this calculation, winding up with "87%," is completely bogus; bonus cookie for the first person to give the proper calculation. (Hint: remember nCr from basic statistics?)

Tim said...

Please note -- to receive the prize, you must have your browser set to accept cookies ... :)

Jayman said...

Mr. Veale, I know what you mean about the hubris. From what I've read from him, Carrier always seems to need to demean his opponents and act as if he is most intelligent man in the world. It makes it very hard to read his work even when I think he might have something reasonable to say.

Duke of Earl said...

Okay, in the binomial coefficient equations.

10!/(3!x7!) = 120.

Thinking
Thinking

120(0.25^3)(0.75^7)=0.25

So the probability that 3 people in a group of ten are named Mark where 25% of the population is named Mark is 0.25

I won't call it 25% because probabilities are not presented in percentages.

Tim said...

A cookie for Duke! Two cookies, in fact! (Is your browser cookie-enabled, Duke?)

Extra credit -- and Duke, perhaps you should just eat your cookies and let someone else have a crack at it -- to what question would Carrier's calculation yield the right answer?

Blue Devil Knight said...

He needs to team up with a mathematician, or a philosopher who specializes in probabilistic inference/probability theory. Who is familiar with common mistakes, of which there are many. I can barely bear to watch.

Tim said...

Don't touch that dial, BDK -- we're just getting warmed up.

Duke of Earl said...

sitting eating his cookies

I trained as an Engineer, so this sort of thing should have been my meat and drink.

Unfortunately I never found a job in that area.

Tim said...

Think of it this way, Duke -- you've just demonstrated that you're overqualified to travel around the US giving lectures on the use of probability to groups of awe-struck atheists.

Duke of Earl said...

Well I once considered doing a degree in philosophy, studying every argument for atheism, and then going around debating Christian apologists.

Not because I expected to win, but at least those apologists wouldn't have to put up with temper tantrums and veiled (and not so veiled) insults.

Blue Devil Knight said...

LOL I guess I'll print it out at work tomorrow.

Tim said...

While a few eager readers are scratching their heads over the extra credit question, let's turn back to p. 3 in Carrier's file, where he tries to explain the meaning of the terms in Bayes's Theorem. There are several issues here, but let's start with a conceptual question. Carrier offers this definition:

~h = all other hypotheses that could explain the same evidence (if h is false)

Question: what role does the phrase "could explain the same evidence" play in the definition of ~h? [Warning: this is a trick question.]

David said...

Lord I am sorry for abandoning the intellectual life after many tedious hours in science and concluding that adventures and loving were better than hard thinking. Now that I have an 18 year old son reading and questioning everything, I am thankful for downloads and blogs like this.

Brenda said...

In the Carrier paper he identifies the background as one variable in Bayesian reasoning but he doesn't really mention it after that. The Background, as I understand it and as John Searle defines it is: the set capabilities, tendencies, habits and dispositions that enable us to cope with the world.

And, and this is important, the Background precedes intentionality. So when Carrier says that "Community X would never make up the embarrassing story about Y being castrated" how does he know? He assumes it must be so as a part of his background. But Carrier does not share the same background as community X from 2000 years ago. He has no way of knowing what they would or would not have done.

In order to know or hypothesize what some person A would *not* have done you both need to share the same background. Even today among contemporary cultures the people of culture A often cannot understand the motivations behind the behavior of culture B.

It's really dangerous to assume that people of other cultures and other times thought and acted the same as you.

Anonymous said...

to what question would Carrier's calculation yield the right answer?

meh, probability was always the branch of mathematics with which I was least comfortable (more of a calculus aficionado myself), but w/e...I guess I'll try:


Working backwards.

Carrier's calculation 1-(.75^7)=.86 takes the form of the basic probability formula, 1-P(A)=P(not-A). So in Carrier's example, P(A)= .75^7

.75^7 is the probability that the first 7 people you meet successively are not named Mark (in much the same way as how P(first seven coin flips being heads)=.5^7) .

Hence, P(not-A)= the probability that it is not the case that the first 7 people you meet successively are not named Mark.


So Dr. Carrier should have asked something along the lines of, "What is the probability that it is not the case that the first 7 people you meet successively are not named Mark?"


Cookie?

(or at least a crumb?)

Brenda said...

"Question: what role does the phrase "could explain the same evidence" play in the definition of ~h? "

Strawman? How could anyone know ~h? Isn't that potentially infinite? One of the reasons that AI researchers have had so little success at programming natural language AI is because the Background assumptions needed for a functional AI grow exponentially. One simply cannot list every possible thing an AI should *not* say given condition X.

Anonymous said...

I don't get it...why would this Carrier bloke publish an entire book on Bayes' Theorem when he obviously doesn't have even a rudimentary grasp of basic probability? Any elementary probability text would've rectified the errors.

I, for one, would take painstaking measures to acquaint myself with all of the relevant literature on a topic - you know, to actually try to know what I'm talking about - before I write a book and send my name off into the public realm. Guess that's considered old-fashioned in these dark days of Dawkins, Hitchens, Harris, et al.

WAR_ON_ERROR said...

@Brenda:

You make a good point that we can't be sure Carrier has construed that bit of background knowledge accurately, however when you say this:

"It's really dangerous to assume that people of other cultures and other times thought and acted the same as you."

Doesn't that cut both ways and end up justifying Carrier's point anyway?

Ben

WAR_ON_ERROR said...

@everyone:

As I understand it, Carrier has had help from mathematicians on his chapter at least. I don't know if that applies to the PDF that's been linked to or not.

