JohanRonnblom wrote:

Tim Hendrix wrote:When I am talking about a specific hypothesis for historicity I am talking about Carriers specific, five-point scenario which falls into the general class of ahistoricity (the general class of ahistoric theories is the negation of "Jesus existed").

As Carrier explains in great detail, a 'theory' such as 'Jesus existed' is completely useless. (...) Unless you somehow specify who this historical Jesus would be, in such a way that he is identifiable, you will achieve nothing.
Now, you may not agree with Carrier's specification. (...). That is not a problem for Carrier. His calculations are obviously only relevant if the reader agrees that his hypotheses are sound and that any other Jesuses are sufficiently unlikely as to be irrelevant.
If you want to argue for another Jesus, go ahead and do that. But do not complain about Carrier's use of BT, because it is completely unrelated to that.

I think you are both misunderstanding what Carrier is doing and what I am pointing out.
Carrier himself defines (and uses) the "minimal historicity" thesis which is denoted by h in OHJ. Logically, the negation of h would be the general thesis of mythicism (the negation of the proposition that Jesus existed) and if we assume we can use probabilities in this context (which both I and obviously Carrier am) it must have a probability. Please don't confuse the negation of h with what Carrier writes as the negation of h since this is a formal error in OHJ as Carrier agrees.

What I am pointing out is that Carriers particular thesis for mythicism, namely that Jesus did not exist AND his five points are correct, is contained within the general proposition that Jesus did not exist. Because it is contained within the general proposition it follows it must have a lower probability. THAT is the point I am getting at, and it has absolutely nothing to do whether I think Carriers particular theory is sound or not.

Let me illustrate this with an example:
Suppose we consider the general proposition that a person is guilty (Jesus existed). The negation of that proposition is that he is innocent (Jesus did not exist). A subset of this proposition is that the person is innocent AND someone planted his DNA at the crimescene (Jesus did not exist AND the five point myth-theory is true).

We must not confuse the probability a person is innocent with the probability he is innocent AND someone planted the evidence at the crimescene, because we can go arbitrarily wrong if we proceed in that manner. What we are dealing with is a more basic matter of logic.

Now, I am sure you will bring up that Carrier says in many places that this is irrelevant and that we can equate the probability Jesus did not exist and that his 5-point myth-theory is true. I want to pre-emt that by saying I often think there is a great deal of difference between what Carrier *says* is true about probabilities, and what he actually demonstrates as being true (or provide a rigorous argument for) in OHJ and this is certainly one of those cases. If you want to advance one of his arguments, please try to formulate it as a statement about probabilities using the rules of probability and the problems should become evidence.
Furthermore, where that discussion will bring us is exactly to the Rank-Raglan prior which we must trust computes the probability of Carriers five-point myth theory. I simply think that is a misunderstanding of what reference classes do and observe that many of Carriers examples do not fit his five-point myth theory; i.e. he is comparing apples and oranges. The problem of using reference classes in this manner is well known and collectively discussed as "the reference class problem".

JohanRonnblom wrote: You may think so but note that you are now making up an absurd Bayesian argument, then proceeding to show why this argument is absurd. It is not Carrier's argument that you criticize, but a complete straw man of your own making.

Well, the gameshow only illustrates a very fundamental point about error analysis and I can't really see why you object so storngly to it. Nevermind, let us leave that aside and focus on the calculations that has to do with Carriers argument. You agree that my two general conclusions are true, i.e. that BT magnifies errors by about a factor of five in the scenario I illustrate and bias is increased by a factor of about 20, again in the scenario I use? In other words, are any of the graphs wrong?

JohanRonnblom wrote: Again, it seems as though you are arguing not just about Carriers use of BT, but against BT in general. Let me ask you, do you believe that it is possible - in general - to use BT to combine several pieces of evidence for or against a proposition, in such a way that the combined accuracy increases, rather than decreases, with the number of available pieces of evidence?

Well, don't think it is my fault that it seems to you that I am making any kind of general argument against BT and I hope you understand why this statement is very puzzling to me.

To answer your question, as I mentioned in my previous post, an analysis of a simply coin is such an example of how BT can (and do!) work and converge, as is nearly any other Bayesian model under the sun including (hopefully) the one I have been building the past weeks on my computer. You can consult the Bernstein-von-Misen theorem for a general statement about convergence.

What I point out is that what Carrier is doing is simply quite different from other applications of BT and (as a rule) his approach is very strongly affected by even weak bias. Please don't take my word for it, verify it yourself with a pocket calculator if you have doubts.

JohanRonnblom wrote:

Tim Hendrix wrote: it is Carrier not I who makes the assumption it is possible for us to guess complicated probabilities with high fidelity. If we assume this is not possible, well, there goes Carrier's entire project.

Do you likewise doubt that it is possible to estimate the probability that Caesar existed? That Abraham Lincoln existed? That Tim Hendrix exists? It does seem that you are getting very close to the philosophical position of radical skepticism, where nothing can be known.

No, I don't doubt we can estimate these probabilities but now I think you are being flippant by choosing these examples. What I point out is that BT, as Carrier uses it, increases uncertainty and is very strongly affected by bias. To take the election night example, even uncertainty of 10s of percent for a very well-defined problem for which we had very good data proved to be woefully insufficient and so again I ask: Is it not reasonable to assume there is quite a great deal of uncertainty in the probabilities Carrier estimate?
The relevant question remains if we can estimate (non-trivial) probabilities without bias, and with as high accuracy as Carrier implies.

JohanRonnblom wrote:

Tim Hendrix wrote: More importantly, I think you miss the point of the section, namely the numerical stability of BT. If you don't like the gameshow just look at the graphs that are structured as Carriers actual calculation. I think you will agree that the overall conclusion holds, that a computation such as Carriers can be inverted by assuming a bias of just a few percent in the (subjectively guessed) probabilities.

This is completely unrelated to Bayes Theorem. Yes, if someone is wrong in all her arguments, then the conclusion will be wrong. This is true for every historian and every Jesus scholar out there. The use of BT, however, allows us to see what someone ranks as her strongest argument. We can then focus our energy on that argument, until we reach an agreement, or at least arrive at reasonably close estimates. And so on for the other arguments.

Well, I don't disagree that the ranking-effect might be usefull, and I have never dismissed the use of BT wholesale. At this point it seems we actually agree that a computation such as Carriers can be inverted by a small bias, you just don't see that as a potential disadvantage?

JohanRonnblom wrote: In addition, we can see that if someone is making a very complex argument that she admits has uncertainty, then it may follow that a series of conclusions reached from those arguments will have increasingly diminished certainties, until they become so uncertain that they are clearly meaningless.

Well, that is basically my entire point, that BT will increase our uncertainty by about a factor of five and bias by about a factor of 20 (in the context of Carriers use of BT). I am confused how I can be so wrong in your view when you seem to agree with what I actually conclude?

JohanRonnblom wrote:

Tim Hendrix wrote:Well, in most cases Carrier just says he can be absolutely certain that the probabilities have ratio 1, i.e. no uncertainty at all...

I can't find any single instance of him making such an assumption. Can you give me an example where he does this?

Yes, that is very easy. His upper and lower estimates agree many times, for instance in case of the Gospels, Josepheus, Thallus, etc. etc. Just consult the final table with his probabilities to verify this for yourself.

