P.S. One way to walk through this is to use iteration. All Bayesian priors are the outputs of inserting prior evidence. So you can run the equation fully as if you “just discovered” f1. Then use the posterior this generated as the prior and run it again for f2, as if you “later” then suddenly discovered that.

Does anything change?

Does having f1 (which now is in b and not e) change P(f2|h) or P(f2|~h), because now it’s P(f2|h.f1) and P(f2|~h.f1)?

Then imagine history went the other way around. Does discovering f2 first and then f1, iterated in the same way, change anything? Or is the end result the same?

If either changes anything, then analyze why (and whether that change is actually warranted). Then we might get somewhere.

No. Or maybe yes.

f1 = “vanishing family”
f2 = “omission of Paul’s trials”

These are dependent probabilities. Although maybe you are using “essentially independent” in a colloquial way and you simply mean dependent probabilities without full causal determinism (like I think you mean at the end).

So, key point is, that doesn’t change their value.

Maybe you think that their being dependent probabilities somehow entails their probability values should change. By itself that is mathematically false. To see this, just try building any coherent argument for changing either of them. What do you think should change about them and why? Work through it and maybe you’ll get what I mean.

Because dependent probabilities still multiply. Exactly as I have them. There is no change to the math required by any dependency between them. As long as their probability value is correctly calculated on their dependency and not on an assumption of their independence, before multiplying them.

That is why f3 remains 1 on both h and ~h: its probability is dependent on other evidence (e.g. the Gospels). We fully expect Acts to include references to historicity derived from the Gospels (particulary GLuke itself) regardless of whether Jesus existed. What we need is something that is not fully causally expected like that.

There are only two things that are: the missing people (not just the brothers, but e.g. his mother and father) and the strange omissions of a historical ministry or any of its events in Paul’s trial speeches. Is either of those 100% expected given the other? No. Therefore, you have to determine what their dependent probabilities are before multiplying them. And it won’t be 100% on h, for either of them.

That’s what I did.

So all you can do at this point is argue for a different assignment of P to either or both, even by appealing to dependency relations. So, let’s see what your argument for a different value is. Then I can show you why you won’t end up with any meaningfully different result. Or you will convince me they should be altered.

As I wrote:

It is not simply the case that P(f1|f2) = 1, as if Luke’s omitting the entire family of Jesus from the church’s history logically entailed that he would omit references to a historical Jesus in Paul’s trials. It doesn’t.

That fully justifies my point.

There is no logically necessary relation between “Jesus existed and Luke omitted mention of his brothers” and “Jesus existed and Luke omitted any mention of that from the trial speeches of Paul.” The order of examination is not relevant (there is no logically necessary relation in the other direction either). Therefore they cannot wash out into a single probability. Their causal dependency is mediated by additional unrelated causes. Thus, you still have to determine the probability of each, when both exist. Yes, you are then calculating dependent probabilities. But dependent probabilities are still distinct probabilities.

So you aren’t making any relevant point here.

Try making a coherent argument that one of those two probabilities should be different than I assign them. Then maybe you will see the problem with what you are claiming.

I reversed f1 and f2 because that’s the order in which you dealt with them in the book.

I disagree with you here though:

“To make a valid critique of this, one has to demonstrate that those would not have the same value. You can’t simply assume that they won’t.”

If you assume independence, the burden is on you to justify it. I think it can be justified though, as I suggested in my last comment.

OK, I think we’re getting somewhere. Let me see if I’ve got your argument:

The odds ratio we want is

P(f1.f2|h)/P(f1.f2|~h) =
P(f1|h)/P(f1|~h) P(f2|f1.h)/P(f2|f1.~h).

You use the approximations

P(f2|f1.h) ≈ P(f2|h)
P(f2|f1.~h) ≈ P(f2|~h)

because f1 and f2 are essentially independent (and to the extent that they’re not, the approximation favors h).

Is that correct?

So, if you wanted a correct equation for h given Acts alone, it would be (excluding f3 now as irrelevant; it is simply a placeholder for the absence of any other determining evidence):

P(h|f1.f2.b)/P(~h|f1.f2.b) = { [P(f1|b.h)/P(f1|b.~h)] x [P(h|b)/P(~h|b)] } x [P(f2|f1.b.h)/P(f2|f1.b.~h)]

Which will have the same value as:

P(h|f1.f2.b)/P(~h|f1.f2.b) = { [P(f2|b.h)/P(f2|b.~h)] x [P(h|b)/P(~h|b)] } x [P(f1|f2.b.h)/P(f1|f2.b.~h)]

To make a valid critique of this, one has to demonstrate that those would not have the same value. You can’t simply assume that they won’t.