I can see that there's some ego battle between atheists and Christian intellectuals here on who really knows math and who has the biggest PhD, but for those of us concerned with who is correct on the actual issues (in things outside our realm of knowledge), I don't see anything here in the comments that is that helpful. That's not a jab, it's just a plea to actually try to explain the issue in non-math expert friendly terms if possible. I'm sure there are both atheists and Christians who would appreciate that.

Ben

Brenda said...

"Doesn't that cut both ways"

Yes it does. That's why it's safer to be in the middle.

WAR_ON_ERROR said...

@Brenda:

Well I don't think Carrier is arguing that it is highly probable that cults are likely to invent embarrassing stories, so it would appear that his position is already the middle?

Ben

Mr Veale said...

" ~h = all other hypotheses that could explain the same evidence (if h is false)"



~h includes hypotheses that lower the probability of the evidence.

(Am I missing your point, Tim?)

Graham

Mr Veale said...

~h is just all the hypotheses that aren't h. That's it. You consider them before you consider the evidence.

That information is concealed in that subtle little word prior

(I think you explicitly say a bit about ~h in the Blackwell paper, Tim, which makes me wonder if Carrier has even read it.)

Graham

mattghg said...

Question: what role does the phrase "could explain the same evidence" play in the definition of ~h? [Warning: this is a trick question.]

Um, is the answer: no role at all? The definition of ~h should just be 'h is false', right?

Tim said...

Anonymous writes:

So Dr. Carrier should have asked something along the lines of, "What is the probability that it is not the case that the first 7 people you meet successively are not named Mark?"

Cookie?


Yes indeed! Please note that this has absolutely nothing to do with ten guys or with three guys: it's all about seven, namely (to rephrase Anon's version) that if you meet seven guys, at least one of them will be named Mark.

Enjoy your cookie!

Tim said...

Graham writes:

~h includes hypotheses that lower the probability of the evidence.

And mattghg writes:

Um, is the answer: no role at all? The definition of ~h should just be 'h is false', right?

A cookie for each of you!

Stay tuned for more great prizes ...

Tim said...

Ben writes:

I can see that there's some ego battle between atheists and Christian intellectuals here on who really knows math and who has the biggest PhD, but for those of us concerned with who is correct on the actual issues (in things outside our realm of knowledge), I don't see anything here in the comments that is that helpful. That's not a jab, it's just a plea to actually try to explain the issue in non-math expert friendly terms if possible. I'm sure there are both atheists and Christians who would appreciate that.

It has nothing to do with the size (?!) of anyone's Ph. D.

In very simple, non-math-expert terms: We are showing, piece by piece -- and we are by no means done yet -- that Richard Carrier is completely out of his depth with respect to the mathematics of elementary probability. He garbles the explanation of elementary concepts, and he fumbles the computation of his own chosen examples.

This is not a matter of scholarly disagreement over the interpretation of some bit of evidence or the relative merits of two competing hypotheses; it is not a wrangle over the preferability of two possible translations of a bit of Greek; it is not a clash of worldviews. It is much simpler than that. Carrier has not crossed the pons asinorum of elementary probability.

Now, there is no shame in this as such. Many fine people, including many good historians, have not mastered the basic mathematics of probability. But for someone who poses as an expert and describes as "crappy," in its use of mathematics, a peer-reviewed article by people who actually do understand what they are doing mathematically, a demonstrable failure to understand extremely basic aspects of probabilistic reasoning must be -- how to put this? -- an inconvenience.

That's all.

Tim said...

Brenda,

Searle is talking about "background" in a completely different sense; he's not dealing with the same term. They're just homonyms here.

Tim said...

There are other problems with p. 3, but since Mr. Veale and mattghg have answered the previous question regarding the definition of "~h," let's pursue that theme a bit. On p. 4, Carrier gives the following definition:

P(~h|b) = 1 – P(h|b) = the prior probability that h is false = the sum of the prior probabilities of all alternative explanations of the same evidence (e.g. if there is only one viable alternative, this means the prior probability of all other theories is vanishingly small, i.e. substantially less than 1%, so that P(~h|b) is the prior probability of the one viable competing hypothesis.

I'm running short on cookies, but I do have a few candy canes and chocolate coins left over from Christmas. So, for a free dip into my Christmas candy: what is wrong with the explanation being offered here?

Patrick Chan said...

Thanks, I've downloaded it!

Jason Pratt said...

BDK,

No, I mean that Bayesian Theory was intended (by Bayes himself) to describe the process of adjusting an inductively inferred expectation of (H)ypothesis being true, according to the arrival of new evidence pro or con and in regard to whatever intuitive weight the thinker has assigned to the new evidence.

It is not really supposed to be a mathematical operation; Bayes constructed it by analogy with math operations, but it isn't really supposed to be about multiplying this or dividing that. My complaint isn't primarily about assigning fractional numerical values to the elements; it's about trying to use the Theorem as a math operation at all.

I don't want to distract from Tim's critiques, however; I only wanted to register my (rather radical) position in case Tim and/or Richard happened to be going the same route (now if not years earlier). They aren't--they both treat the Theorem as an operation to be mathematically resolved for purposes of estimating historical "probability"--so I'll be quiet while they crit each other on that procedure back and forth.


Greg,

Yes I know Swinburne qualifies himself in his published scholarly work (as I already alluded to). It's when he goes out on the road in popular apologetics and releases public relation material for news services (etc.) to run articles with, that he oversimplifies the matter into ridiculousness. The first time I heard about it years ago (from my friend, author and fellow Christian apologist David Marshall), I seriously thought he or maybe Swinburne had to be making some kind of satirical joke about popular apologetics through news articles.