To say the upper and lower estimates agree is to say there is no reasonable way a piece of evidence can be more probable on historicy than on mythicism, i.e. no uncertainty at all. In the case of the Gospels, that's obviously not what most experts on the NT think. In other words, I would claim this is a case of a probability which does have some uncertainty.

JohanRonnblom wrote: That is just how history works. How would this be any better for Ehrman or any other? I think Ehrman is massively biased. The only thing to do about that is to examine the arguments and hopefully our assessments can gradually converge, at least on some issues.

I have noticed that critique of Carrier and OHJ tends to very quickly be answered by pointing out how bad and biased other historians are. Ehrman could be forming his opinion by reading tea leaves and that should not prevent us from discussing OHJ. At any rate, if we assume Ehrman and all other historians on the NT are biased, then it is reasonable to assume Carrier too is affected by some bias. If we put that bias at just a few percent then, well, there you go.

Tim Hendrix wrote: Carrier himself defines (and uses) the "minimal historicity" thesis which is denoted by h in OHJ. Logically, the negation of h would be the general thesis of mythicism (the negation of the proposition that Jesus existed) and if we assume we can use probabilities in this context (which both I and obviously Carrier am) it must have a probability. Please don't confuse the negation of h with what Carrier writes as the negation of h since this is a formal error in OHJ as Carrier agrees.

Are you saying Carrier agrees that there is a formal error in OHJ? Where?

I think I can now explain your mistake in mathematical terms.

Let's define the following sets:
hm: the probability that Jesus was historical and fits Carrier's minimal theory of historicity
hn: the probability that Jesus was historical and does not fit Carrier's minimal theory of historicity
mm: the probability that Jesus was not historical and fits Carrier's minimal theory of myth
mn: the probability that Jesus was not historical and does not fit Carrier's minimal theory of myth

Now, these sets logically cover the entire probability space so:
hm + hn + mm + mn = 1

Carrier has now asserted that hn and mn are 'vanishingly small', certainly less than 1/1000000, and 'can therefore be ignored'. This means, that it is asserted that
hn = mn = 0

Therefore, we have:
hm + 0 + mm + 0 = 1
or hm = 1 - mm,
or hm = ¬mm
We can see that this is true, even though
hm ≢ ¬mm

So, you see, we have equality because of the specific assumptions that are explicitly made, even though we do not have identity. Again you may disagree with those assumptions, but there is nothing wrong with the argument.

Tim Hendrix wrote: Suppose we consider the general proposition that a person is guilty (Jesus existed). The negation of that proposition is that he is innocent (Jesus did not exist). A subset of this proposition is that the person is innocent AND someone planted his DNA at the crimescene (Jesus did not exist AND the five point myth-theory is true).

We must not confuse the probability a person is innocent with the probability he is innocent AND someone planted the evidence at the crimescene, because we can go arbitrarily wrong if we proceed in that manner.

We can indeed equate those probabilities if we assume that it is known with virtual certainty that someone did plant DNA at the crime scene. The only way you can challenge that equality (again, not identity) is by arguing that there is a significant probability that DNA was not planted. You're free to make that argument, but that is something completely different than claiming Carrier's math is wrong.

Tim Hendrix wrote:You agree that my two general conclusions are true, i.e. that BT magnifies errors by about a factor of five in the scenario I illustrate and bias is increased by a factor of about 20, again in the scenario I use? In other words, are any of the graphs wrong?

Not wrong, just irrelevant. When you claim that something is 'magnified', you need to ask what it is a magnification of. To the extent this has any relevance to the topic at hand, of course if you say:
Jesus was a man and he was real and he lived about 2000 years ago and he was a preacher and people followed him and he was executed and Pilate did it, then any bias will of course 'magnify' the probability that you are wrong on this combined statement, versus the individual probability that you are wrong on any of the details.

But, this is not how Carrier uses BT. Rather, he claims that Jesus was mythical because of A, Jesus was also mythical because of B, etc. If you do it correctly you will therefore see that even if Carrier is wrong (or 'biased') on one of his arguments, the combined error of the sum of his arguments will be smaller because he used BT.

Now, of course, if Carrier is wrong on all of his arguments, or sufficiently wrong on most of them, his conclusion will of course still be invalid. But you cannot blame his use of BT for this, because there is no method you can use to arrive at correct conclusions based on incorrect arguments. But, contrary to what you are arguing, his use of BT will reduce, not magnify, the combined error.

Tim Hendrix wrote: What I point out is that what Carrier is doing is simply quite different from other applications of BT and (as a rule) his approach is very strongly affected by even weak bias.

And I say that this is not his approach at all, but rather an absurd model that you have invented yourself and which is completely different from Carrier's argument.

Tim Hendrix wrote: Is it not reasonable to assume there is quite a great deal of uncertainty in the probabilities Carrier estimate?

Of course there is, and Carrier is explicit about that. But, in the end, we must choose: either it is possible to say something about the probability that Jesus was historical, or we can say nothing at all, and therefore must assume that the probability was 50%.

Tim Hendrix wrote: Well, I don't disagree that the ranking-effect might be usefull, and I have never dismissed the use of BT wholesale. At this point it seems we actually agree that a computation such as Carriers can be inverted by a small bias, you just don't see that as a potential disadvantage?

I see that as a massive advantage, because the problem is not with the computation, but with the bias. Remove the computation, and the bias still exists, and still causes the conclusions to be wrong. Ditto for any other author, on any subject. Therefore, we should argue about Carrier's arguments, not about his correct use of BT.

Tim Hendrix wrote:

JohanRonnblom wrote: In addition, we can see that if someone is making a very complex argument that she admits has uncertainty, then it may follow that a series of conclusions reached from those arguments will have increasingly diminished certainties, until they become so uncertain that they are clearly meaningless.

Well, that is basically my entire point, that BT will increase our uncertainty by about a factor of five and bias by about a factor of 20 (in the context of Carriers use of BT). I am confused how I can be so wrong in your view when you seem to agree with what I actually conclude?

Carrier is in fact not making such a chain of arguments. He is not claiming that Jesus was probably x, therefore he was probably y as well, and that means maybe he was z. Rather, he is claiming that because of reasons A, B, C, D and so forth, Jesus was probably x. This is a major difference between Carrier and most other authors dealing with Jesus, whether historicists or mythicists.

Tim Hendrix wrote:

JohanRonnblom wrote:

Tim Hendrix wrote:Well, in most cases Carrier just says he can be absolutely certain that the probabilities have ratio 1, i.e. no uncertainty at all...

I can't find any single instance of him making such an assumption. Can you give me an example where he does this?

Yes, that is very easy. His upper and lower estimates agree many times, for instance in case of the Gospels, Josepheus, Thallus, etc. etc. Just consult the final table with his probabilities to verify this for yourself.

Oh please, just because something is rounded to 1 does not mean he does not concede there is no uncertainty at all. Also, note that these numbers do not represent probabilities that Jesus was not historical, they represent the relative plausibility that a certain text would be written given certain conditions. The number '1' is just chosen to relate to the number representing the plausibility that the text would have been written given certain other conditions. Carrier here chooses 1 to mean 'nothing in the text conflicts with these conditions' and lower numbers to mean that there are details in the text that do not add up well with the conditions.

In every case, you are of course free to make arguments that Carrier is wrong in these claims. Which is really what we should be doing, rather than pretending there is something wrong with his math.