You didn’t “find P(f3|h) to be 1”, you found the ratio P(f3|h)/P(f3|~h) to be 1.

I actually did both. After factoring out the common coefficients of contingency (all the random factors that equally affect the outcome on either theory, e.g. the exact stories and words chosen by the author of Acts, which could have varied endlessly but make no difference to what we are measuring: see notes on p. 289, and directly on point, p. 605), P(f3|h) is indeed 1, as is P(f3|~h), i.e. the text (apart from f1 and f2) is 100% as expected on either theory (even given f1 and f2). That is why their ratio is 1 (as it would still be if we left those coefficients of contingency in, since by definition those have an identical value on either side of the ratio).

So are you saying that in the section of that chapter where you looked at f2, you were assuming f1 and f3?

Yes. And everywhere else in OHJ I break out individual textual evidence from the same author. Including the other way around (e.g. in Ch. 11 on the a fortiori side I assess the evidence of parentage and brothers separately but combine them in the final analysis, so the inclusion of both is stronger evidence for historicity than if we had only one; this is again a dependent probability assumption, since Paul would be the author of both or either).

This should be clear from the wording of the text of those chapters. I nowhere claim these are causally independent of each other. My assessment of their particular odds is always in respect to their own improbability even coming from an author we already assume is inventing history [~h] or recording it [h]. I even have a section in Ch. 9 on what happens if we remove the dependent assumption of any contact with history in Luke (so that he must be inventing regardless of ~h or h); and a section in Ch. 12 on what happens if we do abandon that assumption (pp. 603-04).

Hendrix I noted has a very hard time with colloquial English. I think he doesn’t do well ascertaining the mathematical model of an ordinary sentence in English, and has a tendency to import assumptions as to what is being said in a sentence, that is nowhere being said. Hence, for example, no one has any business assuming an estimate is independent unless it is declared to be. One should always first analyze whether what is being presented is a credible dependent probability because the math requires it to be. Rather than assuming someone is doing something else, you should always first assume that they are doing what they are supposed to be doing. And only if you then find an error can you claim there is one.

Your equation at the end has some mistakes in it, so I can’t answer that specific question, but I think you meant to write:

P(f1.f2.f3|b.h)/P(f1.f2.f3|b.~h) = [P(f1|f2.f3.b.h)/P(f1|f2.f3.b.~h)] x [P(f2|f1.f3.b.h)/P(f2|f1.f3.b.~h)] x [P(f3|f1.f2.b.h)/P(f3|f1.f2.b.~h)]

Which is not correct. Because in a single equation you have to decide where to put each f, whether in e or b; and you can run the equation iteratively, i.e. you’d have a run of equations rather than a single equation like this, where by the end all e ends up in b (since b, like h and ~h, has to have the same content across the equation).

I might have misled you there so let me explain it from a different angle:

Note that here, f3 is a meaningless import (it becomes like “the moon is not made of cheese,” a background fact of no relevance to the outcome), so it really need not be in the equation (any more than “the moon is not made of cheese” should be). It’s here just a placeholder for “there is no other evidence in Acts bearing on the question.” Had that not been the case, f3 would be something else, something specific, and we’d move this f3 to f4, and so on.

So really the only question is, can we evaluate f1 knowing f2 is coming up?

In other words, can we use the method of iteration (p. 240, 509. nn.) and still get the same result—as we should, since order of the presentation of evidence should not affect the outcome in most cases.

In that case, formally we would not have a single equation, but two in succession (the full equation only represents the final end result of all iterations). If it’s the case that we “find” f2 only after we calculated the effect of f1, so that we can’t have conditioned f1 on finding f2, would this change the math? The answer is no. Because the common dependent causes remain the same (h or ~h are the common root cause of both f1 and f2, and h or ~h are already conditional terms when evaluating f1 before we get to f2; likewise b already includes “the entire content of Acts has the same author,” hence both f1 and f2 are already conditioned on that fact as well, and so on). Insofar as f1 and f2 have separate necessary causes besides that, they don’t cross and thus don’t have to be accounted for as common conditions (e.g. on h, possibly “Jesus had no brothers” must be accounted for when evaluating P(f1|b), but it has no effect on P(f2|b); conversely, possibly “Luke unknowingly cut indicative material when he abbreviated Paul’s speeches” must be accounted for when evaluating P(f2|b), but it has no effect on P(f1|b)).

This is what I was explaining with the “number of brothers” counter-example. In that case, there is another common cause: once Luke makes the decision to exclude the family of Jesus regardless of why that is (whether “he had none/they had no historical role then” or “Luke wanted to erase them”; or, “there was no Jesus”), it causes the same outcome for f1, f2, f3, and f4, where these now are individual “missing brothers.” Then, “Jesus had no brothers” must be accounted for when evaluating P(f1|b), and does have an effect, indeed exactly the same effect, on P(f2|b), where “f2” is “second brother” rather than “Paul’s weird speeches.” Thus, we don’t get any further impact on P(h|e) from f2, f3, f4, etc. And that’s why I make no case that they do. The whole fact of missing family exhausts its effect in one go.