JRP

Andrew said...

Downloaded, now what?

Doug said...

Tim asked: "what is wrong with the explanation being offered here?"

I'll bite: "nothing."

Tim said...

Patrick and Andrew: see above for the first few questions (and answers) regarding this document.

Doug: Fail! Anyone else? [Hint: does viability have anything to do with P(~h|b)? If sub-hypotheses under ~h have non-zero probability given b, even though that probability is low, do they still contribute to P(~h|b)?]

Mike Darus said...

P(~h|b) = 1 – P(h|b) simply means there is a 100% chance one of the two is correct. Assigning "viability" to one or the other simply exposes your priors.

Be nice to me. I have no background in this.

Tim said...

Mike,

You're in the zone -- have a peppermint -- but there's something more direct to be said. Every sub-hypothesis under ~h that has a non-zero prior given b contributes to P(~h|b). So to say that if

the prior probability of all other theories is vanishingly small, i.e. substantially less than 1%,

then

P(~h|b) is the prior probability of the one viable competing hypothesis

is just mathematically wrong.

Blue Devil Knight said...

Jason: Bayes' Theorem is a mathematical expression, and it would be a mistake to criticize it for being a formula. YOu seem to have confused Theorem (a technical term from math) and 'Theory' (as in Bayesianism or something). Just dont' use the word Theorem and you will be safe :O

Tim: Frankly I would stop after the first page, as it is clearly an idiosyncratic approach to Bayes.

Peter Bruin said...

P(~h|b) = 1 – P(h|b) = the prior probability that h is false = the sum of the prior probabilities of all alternative explanations of the same evidence (e.g. if there is only one viable alternative, this means the prior probability of all other theories is vanishingly small, i.e. substantially less than 1%, so that P(~h|b) is the prior probability of the one viable competing hypothesis.

Yes, P(~h|b) is the a priori probability of h being false, but in the same sentence he implies that this is somehow related to the evidence, which completely misses the point of a priori probabilities.

Steven Carr said...

TIM
There's a cookie for the first person who can explain why this calculation, winding up with "87%," is completely bogus; bonus cookie for the first person to give the proper calculation

CARR
That is a really bad misunderstanding of binomial probability distribution.

I have no idea what sum Carrier was trying to work out. At first I thought it was 'at least 3 people named Mark', but it isn't that either.

Tim said...

Steven,

We have found something on which we agree!!

(Tim pops a champagne cork and pours Steven a glass.)

Doug said...

Hi Tim,

I claim that you're just picking nits on the viability thing. "Viability" is just Carrier's shorthand for "prior probability greater than zero". Moreover, his "much less than 1%" is just an attempt to be practical. If the contribution to the prior of the complementary subset of hypotheses is "much less than 1%" he is quite correct that for all intents and purposes, the remaining alternatives represent p(~h).

Tim said...

Peter writes:

Yes, P(~h|b) is the a priori probability of h being false, but in the same sentence he implies that this is somehow related to the evidence, which completely misses the point of a priori probabilities.

That is at least a reasonable attempt to understand what he means by the phrase

... all alternative explanations of the same evidence ...

This goes back to the initial point that Mr. Veale and mattghg made with regard to his definition of "~h" on p. 3 (regarding the phrase "all other hypotheses that could explain the same evidence").

So -- a cookie for you too!

Tim said...

Doug,

There's this nifty little symbol that looks like a wiggly equals sign: it means "is approximately equal to." If you mean that, and you know what you're doing, you use that or state in so many words that you're giving an approximation.

Note that this occurs in the section entitled "Explanation of the Terms in Bayes’ Theorem," not in a section entitled "Some Useful Approximations to be Used with Care by People who Know What They are Doing and are Sure that the Approximation Won't Screw Things Up."

Tim said...

Doug,

Apparently one of Richard's mathematically literate reviewers also noticed this problem, since in the version of this paper that appears in Sources of the Jesus Tradition, the definition, given on the bottom of p. 98, has been reworded to remove the language of "viability." It now reads:

... the sum of the prior probabilities of all alternative explanations of the same evidence, which is always the mathematical converse of the prior probability that his true (so P( ~ h|b) and P(h|b) must always sum to 1).

Of course, we still have the awkwardness of speaking of alternative explanations. But it is an improvement.

Mr Veale said...

Durn. I can't access this blog from my Work Computer, and someone got there before me! But I typed some of this up earlier for posting this evening, and I want my candy cane!

Problems:

It's not the sum of prior probabilities of all alternative explanations of the same evidence . We've already mentioned this on the thread, but some of the hypotheses under ~h don't raise (and some lower) the probability of 'e'

P(~h|b) cannot be both 1 – P(h|b) and the sole alternative explanation minus the sum of the priors of hypotheses with vanishingly small probabilities. That is self-evidently incoherent!

To illustrate the problems:(A) suppose can have two alternative hypotheses to which you grant some significant prior probability (say 0.25 and 0.2) ;the rest of the space need not be taken up with a set of hypotheses with "vanishingly small" priors. There could be a hypothesis with a higher prior probability that renders the evidence "vanishingly small". So it doesn't count as a "main alternative", even though a priori its probability is higher than the viable hypotheses.