Tim Hendrix wrote: I have noticed that critique of Carrier and OHJ tends to very quickly be answered by pointing out how bad and biased other historians are. Ehrman could be forming his opinion by reading tea leaves and that should not prevent us from discussing OHJ. At any rate, if we assume Ehrman and all other historians on the NT are biased, then it is reasonable to assume Carrier too is affected by some bias. If we put that bias at just a few percent then, well, there you go.

Again, either we think that there is something we can know about the probability that Jesus was historical, or we can say nothing and the probability becomes 50%. If your argument is that everyone can be biased and therefore every argumentation gives us massively wrong conclusions, then that applies not just to Carrier but to all authors. But, I say, it does not apply equally to all authors, because Carrier is making a much more limited claim than these authors do. Therefore, we do not have the 'magnification' you talk of.

Certainly, authors arguing in favour of historicity could take the same approach and write a book only arguing historicity, not focusing on a thousand different details of Jesus' life. That is the type of books I would like to read.

JohanRonnblom wrote:

Tim Hendrix wrote: Carrier himself defines (and uses) the "minimal historicity" thesis which is denoted by h in OHJ. Logically, the negation of h would be the general thesis of mythicism (the negation of the proposition that Jesus existed) and if we assume we can use probabilities in this context (which both I and obviously Carrier am) it must have a probability. Please don't confuse the negation of h with what Carrier writes as the negation of h since this is a formal error in OHJ as Carrier agrees.

Are you saying Carrier agrees that there is a formal error in OHJ? Where?

Yes, just read his reply to me, but the issue is so obvious it hardly needs mentioning.

JohanRonnblom wrote: I think I can now explain your mistake in mathematical terms.

I will just note you wrote "mistake" here.

JohanRonnblom wrote: Let's define the following sets:
hm: the probability that Jesus was historical and fits Carrier's minimal theory of historicity
hn: the probability that Jesus was historical and does not fit Carrier's minimal theory of historicity
mm: the probability that Jesus was not historical and fits Carrier's minimal theory of myth
mn: the probability that Jesus was not historical and does not fit Carrier's minimal theory of myth

Now, these sets logically cover the entire probability space so:
hm + hn + mm + mn = 1
Carrier has now asserted that hn and mn are 'vanishingly small', certainly less than 1/1000000, and 'can therefore be ignored'. This means, that it is asserted that
hn = mn = 0
Therefore, we have:
hm + 0 + mm + 0 = 1
or hm = 1 - mm,
or hm = ¬mm
We can see that this is true, even though
hm ≢ ¬mm
So, you see, we have equality because of the specific assumptions that are explicitly made, even though we do not have identity. Again you may disagree with those assumptions, but there is nothing wrong with the argument.

Well, I agree with all of that and in fact that is essentially all something I have written in my review in different notation. For me to have made a "mistake" (in your words) would require that I disagreed with something in the above but obviously I don't: IF we assume the probabilities are what Carrier claims they are the conclusion follows. That has never been in question.
My review tries to uncover what Carriers argument implies and why I think Carriers arguments for why the probabilities are what they are are not very good. So back to you: Where is my mistake?

JohanRonnblom wrote:

Tim Hendrix wrote: Suppose we consider the general proposition that a person is guilty (Jesus existed). The negation of that proposition is that he is innocent (Jesus did not exist). A subset of this proposition is that the person is innocent AND someone planted his DNA at the crimescene (Jesus did not exist AND the five point myth-theory is true).

We must not confuse the probability a person is innocent with the probability he is innocent AND someone planted the evidence at the crimescene, because we can go arbitrarily wrong if we proceed in that manner.

We can indeed equate those probabilities if we assume that it is known with virtual certainty that someone did plant DNA at the crime scene. The only way you can challenge that equality (again, not identity) is by arguing that there is a significant probability that DNA was not planted. You're free to make that argument, but that is something completely different than claiming Carrier's math is wrong.

I don't think it is fair that you say that I claim "Carrier's math is wrong" in general as I obviously only question specific arguments he is making and often that critique has nothing to do with broad formal errors. Please stick to the concrete points that I actually make.

With regards to the above: Yes, IF we make certain assumptions, THEN other conclusions follows. However, if you wanted to actually make that argument, i.e. claim that:

P(Innocent AND planted evidence at crime scene) = P(Innocent)

It is reasonable to ask (the police certainly would) WHATmakes that conclusion sound as it obviously is not true in general. Carrier provides reasons which I discuss at length and if you follow the discussion it all comes down to what Carrier believes his Rank-Raglan prior can do.

If you think my discussion is utterly wrong as you seemed to say in your first post (however I do wonder if the discussion was part of what you did not read?) then please point out where the concrete mistakes lie, preferably using math if you are making a formal claim. However, please refrain from implying that I am somehow claiming that if we assume all of Carriers premises are true the conclusion does not follow. That is simply a misunderstanding of what I actually say.
On this point, that Carriers argument is sound if the conclusions are true is no surprise. The same can be said about all of William Lane Craigs arguments for God and the various arguments for the existence of God that uses BT.

JohanRonnblom wrote:

Tim Hendrix wrote:You agree that my two general conclusions are true, i.e. that BT magnifies errors by about a factor of five in the scenario I illustrate and bias is increased by a factor of about 20, again in the scenario I use? In other words, are any of the graphs wrong?

Not wrong, just irrelevant. When you claim that something is 'magnified', you need to ask what it is a magnification of. (...) this is not how Carrier uses BT. Rather, he claims that Jesus was mythical because of A, Jesus was also mythical because of B, etc. If you do it correctly you will therefore see that even if Carrier is wrong (or 'biased') on one of his arguments, the combined error of the sum of his arguments will be smaller because he used BT. (...) But, contrary to what you are arguing, his use of BT will reduce, not magnify, the combined error.

What is being "magnified" in my terminology is the bias of the individual probabilities. I think the example I give using Carriers probabilities is very exact but I can easily supply the script if you have troubles following it.
I don't follow your alternative suggestion for how bias should be treated. Let's suppose we assume a bias of 10% and that the individual probability of historicity/mythicism that goes into BT are roughly the same (in the unbiased case). Can you please provide what calculation you are suggesting as the relevant one which shows that bias has no effect? If you can't, I think I am going to stick with what I have.

JohanRonnblom wrote:

Tim Hendrix wrote: What I point out is that what Carrier is doing is simply quite different from other applications of BT and (as a rule) his approach is very strongly affected by even weak bias.

And I say that this is not his approach at all, but rather an absurd model that you have invented yourself and which is completely different from Carrier's argument.

I am very puzzled by this statement. If Carrier's approach is not to make guesses (based on his personal evaluation of the evidence) of the various probabilities and combine them using the calculation in the book then I don't know what it is. If you wish to claim that's what statesticians are or should be doing, well, I will just invite you to read a book on statistics.

JohanRonnblom wrote:

Tim Hendrix wrote: Is it not reasonable to assume there is quite a great deal of uncertainty in the probabilities Carrier estimate?

Of course there is, and Carrier is explicit about that. But, in the end, we must choose: either it is possible to say something about the probability that Jesus was historical, or we can say nothing at all, and therefore must assume that the probability was 50%.