By contrast, our actual f2 (Paul’s speeches) is causally distinct from f1 (as in, f2’s presence or absence does not of itself affect the presence or absence of f1, but for their common cause, which is included within h and ~h and of course b, e.g. “the content of Acts has the same author,” which is always what P(f1|b) is conditioned on, requiring no knowledge of f2). That’s why I can treat them separately (and indeed, should). And that’s why an iterated method would get the same result, e.g. P(f1|b.h) will have the same value as P(f1|f2.b.h) once we recognize what’s already in h and b, like “Jesus existed” and “the content of Acts has the same author” and “the author of Acts had historical data of some kind, whether or however he used it” and so on.

To argue against this, one would have to demonstrate somehow that f2 would make f1 more likely on h than I estimate it to be (since it can’t make it more likely on ~h; on ~h it is already pegged out at 1), or somehow that f2 would make f1 less likely on ~h than I estimate it to be. Otherwise anything else would reduce, not increase, the probability of historicity (which is also logically possible, but I am assuming that isn’t a critique Hendrix intends to make, but one could do, i.e. someone could argue I under-estimated the impact of f1 on historicity, because of some overlooked causal relationship from f2, although I don’t see how—if I had, I’d have noted it).

So, Hendrix needs to do that. He can’t just say “Carrier overlooked a causal dependency.” He has to actually identify an actual causal dependency I overlooked. That kind of criticism I’d welcome. But you have to actually do it. You can’t just handwave and claim it “must” have happened “somewhere” even though you can’t find any examples of it happening.

“…what I present is P(f2|f1.f3.b.h) … against P(f2|f1.f3.b.~h)…”

So are you saying that in the section of that chapter where you looked at f2, you were assuming f1 and f3?

And in general, in the section of the chapter where you looked at fi, you were assuming the other 2 f’s?

Then you would be calculating

P(f1.f2.f3|b.h)/P(f1.f2.f3|b.~h) =
P(f1|f2.f3.b.~h)/P(f1|f2.f3.b.h) P(f2|f1.f3.b.h)/P(f2|f1.f3.b.~h) P(f3|f1.f2.b.h)/P(f3|f1.f2.b.~h)?

Is that what you did?

No, that’s incorrect. They can be dependent probabilities. And in every case you mention (which are not mentioned in Hendrix in this respect), they are. Perhaps you don’t know how the math works, but the correct way to multiply probabilities is to recognize their dependency. You do not have to assume their independency. If you don’t understand what I mean, I give another example in A Bayesian Brief on Comments at TAM. But I’ll explain using your example here.

First, note that I find P(f3|h) to be 1 all around, even at both margins of error, so it has no effect at all on P(h|e). This is actually because I am providing a dependent probability: if it is already the case that Jesus existed (or didn’t) and still Luke omits 1 and 2 (or also did), then the remaining contents of Acts have no further effect on P(h). This exemplifies how dependent probability works, and how it impacted my values: my P(f3) is already dependent on f1 and f2, and that’s why P(f1) washes out as 1/1, having no effect.

So I will assume you mean to focus on f1 and f2, the only features I find have any effect on the probability of historicity. I’ll proceed on that assumption.

Given that Jesus didn’t exist, the probability of those two observations (f1 and f2) is high-ish; but if Jesus did exist it is low-ish. But the probability of having only one of them (say, f2) is not as high; likewise, if Jesus existed, not as low. Our math has to account for this, because it’s not the case that, say, P(f2|f1) = 1; after all, Jesus could not have existed and still Luke made up references to his historicity in Paul’s speeches, and Jesus could have existed and somehow those speeches still lacked, or Luke removed (perhaps seeking brevity?), references to the fact. The probability of either is not zero, even given the absence of brothers (which can have its own separate cause—like, the historical Jesus had no brothers), and so it cannot be the case that P(f2|f1) = 1 or that P(f1|f2) = 1. Therefore, when I give a probability for any segregated item in Acts (which is really, only those two), it is the dependent probability; formally, what I present is P(f2|f1.f3.b.h) and P(f1|f2.f3.b.h), against P(f2|f1.f3.b.~h) and P(f1|f2.f3.b.~h). The math then follows, correctly as I calculate. Hence I am already running dependent probabilities. I am not running independent probabilities.