(B) Suppose the police have a large pool of suspects for a particular crime - 52 hit men in town for a convention. Say they grant a prior probability of 0.005 to each one of 50 of those hit men(they are garrotte men, and the victim was poisoned.)
The remaining two hit men are poisoners and are a prior of 0.25 and 0.5 (for the sake of argument we'll assume that it had to be one of the hit men at the convention.)
But suppose that the evidence for one of the garrotters turns out to be so good it renders 'e' to be 0.9999999999. Suppose that the two poisoners at the conference have alibis that are so solid, along with other exculpating evidence, that they only give 'e' a probability of 0.0000000000000001. Then we update the prior probabilities, and find that those hypotheses with priors less than one should not have been dismissed....

A & B show that P(~h|b) does not fit Carrier's description, and that failing to recognise this can lead to significant errors.

Given that the burden of the McGrew's article was to show that p(e|H) can overwhelm even vanishingly small probabilities, and given that the McGrew's article also talks about the hypotheses covered by ~h, I can only conclude that Carrier has not even bothered to read the article he dismissed.

Graham

Bobcat said...

My post above was supposed to have snark and sarcasm brackets attached. Oh well, I trust they were obvious.

WAR_ON_ERROR said...

@Tim:

The "size of one's PhD" was a non-literal reference to a convoluted academic pomp war. I pointed out that Carrier says he's had his stuff vetted by qualified people who generally approved of it. Are you contesting that?

Nitpicking what is meant to be an intro to a difficult math subject as though even textbooks don't have remedial errors simply doesn't prove your point, even if you come up with a few more examples. If you come up with something valid, Carrier will just correct the text (and it looks like you are critquing an older version that's already had revisions anyway?). You aren't going to refute Bayes' theorem or its application to history as I'm sure you agree.

Ben

WAR_ON_ERROR said...

@Bobcat:

Carrier told me in an email that he no longer believes that Lydia had any deliberate intention to mislead her readers. So yes, he is capable of mutual interpretative charity.

Ben

Tim said...

Bobcat,

I think I found your tags:

<sarc> ... </sarc>

and

<snark> ... </snark>

:)

James Anderson said...

Ben,

So Carrier's revised position is that the McGrews still misled their readers, but did so unintentionally?

I guess we should be grateful for small mercies, but that still doesn't square with the two paragraphs from the essay that Lydia reproduces here.

Tim said...

Graham,

Almost perfect: for

p(e|H)

put

the ratio P(E|H)/P(E|~H).

I certainly agree that Carrier has not publicly shown any evidence of having read our paper. Given the errors in his own tutorial, I have grave doubts that he is able to read it with understanding.

And you're in luck -- a fresh batch of cookies just came out of the oven! Here, have two while they're warm.

Tim said...

Ben,

I pointed out that Carrier says he's had his stuff vetted by qualified people who generally approved of it. Are you contesting that?

I'm contesting that there has been any competent vetting of his online version of his own work -- the only version available for the general public, and a version that is referred to in the footnotes to his published articles. (See footnote 14 on p. 100 of Sources of the Jesus Tradition, where he is still advertising it as "a tutorial in using Bayes's Theorem.")

But you're missing the central point. Anyone, even a complete mathematical dunce, can have his work cleaned up to some extent by competent people coming along and correcting his errors. The question is whether this person should, in propria persona, put himself forward as competent to criticize the mathematics in peer-reviewed professional work and to offer tutorials on Bayes's Theorem to others.

Victor Reppert said...

OK, I'm going to have a little fun here too. This is my blog, and so he doesn't get to have all the fun.

RC: 2. Any argument you make, read, or hear can be modeled with Bayes’ Theorem (if the argument is valid and sound, although if it isn’t, Bayesian modeling will also expose that fact, too). So to practice and thereby develop more understanding of Bayes’ Theorem, a good plan is to develop a Bayesian analysis of any empirical argument you find
convincing (or unconvincing) in order to see how that everyday argument would be represented correctly in Bayesian terms. With enough practice at reverse-engineering standard arguments into Bayesian form, you will be better equipped to directly engineer Bayesian arguments from the ground up.

VR: Why am I doing a facepalm here?

Victor Reppert said...

I meant "Tim doesn't get to have all the fun."

Mr Veale said...

It's worth pointing out that these aren't minor errors. And that I have a phobia of maths and can spot that there are major errors in Carrier's presentation.

My friends would find this hilarious.

Graham

steve said...

WAR_ON_ERROR said...

"Carrier told me in an email that he no longer believes that Lydia had any deliberate intention to mislead her readers. So yes, he is capable of mutual interpretative charity."

After the fact. After his hand is forced.

Tim said...

VR: Why am I doing a facepalm here?

I don't want to steal anyone else's cookies, but might it have something to do with the fact that you've taught logic a time or two and know what, er, certain terms actually mean?

Mr Veale said...

Any argument you make, read, or hear can be modeled with Bayes’ Theorem (if the argument is valid and sound, although if it isn’t, Bayesian modeling will also expose that fact, too).

If it is valid and sound the probability is 1. We don't really need to see what extra evidence does to the argument. The point being that Bayes Rule model inductive, not deductive reasoning.

Let's see Richard model this using Bayes Rule.

1. (A → C) • ~D (premise)
2. (B • A) • C (premise)
3. A → C
4. ~D
5. B • A
6. C

Only the first paragraph, but how am I doing?

Graham

Mr Veale said...

And how exactly would Bayes Rule expose, say, a case of affirming the consequent?

Victor Reppert said...