Two points: Firstly, if someone asks me to guess the weight of a tree and I can't tell if it's weight is more or less than 1000kg then the right conclusion is to say that my best estimate is 1000kg. But that is not the same as saying that the highest (and lowest) estimate of the tree's weight is 1000kg, which is what we are discussing (Carriers optimistic/pessimistic estimates of the various probabilities).
Secondly, many scholars of early Christianity's claim that it IS possible to conclude something about the historicity of Jesus using evidence from the Gospels. In other words, at the very best Carriers estimate of probability are vastly more critical of Jesus historicity than what most scholars seem to believe. If Carrier is systematically too critical, well, you can see elsewhere what I believe will happen with such systematic bias.

JohanRonnblom wrote:

Me: Well, in most cases Carrier just says he can be absolutely certain that the probabilities have ratio 1, i.e. no uncertainty at all...
You: I can't find any single instance of him making such an assumption. Can you give me an example where he does this?

Me: Yes, that is very easy. His upper and lower estimates agree many times, for instance in case of the Gospels, Josepheus, Thallus, etc. etc. Just consult the final table with his probabilities to verify this for yourself.

You: Oh please, just because something is rounded to 1 does not mean he does not concede there is no uncertainty at all.

You asked me for an example of where the probability ratio was one in both the optimistic and pessimistic case and I have given examples. Now you are moving the goalpost by saying that Carrier might hypothetically believe there is still some uncertainty (he just did not write that in OHJ...), or that we should really be asking about other probabilities, etc.
I am sorry but that is disingenuous: Carriers upper and lower estimates agree in many cases as I originally wrote, which is to say he assumes no uncertainty in those terms. You might find that perfectly reasonable but that does not mean it is not worth pointing out what the optimistic/pessimistic case actually corresponds to...

JohanRonnblom wrote: Which is really what we should be doing, rather than pretending there is something wrong with his math.

"Pretending", "wrong with his math". Please stick to what I have actually written rather than defending OHJ from these broad and inaccurate accusations I have never made. If you can point to a place where I claim Carriers math is wrong, and it is not, please post that quote and explain why I am wrong. Otherwise, I am just going to call this for what it is: a lazy strawman.

JohanRonnblom wrote: either we think that there is something we can know about the probability that Jesus was historical, or we can say nothing and the probability becomes 50%. If your argument is that everyone can be biased and therefore every argumentation gives us massively wrong conclusions, then that applies not just to Carrier but to all authors. But, I say, it does not apply equally to all authors, because Carrier is making a much more limited claim than these authors do. Therefore, we do not have the 'magnification' you talk of.

I don't make general but specific claims: If we take Carriers computation, a bias of about 4% (and I define what that means) will invert the result of the computation. That is not the same as saying "everyone is biased and therefore every argumentation gives us massively wrong conclusions" (if you can't see the difference I will be happy to explain it to you). If you notice, I don't even conclude that means BT is useless, or even that Carriers computation is useless: I just point out to the reader that this is a feature of the computation in OHJ.

We cannot have a reasonable discussion if so many of the things I say are turned into hyperbolic strawmans which you then knock down. You claimed earlier that "He does at some point list some bullet points, which is helpful, and as far as I can determine they are all invalid". Well, those bullets are all the major points of my review. If all points I make in my review are invalid, you should not have to change what I say in order to find an actual error.

DB Roy wrote: What you'll find from Flann5 and some others here is that they don't get it. Bayesian analysis they simply cannot grasp. You explain it to them and the next post they are right back at it again as though you had explained nothing.

If his use of Rank-Raglan is an example, then Carrier doesn't understand Bayesian analysis either. Prior probabilities should be based on comparable instances, for example, while it is obvious that the figures he has picked out are not at all comparable.

Bayesian analysis is a method for updating probability calculations when you have prior information that is at least as solid as the sample you are working from. In general, you should be looking at a series of samples from the population currently being sampled, or some other way of knowing what the population probability really looks like.

If you are using uninformed priors, then at best you can use probability arguments from logic, such as the probability of a six-sided die landing on a six. If you then get a sample with 2 out of 3 rolls coming up as a six, you might want to recalculate the probability that the next one will be a six, given that the die may be loaded and the sample is informative, but you start out with priors saying, well, it can happen by chance that you get 2 rolls out of 3 coming up as a six.

The best discussion of this matter that I have seen is in Kahneman's "Thinking, Fast and Slow" because several of his key findings rely on Bayesian analysis to understand why the average person's thinking is flawed by overestimating the informativeness of their (usually small) sample.

One of the most frustrating things about reading Carrier is that he slips back and forth between his interpretations of "historical". He begins by accepting the possibility that a person may have existed and the stories about him dressed up with legend, which is quite honest of him. But then he jumps from this issue to a more general one, such as in his use of Rank-Raglan, as to whether the character is as represented.

He could have used Pythagoras, for example, or Horatio at the bridge, or yes, King Arthur and Robin Hood. Many historians believe Robin Hood was probably a real person, despite probably having accrued many invented legends. Similarly with Romulus, King Arthur and Roland. But instead Carrier uses a sample of highly mythologized characters such as Zeus and Odin, and further neglects the possibility that, say, Theseus, Perseus and Jason were in fact real people later dressed up with fanciful stories.

We know, for example, that there were historical elements to the Theseus legend which were considered fictional and fanciful before Schliemann dug up Knossos. How about a sample of "legendary figures about whom little is known"?

DB Roy wrote:I stopped playing the game because you're just explaining the same position over and over again and they are either not very bright or they are amusing themselves making you jump through hoops. I think they actually believe that if they wear you down to the point where you just give up and stop arguing your position then that means they won. Either way, nothing you say will sink in very far or they'd have to confront the truth which they cannot do. The idea of trying to prove a low probability for Carrier's existence is an example of this stupidity. The probability is obviously very high but they don't get it. I don't know how they couldn't get it but they don't.

It's called reductio ad absurdum. Over and over Carrier argues tendentiously. One never gets the feeling from him that he is after an hones evaluation of the evidence. Just about pretending to give one.

So many options I could not find "delete".

Tim Hendrix wrote: Regarding the first quote, the specific vs. general hypothesis of historicism, that's exactly the one mistake I agree is in my review: I summarized OHJ falsely by assuming Carrier compares a specific theory of historicity vs. a specific theory of mythicism and not the general idea Jesus was historical vs. a specific hypothesis of mythicism (as he does). However, the difficulties I highlight regarding comparing specific or general theories of historicity/mythicism obviously still stands as does everything I say following this as all my arguments relate to Carrier's specific theory of mythicism.

I probably should not be getting involved without reading the relevant reviews, etc., but I did want to highlight a few of the points made, which I consider to be at the heart of the matter.

Tim Hendrix wrote: it is Carrier not I who makes the assumption it is possible for us to guess complicated probabilities with high fidelity. If we assume this is not possible, well, there goes Carrier's entire project.

More importantly, I think you miss the point of the section, namely the numerical stability of BT. If you don't like the gameshow just look at the graphs that are structured as Carriers actual calculation. I think you will agree that the overall conclusion holds, that a computation such as Carriers can be inverted by assuming a bias of just a few percent in the (subjectively guessed) probabilities.

If he was better at doing sensitivity analysis, he might have more credibility. But since his project is to impose priors rather than just subjectively assess evidence, he doesn't want to do a careful analysis of the sensitivity of conclusions to his priors.