Another way to think of it is this: given that Jesus didn’t exist, we expect f1 and f2 given that Acts contained relevant sections f1* and f2*. For example, if Acts had no speeches of Paul, then we couldn’t say how likely their content would be on h or ~h and thus we’d lose the effect of f2 on the total probability of historicity, which means any negative effect I find it has would wash, and the probability of historicity would go up. The question then is: by how much would it go up? Well, by exactly as much as I assess it goes down—given that we do have f2* and it does have those omissions.

This is how cumulative evidence works. The more evidence in the same text for any hypothesis when that evidence is not strictly entailed by the previous evidence, the higher P(h|e) must go. You can’t claim “as soon as you have any single item of evidence in that text, then your P(e|h) is always fixed at that and never goes up no matter how many other items of evidence we find in that text supporting h.” Obviously, as some items of evidence in the text could be stronger than others. So they can’t all be the same, even individually, much less in sum. And this is because the dependent probabilities can be lower than one. So you have to work out when they are.

For example, it would be invalid to argue P(f1|e) should go up because Jesus had (so Mark says) four brothers and four missing brothers is more improbable than one missing brother. Because the missing of brothers is an all or nothing effect; the probability is therefore the same no matter how many there are supposed to have been. Once you have one brother prominently mentioned in the public history in Acts, P(f1|e) would swing in favor of historicity, the reverse of what I found. But that’s what I found only because (in fact) no brother is mentioned. That the other brothers vanish is then no more likely on either historicity or ~historicity. That’s how dependent probability works. Whereas, once no brother is mentioned, P(secondbrothermissing|allbrothersmissing) = 1, so adding a second brother’s disappearance has no further effect (because any probability multiplied by 1 remains unchanged). That’s also how dependent probability works.

But since Luke could have made-up stories about the brothers of Jesus, their being in the public history of Acts would only increase P(h|e), it would not make it equal to 1. And since Luke could have left them out for other reasons even if Jesus existed, their not being in the public history of Acts would not make P(h|e) equal to 0. So we still have to work out what the differential effect is. We can’t just set it to “1” or “0.”

And this effect is independent enough of whatever Luke decided to do (or had materials to use) for the speeches of Paul, such that P(f2|f1) does not equal 1 and thus, unlike stacking up missing brothers, their cumulative presence does have a differential effect on the probability. It short, it’s still less likely on historicity that Luke would do both, than that he would do only one or the other. So we still have to calculate how much less. The way we do this is with dependent probabilities, exactly as I do.

In colloquial terms, Acts has more oddities in it than it could have had; their cumulative effect must therefore be assessed. It is not simply the case that P(f1|f2) = 1, as if Luke’s omitting the entire family of Jesus from the church’s history logically entailed that he would omit references to a historical Jesus in Paul’s trials. It doesn’t. We thus have a causal question to answer: why did Luke do each of those things? Since his reasons could be different in each case, the probabilities are partially independent, and thus we have to ascertain their actual dependent probability, which is not going to be 1.

Another way to think of it is this: if we want to be able to test these things, we need to tease them out so we can see the effect of removing them on the conclusion. This is, I suspect, what Hendrix does not understand. So, what would P(Acts|h)/P(Acts|~h) be, without f2? That is, keep f1, but have a copy of Acts lacking f2. Obviously, P(Acts|h)/P(Acts|~h) should go up in such a case. Thus, even as a probability dependent on the presence of f2, P(f1|h)/P(f1|~h) still has a different value than P(f1.f2|h)/P(f1.f2|~h). We need to assess what value that is. And that’s what I do.

That’s why my estimates of the impact of this evidence (f1 and f2) are so low: they are taking into account their possible causal conjunction. After all, maybe Luke had separate or the same reasons to do f1 and f2 even though Jesus existed. Whereas if Jesus didn’t exist, then f1 and f2 are fully expected, and thus their dependence is irrelevant, because they are already topped out in probability, and thus have the same value even multiplied together (because 1 x 1 is 1); it’s therefore only their improbability on h that has any effect on the total probability. So then we have to assess whether the conjunction of f1 and f2, given h, is just as likely as f1 or f2 alone, and what I find is no, it is not. The conjunction is still less likely than either alone. Unlike in the case of f3, where I find the conjunction, given the same dependence, is as likely either way and thus has no effect. And all I am doing in OHJ is displaying that fact mathematically.

No, I’m referring to way you combine the probabilities of individual data points. For example, on the last page of chapter 9 you summarize 3 data points, let’s say

f1 = “vanishing family”
f2 = “omission of Paul’s trials”
f3 = “remainder of Acts”.

In the course of the chapter you have arrived at your values for each of P( fi | b), i=1,2,3. Then you get

P( f1.f2.f3 | b ) = P(f1|b)P(f2|b)P(f3|b).

But this is only valid if you assume that the fi|b are independent of one another.