Exactly, Graham. Bayes' theorem is a rule in inductive reasoning, while valid and sound are rules in deductive reasoning. Therefore Bayes' theorem is not, and cannot be, a rule for determining validity and soundness. Deductive validity is a matter of a conclusion following necessarily from the premises, and soundness is a matter of those premises being true. This is not about probability at all.

Jason Pratt said...

Sidenote to BDK: yes I know; you could say my complaint is that even though it is routinely called "Bayes's Theorem" (including for example the title of the textbook Swinburne edited on the topic, which is sitting on my shelf) it isn't supposed to be a theorem at all.

Anyway, enjoying the comments, if a little perversely so. {g} But I'm honestly trying (if possible) to learn better from them, too.

JRP

BBB said...

Victor,

Sorry if this is a real stupid question, but what if some of the premises in a deductive argument are only somewhat certain (say 75%, 15% or whatever). How would we discover the probability of the conclusion if the premises are not certain, but only more or less probable? Or is this simply not something we can do?

Tim said...

BBB,

It's not a stupid question at all. The answer is that in a simple valid deductive argument (simple, in the sense that every premise is required for the derivation to go through), there is a definite relation between the probability of the premises and the lower bound of the probability of the conclusion.

Have a look at this and tell me if it helps. If not, I can try to unpack it further.

Victor Reppert said...

For a little comic relief....

http://www.youtube.com/watch?v=2eMkth8FWno

Blue Devil Knight said...

This comment thread has been something of an exercise in pedantry, perhaps deserved by Carrier because of his baseless comments.

Stepping back, I think he could salvage his approach with some tweaking, working through it with some people who know their stuff really well instead of the usual echo chamber into which he speaks.

But it alas looks exactly as I feared: an instance of the sloppy scholarship he was criticizing in all the other jesus-as-myth scholarship.

Mr Veale said...

I'm finding this thread very helpful! Please continue!

Blue Devil Knight said...

Jason it would be like saying 1+1=2 is not a theorem in standard axiomatizations of arithmetic. Just not something you would ever want to claimm.

Victor you could be unfair a bit to Carrier here. You can recast most logical notions in probabilistic terms (as in probabilistic logics).

Peter Bruin said...

Jason Pratt: BDK is right that Bayes's Theorem is a theorem (whether you like it or not). It is a basic theorem about the mathematical concept of a probability space.

That said, one must be very careful (much more careful than Carrier) when modelling historical events using probability theory. I am guessing that this is the point you want to make.

In fact, even saying that it makes sense to assign a probability to a historical event is a controversial claim about the interpretation of probabilities. I have the impression that Carrier doesn't realise this.

A different problem is that when you try to assign such probabilities, it is very easy to become thoroughly confused. It seems to me that this has also happened to Carrier.

Jake Elwood XVI said...

I think Dr McGrew's comments at January 07, 2011 5:50 AM says it all.

BDK is on to something. It will wear thin soon, and then it will just look like the side mocking Dr Carrier are just poor sports.

It is clearly shown that Dr Carrier is out of his depth and as such his own comments about his confidence and abilities make ridiculing him too easy. However maybe we should stop. Maybe we should let Dr Carrier continue is already sound effort in ridiculing himself.

I just feel sorry for him know.

Peter Bruin said...

BDK: I don't think it is pedantry; small differences in the formulation and interpretation of probability theory (and statistics) do sometimes have huge and counterintuitive consequences.

It seems to me that there are really fundamental problems with Carrier's approach. For example, one would have to formulate what we mean by the a priori probability that Jesus existed, disregarding any evidence we have about him. I am at a loss even to identify whom I mean by Jesus without referring to the evidence.

Bobcat said...

WOE,

You wrote, "Carrier told me in an email that he no longer believes that Lydia had any deliberate intention to mislead her readers. So yes, he is capable of mutual interpretative charity."

That's good to hear, but unless he emailed the McGrews (to make sure he understood the article) before publicly lambasting the article, he seems not to be living up to the standards that he demands others live up to with regard to him. Of course, it is possible that he no longer thinks that people should live up to those standards with regard to him, and that what he wrote in Sense and Goodness without God was a bit of youthful exuberance.

BBB said...

Tim,

wow, so you add up the uncertainties of the premises to get the uncertainty of the conclusion, subtract that from 1, and you have the lower bound for the probability of the conclusion? Is that a correct summary?

For some reason I thought you would multiply the probability of each sentence together (say if P1's probability is 50%, and P2 is 50%, the conclusion would be 25%). I mean, that's how I would do it if i wanted to know the probability of two coin flips resulting in 2 heads (or 2 tails).

I think I understand the method of your paper, let me know if I don't. I'm out of my league here, but I want to understand. If nothing else, I'm gonna start taking statistics courses this semester, so maybe that'll help ^_^.

Landon Hedrick said...

Tim,

You should conduct an online reading group of Carrier's book Sense and Goodness Without God.

Anonymous said...

For those interested in an entertaining demolition job of Carrier's Sense and Goodness Without God, David Wood provides one here:

http://www.answeringinfidels.com/index.php?option=content&task=view&id=86

Victor Reppert said...

Landon: The trouble with your suggestion to Tim is that this would require him to read Sense and Goodness Without God.

Tim said...

Read an entire book by this guy? I have an urgent need to schedule a root canal.

Tim said...

BBB,

so you add up the uncertainties of the premises to get the incertainty of the conclusion, subtract that from 1, and you have the lower bound for the probability of the conclusion? Is that a correct summary?

Just about. When you add the uncertainties of the premises, you get the maximum uncertainty of the conclusion. Subtracting this from 1 gives you, therefore, the minimum probability.