Tim Hendrix wrote:Notice Carriers way of treating this problem is simply to say, in one form or another, that he is aware of his biases and believe he has handled them adequately. The problem is that I simply don't think his estimate of the range of probabilities would correspond to what other, reasonable historians might assign. For instance, I think it is very likely that Ehrman or other NT experts believe that (say) the Gospels are more to be expected on historicity than on mythicism.

For example, where is his "assessment" of the probability that euhemerization could occur without leaving a trace - that is, that an entire church could migrate from believing in a completely mythical Christ to studying and proclaiming the historical details in the gospels, without leaving a single trace of the conflict likely to follow from such a project?

This is always the weakness of conspiracy theories: they are left claiming that the coverup was complete because the institutions of power are all-powerful and completely intolerant of inconvenient truths. Now, my priors are that such massive coverups are sort of possible. But it is going to be hard to put a population probability on them given that we have defined away the possibility of evidence of such a case. The best we can do is assess the "near misses" - how close did Stalin come to wiping out the evidence of the Ukrainian genocide, say, and how difficult would that have been, and how much power did he have to do it? Undoubtedly small coverups have successfully occurred, and as a consequence we have no evidence of them, but, you know, as "1984" pointed out, honesty is capable of incredible pushback.

Tim Hendrix wrote:Well, if it seems like that I think it is because you are not aware of how Carrier's way of using BT differs from common statistical practice:
Under normal circumstances, BT is applied in situations where we have a well-defined model and data, nearly always in the form of repeated observations of some form, which relates to the parameters of the model. A simple example might be a model of a coin where the parameter would be the probability the coin comes up heads and the data are a sequence of coin flips.

In this case, the probabilities have a well-defined meaning and the presence of actual data means the posterior density of the parameters will converge (assume the model is well-specified). Furthermore, we can (and should!) test the model. That is NOT the case in Carrier's work. His probabilities are all guessed (or for the prior, based on an obvious misapplication of reference classes) which is why we should be very concerned about numerical stability and ask what happens if we are slightly biased.

Bravo. I hope this makes sense to other readers, and that my example of rolling a die gives some further fleshing out of the issues in use of BT, but you have identified the crucial issues here: guessing priors (rather than having strong reasons for thinking they match the actual population probability) and the failure to carefully assess sensitivity of results to overaccurate assumptions.

JohanRonnblom wrote: At the end of the day, the usefulness of BT in OHJ (or history) is, in my opinion, not to get extremely accurate mathematical calculations, but to get clarity in the discussion. Which evidence is most consequential? Which disagreements on facts and arguments are most important?

Well, fine, and if Carrier had stuck to considering his arguments as "one way of going about it" as an illustration of the need for such clarity, I would be applauding him.

JohanRonnblom wrote:There is, I would add, one point where I agree with you: Carrier's application of a reference group for prior probability is a little odd and in my opinion not convincing for several independent reasons. There appears to be no obvious reason to choose that group, which leaves open the possibility that it was chosen out of thousands of possible reference groups simply because it happened to give numbers Carrier liked. There is no obvious causal reason for why being in this group would make Jesus unhistorical, which makes the conclusion seem spurious especially given the very small data set. Perhaps we could do this if we were dealing with a very large medical trial, even where we had no theoretical understanding of how a cure worked, but we would still be suspicious. And, as I have argued extensively, I do not believe that Jesus fits all that well into the chosen reference class. But this is not a flaw in Carrier's use of BT, but in the quality of his argumentation.

Tim Hendrix wrote:I wholeheartedly agree with you on this point. The typical response would be that this prior probability is "only the prior" and it is therefore not such an important concern. I believe (as I also express in my review) that this step should give us every reason to be concerned: Carrier is not simply using his reference class to assign the probability Jesus did not exist, but the probability Jesus did not exist and fit his 5 point hypothesis. That is relevant because his 5-point hypothesis is quite specific (i.e. a-priori unlikely) and fits the evidence like a glove, i.e. it will serve as to introduce bias in favor of mythicism which connects this point to the above discussion about numerical stability:

I see some of my concerns have been addressed already.

Talk of gloves reminds me of a similar abuse of evidentiary arguments. In the OJ trial, the hypothesis "all that is going on is a frame-up by a rabid racist cop" seemed very credible (high prior prob) to Black jurors, while the alternative "rabid racist cop is overzealous but the rest of the evidence makes a big difference" seemed more likely to White jurors. One can frame the effect of priors strongly by the mode of argumentation, and which possibilities get considered is a major portion of that framing.

Tim Hendrix wrote: It is after all much more difficult to explain why people wrote about Jesus as existing in the Gospels some 40 years after his supposed existence if you assume Jesus just did not exist (general hypothesis of mythicism), than if you assume he did not exist but people later believed he did (implied in Carriers 5-point hypothesis). Carrier has several arguments in Chapter six to explain away this move, none of which I think works and can easily be abused in other, similar situations.

I will give you two reasonable hypotheses for the delay in writing down the gospels. First, there is the usual one: the early church expected imminent parousia, the second coming and overturning of the order of the world. Until at least after the First Judean Revolt, it would have seemed a bit bizarre to be assuming a written record was valuable.

Second is the question of "scripture": until Paul's letters began to be circulated widely, the assumption would have been that LXX "scripture" had privileged status, and there was no reason to construct something with equivalent status for the NT. But as people saw the usefulness of the Epistles, they might reasonably have turned to setting down in writing many of the sayings and stories being circulated in the church.

I think those are the kinds of perspectives which mainline historians (e.g. Ehrman) have treated as most credible.

Tim Hendrix wrote:

JohanRonnblom wrote: Are you saying Carrier agrees that there is a formal error in OHJ? Where?

Yes, just read his reply to me, but the issue is so obvious it hardly needs mentioning.

I can't be bothered to read it in full, but a quick search reveals that he explicitly claims exactly the opposite: "[Hendrix' review] identifies no actual errors of logic, mathematics, or fact pertinent to the actual argument in OHJ".

Tim Hendrix wrote: I don't think it is fair that you say that I claim "Carrier's math is wrong" in general as I obviously only question specific arguments he is making and often that critique has nothing to do with broad formal errors. Please stick to the concrete points that I actually make.

Very good. But that means you are not a mathematician disagreeing with an historian on math. Now you are a mathematician disagreeing with an historian on history. Of course an amateur can still make good points. I'm an amateur on both subjects myself. But it is still valuable to be clear about this distinction. And, more importantly, you're not really bringing any arguments. You're basically just jumping up and down, screaming "but what if Carrier is wrong!?".

Tim Hendrix wrote:Carrier provides reasons which I discuss at length and if you follow the discussion it all comes down to what Carrier believes his Rank-Raglan prior can do.

No, it does not. Even if one disagrees with his use of prior, as I do pretty much completely, that does not invalidate his entire argument. Yes, it removes what he has numerically stated to be his strongest argument. But he has other arguments. Additionally his arguments and his scoring of his arguments are separate points: for instance I believe he scores his arguments against historicity from the Pauline epistles absurdly low.

Tim Hendrix wrote: I don't follow your alternative suggestion for how bias should be treated. Let's suppose we assume a bias of 10% and that the individual probability of historicity/mythicism that goes into BT are roughly the same (in the unbiased case). Can you please provide what calculation you are suggesting as the relevant one which shows that bias has no effect? If you can't, I think I am going to stick with what I have.

Of course bias has an effect. As I've said many times, bias has an effect regardless of whether you use BT or not. But you have claimed that Carrier's use of BT 'inflates' the bias. I have done a numerical experiment, using the actual calculation Carrier uses. It shows that this is not the case at all. Rather, because he uses BT he will get closer and closer to the correct conclusion when he adds more and more evidence. Obviously, this is not true if he is too biased or too wrong, but no method can save him, or anyone else, in that case. So that is definitely not a reason to criticise him for his use of BT. It's just a truism that someone who is wrong is wrong.

Tim Hendrix wrote: If Carrier's approach is not to make guesses (based on his personal evaluation of the evidence) of the various probabilities and combine them using the calculation in the book then I don't know what it is.

That is the method of every scholar on every subject. If you think he is wrong, explain why, and don't try to make it look like a mathematical argument.

Tim Hendrix wrote:If you wish to claim that's what statesticians are or should be doing, well, I will just invite you to read a book on statistics.

Of course it is. Give me an example of any real-world application of statistics without doing this and I'll tell you how. The difference between history and more 'hard' sciences is that in history we have sparse evidence and rarely do we get additional data to test our hypotheses. This makes history a subject that is in fact more vulnerable to bias and error. But that is a fundamental problem inherent in the subject of history, and not Carrier's fault.

Tim Hendrix wrote: Two points: Firstly, if someone asks me to guess the weight of a tree and I can't tell if it's weight is more or less than 1000kg then the right conclusion is to say that my best estimate is 1000kg. But that is not the same as saying that the highest (and lowest) estimate of the tree's weight is 1000kg, which is what we are discussing (Carriers optimistic/pessimistic estimates of the various probabilities).

No, that is not at all what we are discussing. Rather, the 'high' estimate in this case would be something like saying he can't believe a tree would weigh more than 2000000 kg, but let's add a zero to be on the sure side, so 20000000 kg. Of course you may disagree with that and say that such trees could exist, but then - again - bring some arguments, don't just complain that you are disagreeing without bothering much about the reasons.

Tim Hendrix wrote: Secondly, many scholars of early Christianity's claim that it IS possible to conclude something about the historicity of Jesus using evidence from the Gospels. In other words, at the very best Carriers estimate of probability are vastly more critical of Jesus historicity than what most scholars seem to believe. If Carrier is systematically too critical, well, you can see elsewhere what I believe will happen with such systematic bias.

I also think that it is possible to conclude something using evidence from the Gospels. While any of the Gospels could plausibly have been written about an historical person, I find it quite improbable that all four of them could have been written and preserved in the way they evidently have been, had Jesus been considered an historical person by the writers. This is because if they believed that they disagreed with their brothers in faith about facts, the texts would surely polemicize against the earlier texts on some of the many, many issues where they disagree. The authors were obviously aware that other members of the sect were all reading the earlier Gospels, and to simply state completely different turns of events without in any way explaining why this version would be more correct than the other version is, in my view, not compatible with a genre claiming to be about literal truths.

This is different from the rapid legendary development Carrier discusses, because in such cases, I believe, the people constructing the legends are in fact very much concerned about getting their perceived truth out and polemicizing against any alternative claims. So, we can draw the conclusions not from the plausibility or implausibility of the legends, but rather from the genre we judge the legends to belong to.

But anyway, all that you are saying is that Carrier disagrees with other scholars, which has never been in dispute.

Tim Hendrix wrote: You asked me for an example of where the probability ratio was one in both the optimistic and pessimistic case and I have given examples. Now you are moving the goalpost by saying that Carrier might hypothetically believe there is still some uncertainty (he just did not write that in OHJ...), or that we should really be asking about other probabilities, etc.

Not moving the goal posts at all. You clearly have not read or understood what Carrier writes (many many times). For instance:
"In reality we could assign lower estimates to P(e|¬h) in every case (to reflect how much the evidence might not be exactly what we'd expect if minimal mythicism were true), but we would then have to lower every corresponing value for P(e|h) according to th ratios determined in Chapeters 8, 9 and 11 (since the evidence isnever 'exactly' what we expect on historicity, either). See note at the end of Chapter 8, §1; and n. 10 in this chapter." [OHJ, p. 597]

So, these numbers are not anything at all like claiming 'absolute certainty'. They are just ratios.

Tim Hendrix wrote: I don't make general but specific claims: If we take Carriers computation, a bias of about 4% (and I define what that means) will invert the result of the computation.

It will not if you use his actual computation rather than your own invention, which he does not use. As I show in my simulation, his actual use of BT does not have this problem.

Also, a bias of 4% is ridiculously low in a field such as history. But note that this bias thing also goes for yourself. If you think Carrier is so 'obviously' wrong, that may be a result of your bias.

Harry Marks wrote: I think those are the kinds of perspectives which mainline historians (e.g. Ehrman) have treated as most credible.

Ehrman is simply not an historian at all. He is a theologian by education, degree and publication history. Though he prefers the title 'scholar of religious studies', which is often used to distinguish intra-religious theologians from those attempting to dispassionately describe religions. These are respectable academic fields, but not history. Ehrman simply has not studied history academically, does not work in a faculty together with historians, and does not publish in journals where historians publish.

JohanRonnblom wrote:

Tim Hendrix wrote:

JohanRonnblom wrote: Are you saying Carrier agrees that there is a formal error in OHJ? Where?

Yes, just read his reply to me, but the issue is so obvious it hardly needs mentioning.

I can't be bothered to read it in full, but a quick search reveals that he explicitly claims exactly the opposite: "[Hendrix' review] identifies no actual errors of logic, mathematics, or fact pertinent to the actual argument in OHJ".

Well, I am aware that Carrier writes that. Here is the example I had in mind: Carrier defines h as the hypothesis that Jesus existed (OHJ, p. 30). Then ~h would (formally) be the logical negation of h, however in OHJ "~h" is defined as the list of propositions in OHJ, p.53. From a formal point of view that is false as the list of propositions are not logically equivalent to the negation of h. Carrier is aware of this abuse of notation and mentions it in OHJ.
To meet a potential source of confusion, if you are only relying on OHJ and Carrier's response to me, I want to point out that Carrier makes a flatly wrong claim about what I say on this issue of ~h vs. "~h" in his response to me. However, I think that would be off topic for this book review.

JohanRonnblom wrote:

Tim Hendrix wrote: I don't think it is fair that you say that I claim "Carrier's math is wrong" in general as I obviously only question specific arguments he is making and often that critique has nothing to do with broad formal errors. Please stick to the concrete points that I actually make.

Very good. But that means you are not a mathematician disagreeing with a historian on math. Now you are a mathematician disagreeing with a historian on history. Of course, an amateur can still make good points. I'm an amateur on both subjects myself. But it is still valuable to be clear about this distinction. And, more importantly, you're not really bringing any arguments. You're basically just jumping up and down, screaming "but what if Carrier is wrong!?".