Multiplying the probabilities of the premises together will give you the right result if the premises are probabilistically independent. But that isn't an assumption we're always given, so the more general rule is to compute the maximum uncertainty, get the minimum probability from that, and use that as your lower bound.

Hope you enjoy your class!

Landon Hedrick said...

Come on guys, it would be fun. I think you could do better than David Wood. Just mull it over.

As a side note, Tim, I have a question. Evidently you don't think Carrier is competent with Bayes Theorem, but do you at least agree with his claim that good historical method depends on it? Basically I'm wondering the following: Suppose Carrier fixes up all of his mistakes and presents Bayes to everyone's satisfaction. Would you agree with his thesis in volume one? (Obviously you would deny that the evidence favors mythicism, which is the subject of volume two.)

Tim said...

Landon,

Seriously, about reading a book by Carrier -- I've got other things to do.

I do think a broadly Bayesian approach is useful in the analysis of some historical arguments. I very seriously doubt that I would agree with Carrier on many of the details.

I honestly don't understand what several people are saying about "cleaning up" Carrier's material. This whole thing started because he chose to describe a paper I co-authored as "crappy" because we didn't do something that we expressly said we weren't going to do. (We even explained why.)

So the question is not, "Well, could someone come along and clean up all of Carrier's messes, and would the resulting math then be right?" Sure; one can start with literally anything -- say, take a four-year-old's math worksheets ("Count the ducks ...") and claim that they are a proper calculation of the ground state of a particular isotope of oxygen -- and then have a professional physicist "clean it up." Once it's cleaned up, it will, of course, be right. But that proves nothing about the physics knowledge of the person who originally filled out the worksheets.

Rather, the question is, "Why on earth would anyone take Richard Carrier seriously on this topic when he's shown himself to be wildly incompetent?" I'm still waiting for a good answer to that one.

Landon Hedrick said...

I understand your frustration. Carrier was wrong to say what he did about your resurrection article, and he should probably publicly apologize for it. (As some people have already noted, he requests that everyone take a principle of charity with his work and contact him to make sure they have understood. But he evidently did not contact you or Lydia before making the erroneous criticism and calling it "crappy".)

As for "cleaning it up," I get your point. I don't know what other people mean by it, but all I meant to ask was whether or not you'd agree with using Bayes for history in the way Carrier proposes. And if so, then I was thinking you might agree with the main thesis of his Bayes book. When I said "if he gets it cleaned up" (or whatever it is I said) I just meant: assuming he gets Bayes' theorem straightened out for his book on Bayes' theorem. Evidently he shouldn't have gone public will all of this Bayes stuff until he got it a little more clear in his own mind.

I don't have an answer to your question. As for Carrier's book, point taken.

Blue Devil Knight said...

Peter Bruin: yes the issues are important but the style of dismantling him is a bit pedantic and smug.

That's not to say Carrier doesn't deserve a bit of it, or that I wouldn't bring down a sledghammer on him if it were me.

Some of the most enjoyable times in blog life are bringing the hammer down on someone who is an overconfident smug arrogant jerk that has no idea what he is talking about, and you can show it. (Note if someone is wrong and isn't a jerk about it, that is a completely different story!)

But that's just petty little me.

Blue Devil Knight said...

Sense and Goodness without God: it would be fun to do the first few chapters on epistemology, I felt like I was reading Ayer again.

Landon Hedrick said...

BDK,

Did it feel like Ayer because of the tone or because of the content? Both maybe? I think Carrier is a fan of Ayer (or, at least according to pp. 6-7, is impressed by Ayer's philosophical genius).

When I invited Carrier to give a talk at my school about some of his work (the morning of the Craig-Carrier debate I organized), one thing I asked him was whether he really thinks three and a half pages is sufficient to demolish Plantinga's epistemology. I wonder what Plantinga would think. He'd probably just chuckle and go climb a mountain, that guy.

Victor Reppert said...

I see two major issues with respect to the Bayesian analysis of Resurrection history. One has to do with the left side of the theorem. Is there some way of showing that everyone ought to go into the discussion with such a low prior for any miracle that we can virtually guarantee that nothing coming out of the ancient world will be sufficient evidence.

The usual approach to getting that result is via some form of frequentism. There have been 96,100,000,000 persons who have ever lived, there has only been a very few resurrection reports that have so much as surface credibility, so therefore the one in 96.1 billion, and therefore resurrection has to lose even to the swoon theory. But these lead-footed methods don't work for various reasons that have been pointed out quite often. Earman is, I think, the guy you have to get around if you think you can prove something like this about the prior for miracles.

If you say "Extraordinary claims require extraordinary evidence," (which invariably implies evidence that you're never going to be able to get out of the ancient world), then you have to figure out how do define extraordinary in this context. Does it mean historically unique? In that case you have Indian Prince worries, and there's even a problem believing media reports that Obama won the election. If you think it means contrary to the laws of nature, then why do you presume that every event has a natural cause? Doesn't that beg the question from the beginning. Whether a miracle-working God exists is part of what's at issue here. The skeptic needs an account of extraordinariness that doesn't beg the question and pins an unmanageable prior on all miracle claims for everyone.

The other issue looks at the right side of the theorem, and asks if
the evidence surrounding the resurrection is more like what we should expect if the miracle happened or more like what we should expect if it didn't happen. My approach on this is to say that prior probabilities on the matter are certainly going to vary, and that nevertheless we can see if the evidence confirms the miracle story of disconfirms it. Unless you have an argument that shows that no one should have a manageable prior for miracles, you can and should ask this question.