Again I think you are at a disadvantage by trying to characterize my review without having read it. As I lay out in the introduction: In this review I will try to examine the Bayesian content of Dr. Carrier's argument from the mainstream view of what constitutes valid and invalid application of Bayesian reasoning. Similar to how you can make a wrong computation in physics without doing something mathematically invalid, you can also apply BT in a way that is problematic without violating BT.
I will be happy to discuss this point further, however, I think it is OT with respect to this forum. I also disagree that I am not bringing any argument and wish to point out that this is again simply a sweeping statement. Furthermore, I don't think it is fair to say I am "jumping up and down" and "screaming".
[/quote]

JohanRonnblom wrote:

Tim Hendrix wrote:Carrier provides reasons which I discuss at length and if you follow the discussion it all comes down to what Carrier believes his Rank-Raglan prior can do.

No, it does not. Even if one disagrees with his use of prior, as I do pretty much completely, that does not invalidate his entire argument. Yes, it removes what he has numerically stated to be his strongest argument. But he has other arguments. Additionally, his arguments and his scoring of his arguments are separate points: for instance, I believe he scores his arguments against historicity from the Pauline epistles absurdly low.

I did not claim it invalidated "his entire argument" as it is clearly irrelevant to the historical perspectives he brings up in his book (and which, as you noted, neither of us a very qualified to discuss). However, Carrier's examination of the evidence takes the form of: "GIVEN my 5-point myth theory is correct THEN the probability of <The gospels> is (so and so)". The use of a particular theory of historicity, therefore (plausibly) affect all terms in the computation as I think it is self-evidently true elements of the 5-point hypothesis make some of the evidence easier to explain. For instance, it is easier to explain the gospels (stories of a man on earth) if you assume that early Christian communities believed in such a man on earth than if you do not make such an assumption. The use of a particular hypothesis (rather than a specific) must carefully be accounted for when the prior is estimated and therefore any issues with the prior potentially bias the entire computation.
[/quote]

JohanRonnblom wrote:

Tim Hendrix wrote: I don't follow your alternative suggestion for how bias should be treated. Let's suppose we assume a bias of 10% and that the individual probability of historicity/mythicism that goes into BT are roughly the same (in the unbiased case). Can you please provide what calculation you are suggesting as the relevant one which shows that bias has no effect? If you can't, I think I am going to stick with what I have.

Of course bias has an effect. As I've said many times, bias has an effect regardless of whether you use BT or not. But you have claimed that Carrier's use of BT 'inflates' the bias. I have done a numerical experiment, using the actual calculation Carrier uses. It shows that this is not the case at all. Rather, because he uses BT he will get closer and closer to the correct conclusion when he adds more and more evidence.

I am happy you bring up a concrete example we can discuss. The problem with your example is that you are not using Carriers estimates of the probabilities, but rather consider another, unrelated, example. If we use the actual computation Carrier uses, and your way of defining "bias" as additive, then it does invert the conclusion. I think this graph is correct but be aware I have not checked it very well:

JohanRonnblom wrote:

Tim Hendrix wrote: If Carrier's approach is not to make guesses (based on his personal evaluation of the evidence) of the various probabilities and combine them using the calculation in the book then I don't know what it is.

That is the method of every scholar on every subject. If you think he is wrong, explain why, and don't try to make it look like a mathematical argument.

Tim Hendrix wrote:If you wish to claim that's what statesticians are or should be doing, well, I will just invite you to read a book on statistics.

Of course it is. Give me an example of any real-world application of statistics without doing this and I'll tell you how. The difference between history and more 'hard' sciences is that in history we have sparse evidence and rarely do we get additional data to test our hypotheses. This makes history a subject that is in fact more vulnerable to bias and error. But that is a fundamental problem inherent in the subject of history, and not Carrier's fault.

I am a bit confused what you mean by the phrase "without doing this" and what you request of me. If you wish to claim that guessing probabilities of events and combining them with BT well describe applications of BT I have to disagree. Applications of BT rely on models containing parameters with a well-defined meaning (technically, we are not guessing the likelihood of the evidence as the probability is parameterized by the model) and where we know various concentration results. As for an example, see for instance the first example in "Bayesian Data Analysis" starting at 2.1: http://bacbuc.hd.free.fr/WebDAV/data/Bo ... alysis.pdf.

JohanRonnblom wrote:

Tim Hendrix wrote: Two points: Firstly, if someone asks me to guess the weight of a tree and I can't tell if it's weight is more or less than 1000kg then the right conclusion is to say that my best estimate is 1000kg. But that is not the same as saying that the highest (and lowest) estimate of the tree's weight is 1000kg, which is what we are discussing (Carriers optimistic/pessimistic estimates of the various probabilities).

No, that is not at all what we are discussing. Rather, the 'high' estimate in this case would be something like saying he can't believe a tree would weigh more than 2000000 kg, but let's add a zero to be on the sure side, so 20000000 kg. Of course you may disagree with that and say that such trees could exist, but then - again - bring some arguments, don't just complain that you are disagreeing without bothering much about the reasons.

Well, I agree that if we should say that the tree likely weights between so and so kg that requires an argument, yes. However, my point is much more basic: If we don't know the weight of a tree, i.e. that we don't know if it is more or less than 1000kg, it is still reasonable to recognize that the trees weight must fall within certain limits. What those limits are, I agree, require an argument.

On to Carrier. Carrier claims that the highest, and lowest, estimate of the probability of the Gospels agree (i.e. the ratio is 1 in both cases). That is what I don't think is fair. As you yourself note you estimate these rates as being different (i.e. the gospels are FOR mythicism), and many others believe they are FOR historicity. My point is then simply that the range this probability can take does not have length 0. I don't know what that range is, I agree, and an assertion of knowledge require an argument. That is why I only make general comments about how such ranges influence our certainty in the final conclusion, in this case, the ranges are magnified by about a factor of five.

JohanRonnblom wrote:

Tim Hendrix wrote: You asked me for an example of where the probability ratio was one in both the optimistic and pessimistic case and I have given examples. Now you are moving the goalpost by saying that Carrier might hypothetically believe there is still some uncertainty (he just did not write that in OHJ...), or that we should really be asking about other probabilities, etc.

Not moving the goal posts at all. You clearly have not read or understood what Carrier writes (many many times). For instance:
"In reality we could assign lower estimates to P(e|¬h) in every case (to reflect how much the evidence might not be exactly what we'd expect if minimal mythicism were true), but we would then have to lower every corresponing value for P(e|h) according to th ratios determined in Chapeters 8, 9 and 11 (since the evidence isnever 'exactly' what we expect on historicity, either). See note at the end of Chapter 8, §1; and n. 10 in this chapter." [OHJ, p. 597]

So, these numbers are not anything at all like claiming 'absolute certainty'. They are just ratios.

I have at all times recognized that these statements are about ratios, simply see my very own quote above where I state exactly that. What Carrier assumes is that the ratio of P(Gospels|h) / P(Gospels|~h) = 1 in both the optimistic and pessimistic scenario. Mathematically that is equal to the assumption P(Gospels|h) = P(Gospels|~h) in both the optimistic and pessimistic scenario. Can I prove to you mathematically that is impossible? No! of course not! Nobody can!. However, what I point out is that this is a remarkable assumption and would seem to be in conflict with the wide range other scholars (and yourself) assign to this ratio.

JohanRonnblom wrote:

Tim Hendrix wrote: I don't make general but specific claims: If we take Carrier's computation, a bias of about 4% (and I define what that means) will invert the result of the computation.

It will not if you use his actual computation rather than your own invention, which he does not use. As I show in my simulation, his actual use of BT does not have this problem.

Also, a bias of 4% is ridiculously low in a field such as history. But note that this bias thing also goes for yourself. If you think Carrier is so 'obviously' wrong, that may be a result of your bias.

Well, there is some hubris to that statement. Your simulations do not use his probabilities as you write yourself, mine does. If we do use Carriers probabilities, your conclusion simply does not hold as you can check yourself. Once again I stress this is a basic point of error analysis.
I will be happy to send you my code so you can replicate the results if you send me an email to [email protected].

You are right we both have to admit we may be biased, but that is why I try to limit myself to concrete points such as how bias affects the computation. Let us assume a bias of 4% is ridiculously low. To get a ballpark figure, what do you think is a fair estimate? Now that we have agreed bias should be taken into account, perhaps we should focus on how that should be done practically?

Tim Hendrix wrote: Well, I am aware that Carrier writes that. Here is the example I had in mind: Carrier defines h as the hypothesis that Jesus existed (OHJ, p. 30). Then ~h would (formally) be the logical negation of h, however in OHJ "~h" is defined as the list of propositions in OHJ, p.53. From a formal point of view that is false as the list of propositions are not logically equivalent to the negation of h. Carrier is aware of this abuse of notation and mentions it in OHJ.

This is getting repetitive. I already showed here that this is not false because Carrier does not need mythicism [according to Carrier's hypothesis] to be logically equivalent to the negation of historicity [according to Carrier's hypothesis]. He only needs it to be numerically equivalent. He has clearly stated the assumptions he makes that make this true. So there is no formal error. You responded to me saying you agreed. And now you are back claiming he made a formal error.

Tim Hendrix wrote: The use of a particular theory of historicity, therefore (plausibly) affect all terms in the computation as I think it is self-evidently true elements of the 5-point hypothesis make some of the evidence easier to explain. For instance, it is easier to explain the gospels (stories of a man on earth) if you assume that early Christian communities believed in such a man on earth than if you do not make such an assumption. The use of a particular hypothesis (rather than a specific) must carefully be accounted for when the prior is estimated and therefore any issues with the prior potentially bias the entire computation.

Bias the computation? How? Yes, obviously if your hypothesis is: "Some people believed that Jesus was a man" that explains stories of a man called Jesus better than if your hypothesis is "Bread is a planet made of jellyfish". But for the n:th time: if you have issues with Carrier's hypotheses, then state these issues. Don't just keep stating that "oh maybe there is something really fishy you know with Carrier's hypotheses that poisons everything, but oh I'm not going to say what is wrong with them, but just be really sure you can't trust anything Carrier says because you know he might just be really wrong here, really biased guy that Carrier is."

Tim Hendrix wrote: I am happy you bring up a concrete example we can discuss. The problem with your example is that you are not using Carriers estimates of the probabilities, but rather consider another, unrelated, example. If we use the actual computation Carrier uses, and your way of defining "bias" as additive, then it does invert the conclusion. I think this graph is correct but be aware I have not checked it very well:

You could at least check it well enough to see that neither Bob or Sue appear in Carrier's estimates of the probabilities or in his computation. I'm using Carrier's formula in the same way he uses it. If I use Carrier's numbers, of course I will get Carrier's results. We can't use his numbers because we do not know what the correct estimates of the probabilities are. So I use a thought experiment of a hard case where we know the correct estimates, which allows us to see the effect of error and bias.

But I think I can see what you're doing: you're simply assuming that every argument Carrier uses is always wrong, therefore the more arguments that are brought into the discussion, the more wrong Carrier is, and the more certain can we be that the opposite of what Carrier argues must be correct. In this reasoning, if we now say that arsenic must be healthy because kittens are cute and Carrier disagrees that we can draw this conclusion, well we are assuming that Carrier is always biased so therefore arsenic must definitely be healthy! This is just a fundamentally illogical argument.

Tim Hendrix wrote:If you wish to claim that guessing probabilities of events and combining them with BT well describe applications of BT I have to disagree. Applications of BT rely on models containing parameters with a well-defined meaning (technically, we are not guessing the likelihood of the evidence as the probability is parameterized by the model) and where we know various concentration results. As for an example, see for instance the first example in "Bayesian Data Analysis" starting at 2.1: http://bacbuc.hd.free.fr/WebDAV/data/Bo ... alysis.pdf.

In this example they are guessing that the probability that a newborn baby is female is described by a binomial model. This is a good enough guess for most purposes, but it is not precisely true. In reality, it is much more complicated than that. Indeed, it is in the modelling that the guesswork usually happens. It's no different for Carrier.

Tim Hendrix wrote: On to Carrier. Carrier claims that the highest, and lowest, estimate of the probability of the Gospels agree (i.e. the ratio is 1 in both cases). That is what I don't think is fair.

Then make an argument for that. Someone might say that kittens are a valid argument. It is clearly not Carrier's job to make everyone elses argument for them. He is simply stating which probabilities he personally finds to be within reason.

Tim Hendrix wrote: I have at all times recognized that these statements are about ratios, simply see my very own quote above where I state exactly that. What Carrier assumes is that the ratio of P(Gospels|h) / P(Gospels|~h) = 1 in both the optimistic and pessimistic scenario. Mathematically that is equal to the assumption P(Gospels|h) = P(Gospels|~h) in both the optimistic and pessimistic scenario.

What this means is simply that Carrier has not found any argument that he believes holds any water for why the Gospels would either prove or disprove historicity. Now, if you think there is any argument that he has either overlooked, or that he is treating wrongly, then it is very easy for you to put in some different numbers, and Carrier is inviting you to do exactly that. But, really, you need to bring some argument, and it should better be one that Carrier hasn't already treated (or a rebuttal to his counter-arguments, etc).

I think that Carrier has fairly assessed the arguments brought forward by other scholars, and frankly those arguments are so weak that they must be dismissed out of hand.

As for my argument, well I have not seen anyone else bring this up, so I can't blame Carrier for not taking it into account.

Tim Hendrix wrote: Well, there is some hubris to that statement. Your simulations do not use his probabilities as you write yourself, mine does. If we do use Carriers probabilities, your conclusion simply does not hold as you can check yourself. Once again I stress this is a basic point of error analysis.

It's not error analysis at all, but just bogus numbers thrown into a hat. If you apply that 'method' to any subject, you will find that the more reasons we actually have to believe that something is true, the more easily we are just victims of bias. If we have 1000 independent samples of DNA from the crime scene, analyzed by 1000 different labs, well, then in your model that makes the conclusion more uncertain than if we have only 1 sample, because any 'bias' will inflate the error.

No, really, you need to give some arguments for why you think Carrier is wrong. Because all you are saying here is that if Carrier is wrong, then he is wrong. It has absolutely nothing to do with BT or Carrier's use of BT. It's just a fundamental truism.

Tim Hendrix wrote: Now that we have agreed bias should be taken into account, perhaps we should focus on how that should be done practically?

As I showed in my simulations, the effect of bias will decrease as we add more evidence. Therefore, we should find as much evidence as possible and engage in a rational discussion about the consequence of each piece of evidence. Eventually, this should lead to a narrowing of the discrepancies between how people estimate the probabilities involved. Some disagreements will remain, but some will get resolved and others will be mitigated. This is how all science is done.