Now notice that I have nowhere said that there is some definite conclusion that everyone will come to, that it is 94% likely that the Resurrection happened based on historical evidence. I am interested in whether the case for the Resurrection confirms it, even if many unbelievers fail to find such evidence "extraordinary" enough.

Luke said...

Tim,

Are you still handing out cookie? I want one!

Anonymous said...

I'm glad Tim mentioned the point about Carrier fixing/saving/'cleaning up' his book after it being exposed that he doesn't know what he's talking about. I'll add another point.

Apparently, Carrier (deeply?) misunderstood Bayes and reasoning related to it. Are people honestly saying "well we know what conclusion Carrier wants, surely there's a way to get to it, he just has to tweak things even if he totally botched the understanding on his first time around"? Is that what's going on here?

Because if so, why even bother with correcting things? Just do this: "Well, Carrier's reasoning was entirely off-base. But I'm sure he's right anyway. So I'll treat all arguments to the contrary as refuted."

There. I saved you time and money.

Vlastimil Vohánka said...

Mr Veale/Graham,

Please, let us know in case you and/or David (Glass?) post something on the issue (as you indicated above).

(We met in Leuven in 2009, right?)

Blue Devil Knight said...

Anon: I was just saying he could tweak his formal treatment/expoosition of Bayes' Theorem so it isn't as confused conceptually. I'm setting aside any consideration of conclusions applications of Bayes' Theorem, which hasn't come up in this thread.

I've seen enough silly probabilistic arguments from creationists that I tend to not take any of these seriously, frankly (either against or for the resurrection). I'm curious of any success stories of such probabilistic arguments coming from philosophy/history, that actually convinced people in ways that more standard techniques wouldn't have.

It comes off as science envy to me, but I don't know enough about historical method and such to say much. I am not saying historians shouldn't use statistics, incidentally (e.g., it would be silly not to, for instance, to use large datasets such as mean income of various political factions if the data are available, and if you can get a p-value, and think you have a good sample of the population, it would only be responsible to use statistics).

There is something about these Bayesian arguments that, while giving off a patina of formality, in fact they tend to be used to express one's pre-existing biases, not test them. I call it quantitative confabulation.

Again, I could be convinced otherwise with a clear success story.

Mr Veale said...

Vlastimil

You certainly met Dave, but not me. I'm just a high school teacher, and I doubt that the Leuven Conference would have been enthralled by a paper on my opinions of detention slips and school dinners!
I'll certainly give you a heads up on anything that we publish. It would be on a semi-popular or popular level.

Graham

Mr Veale said...

Tim/Victor

I've been enjoying the online tutorial on Bayes' Theorem. I think that it's a beneficial exercise.

I also think that Carrier's case is beyond rescuing. Once his understanding of Bayes Theorem is corrected his case against the Resurrection collapses. So I can't see how a mathematician or philosopher can save his work.

I know highly educated and intelligent men and women who take Carrier and Price at face value. They assume that New Testament scholarship exists in a closed environment, with its own rules and procedures. Carrier and Price expose these rules as flawed. Or so the story goes. It is important to show just how poor their arguments are.

It is also an excellent illustration of why we should be wary of relying on blogs and websites rather than peer reviewed books and journal articles for our ideas.

So this thread is much more than an exercise in character assassination. I hope that it runs on for a while yet!

Graham

Blue Devil Knight said...
This comment has been removed by the author.
Blue Devil Knight said...

It is also an excellent illustration of why we should be wary of relying on blogs and websites rather than peer reviewed books and journal articles for our ideas.

I think there is a lot of truth to this. Articles are even more important than books, as peer review for books can be pretty lax.

Blue Devil Knight said...

I also think that Carrier's case is beyond rescuing. Once his understanding of Bayes Theorem is corrected his case against the Resurrection collapses.

Not necessarily: he could have applied the formalism correctly even if he doesn't understand what he's doing with it. This happens all the time in neuroscience when people throw information theory at brains: they get the right number out, but don't understand the math.

Now, we know he got some numbers wrong in this article, but we can't say at the outset how the numbers will fall when he straightens his act out, until the work is done.

Again, I say this with suspicion of this entire Bayesian approach I mentioned above. It's so susceptible to quantiative confabulation that I tend to steer clear of these arguments.

David B Marshall said...

Having read Richard Carrier's Sense and Goodness(not all; unfortunately it turned out I had a root canal later that week, too), and debated him in the Amazon forum for the book, much of this seems oh, so familiar. As no-it-alls go, Richard shames even yours truly. He is also an expert on the Origin of Life, Christian history, God arguments, early Christianity, and repudiates the whole field of philosophy as unworthy of his genius. (This even irritated a hard-core atheist philosopher of my acquaintance.)

In short, Dr. Carrier is bidding fair for the title of "the Ayn Rand of the New Atheism."

To give Richard his due, though, while I agree his chapter on the Resurrection in Christian Delusion flopped (see, again, my review on Amazon), I think he did make some valid points about the history of science. He does show, I think, that Stark and Jaki understate the achievements of ancient Greek science. (Though interestingly, his argument can be taken as further supporting the link between theism and science.)

Perhaps Carrier should be an object lesson for all of us who make big arguments. He is smart. He is capable (on occasion) of studying an issue with reasonable thoroughness, and making a decent argument. But he spreads himself way too thin, and thinks a little of it goes further than it actually does. He often flies too high, and gets his wings scorched.

Jason Pratt said...

David: {{I think he did make some valid points about the history of science.}}

True. Which shouldn't be completely surprising, since that's his actual professional speciality (i.e. what he has a degree in and what he gets paid to teach. If I recall correctly...?)

Btw, hi David! {g}

JRP

Jason Pratt said...

Peter,

I would make those points, too (whether in regard to Richard or in regard to Christian apologists trying to appeal to Bayes Theorem for their own purposes.) But no, my more fundamental critique does in fact go back to the whole original purpose and methodology for the "Theorem".

I have noted I could be wrong about that. But my inductive estimation of likelihood about whether I'm right or wrong about that, based on current evidence before this thread, hasn't yet changed based on my assessment of new evidence presented in this thread.

That's the kind of inductive assessment that the so-called "Theorem" was supposed to be modeling originally. It isn't primarily a mathematical operation at all. My complaint, BDK, is that even experts in the field (starting with Reverend Bayes himself, while trying to provide a quantitative description of inductive inferential procedure) commonly treat that procedure as being a mathematic theorem like 1+1=2. Or rather like 0.3*0.5/0.25=60%. So I should be 60% sure the Res happened! Or didn't happen as the case may be! Or maybe I should believe there's a 60% chance it did or didn't happen! There has to be some way to get a legitimate numerical expression there using my artificially fictional percentage estimates, thus demonstrating the true rationality of my belief or disbelief in the Res! Or in God's existence! Or whatever the topic is! Look, it's 60%!!!

But that isn't how inductive expectations actually work. (Which I think Bayes himself was aware of, because his theory, expressed in pseudo-mathematical terminology like a theorem, works quite well as a description of belief adjustment in light of new evidence.)

JRP

David B Marshall said...

Hi, Jason. This is the first time I've posted here -- am surprised how many people I know!

Carrier is often a pretty sloppy historian, too, though.

AIGBusted said...

Out of curiosity: Is the "Tim" here one and the same as Timothy Mcgrew?

Victor Reppert said...

Yes.

Tim said...

BDK,

My purpose in starting this examination was simply to demonstrate, as directly as possibly, without resorting to a war of credentials, that Richard Carrier was not a trustworthy guide to the mathematics of probability and that his comments regarding our paper in the Blackwell Anthology were not credible. Unfortunately, there is no other direct way to do that than by citing his own mistakes.

However, there is no longer any need to go further (which I was prepared to do), since Richard has apologized to Lydia for his remarks and has admitted that, on the mathematical front, he was wrong and we were right.

Naturally, we still disagree on the substantive facts; he thinks we're wrong, and we think he's wrong. I'd be happy to discuss that with anyone, as time permits. But there is no longer any need to establish our point regarding the relevance of our mathematical calculations. That point has been conceded.

Bon appetit!

Mr Veale said...

Tim
That is an honourable and gentlemanly act.
Graham

Mr Veale said...

Tim
Your actions are honourable precisely because you could enumerate many good reasons for continuing to dissect Dr Carrier's paper
Yet, you emerge as the more honourable man, and one's integrity is more important than any academic critique.
Graham

kilo papa said...

If some "being" had the astonishing power to bring into existence every single "thing" in the billions of galaxies and billions of planets that now exist, then isn't the idea of bringing a three day old corpse back to life the equivalent to a two-bit magician pulling a rabbit out of a hat?
I saw magician David Blaine bring a dead fly back to life. That was more impressive than Jesus' supposed resurrection.
I mean really, you Jesus nutters and your silly superstitions. How truly loony you are.

Vlastimil Vohánka said...

Psychological impressiveness is one thing, probative (probabilistic) force another. Maybe you mean "extraordinary" claims require "extraordinary" evidence. But this is, again, ambiguous, and it is far from clear the evidence for the resurrection is not "extraordinary" in the (probabilistic) sense discussed here. Theory of relativity, a grandscale theory, was established empirically by quite ordinary and humble evidence (observations). Finally, why would God need to go more spectacular? Maybe he does not want to. Maybe because he wants us to do some serious work for ourselves.

Unknown said...

http://www.scribd.com/doc/125993290/Faking-Jesus

Tim said...

In case anyone needs it, here's a summary of the first few pages of this review that Anonymous just posted of Richard Carrier's book, which takes an extraordinarily long time getting to anything like a review:

Jesus Studies is just downright mean and has nothing to do with history.

The gospels are not history; they were written decades after the events they relate and nobody knows who wrote them. Plus, there are textual variants. They weren't even written in Palestine. Jerusalem got destroyed.

An average lifespan in the first century was probably less than fifty years, particularly if you consider infant mortality and use the mean.

People couldn't remember Jesus very well -- the gospels don't even describe his personal appearance. The gospels contain no eyewitness testimony. Eusebius says that Mark "had not heard the Lord, nor had he followed him ..."

Nobody knows what Jesus thought. Nobody knows the year of Jesus' birth or death. Nobody knows how long his career lasted.

The gospel of Mark ends without a resurrected Jesus. Pay no attention to Bruce Metzger and others who tell us that the original ending is likely lost.

The resurrection appearances read like ancient ghost stories. Maybe Jesus never existed at all.


[Sorry, this is where I fell asleep.]

Tim said...
This comment has been removed by the author.
Tim said...

Glanced at the rest of it -- I was mistaken, as it isn't even a review of Carrier at all. The relevance to the OP is quite remote.

Jim Lippard said...

It looks like the linked-to document by Richard Carrier has been updated. Has it been corrected for the issues discussed in the comments here?

Quixie said...

You snore.

Quixie said...

People crying "hubris" … it's friggin hilarious.