Carrier's numbers and math in OHJ

[GakuseiDon]

Going back to Carrier's calculations:
The people which are polled are four in number, and named:
Extra, Acts, Gospels and Epistles: each one of them voting (let's say) 20% for MH (1/5).

For Carrier, with his multiplications, the conclusion of that poll would be 0.16 % for MH (quite a bit different of 20%!).

Where did I go wrong?

By using the wrong example. Polling uses averages. But we are not looking for an average here. Think of it being a horse race. In Race A, a horse has 1/5 odds. In Race B, another horse has 1/5 odds. If you heard that both horses had won their respective races, how would you calculate the odds of that happening? Would you use an average of those odds, or multiply them together?

Going back to your example: Let's assume that, after examining the evidence for Extra, Acts, Gospels and Epistles, we conclude that the odds for each is 1/5 (that is, 1 (MH) / 5 (MM). What are the odds when looking at all those four elements?

We start with Extra: After doing our examination, we conclude that MH is 1/5 of MM. That is, MM is five times more likely to explain the contents of Extra. If that is the only element we examine, that would provide final odds of 20% for MH as a percentage of MM. (This is a very bad result for MH!)

But we then look at Acts: After doing our examination, we conclude that Acts also shows that MH is 1/5 as likely as MM.

So what does that do to the results for both? How do you calculate the odds of both those elements together?

We have three scenarios:
(1) We add the results together. 1/5 + 1/5 = 2/5. But obviously that makes no sense here in this context.
(2) We average the results. So (1/5 + 1/5) / 2 = 1/5. This seems to be something like your approach in your example above. So, even though Extra is 1/5 (MH/MM) and Acts is 1/5 (MH/MM), the odds favoring MH do not change. (This is like trying to average the odds of horse races, so the wrong approach IMO)
(3) We multiple the results. So (1/5 X 1/5) = 1/25. This is what Carrier has done. So Acts' 1/5 strengthens the case for MM based on Extra, and does so dramatically. MH drops from 20% of MM, down to 4% of MM. And we still have two more elements to go!

So do we average the results (like for a poll), or do we multiple the results (like for horse races)? That's the question you need to answer.

Let's say instead that for Acts, the result is 1/1. That is, MH is no more likely than MM to explain the contents of Acts. What does that do to the results? Using the same 3 scenarios:
(1) We add the results together. 1/5 + 1/1 = 6/5. Again, doesn't make sense in context.
(2) We average the results. So (1/5 + 1/1) / 2 = 3/5. Thus MH for both elements is now about 60% of MM, simply because Acts doesn't favour MH over MM.
(2a) Or do we use 0/0 for Acts? In that case, the odds become 1/10. So, even though Acts gives us nothing either way, the odds for MH weaken by half.
(3) We multiple the results. So (1/5 X 1/1) = 1/5. Thus Acts makes no difference to the odds coming from Extra.

Or is there another scenario that I am missing?

[Peter Kirby]

It needs to be remembered that Carrier on that page is calculating P( e | h ), not P( h ) and not the posterior probability P( h | e ). At one point Bernard seems to have remembered that, but lately he seems to have left that at one side. Carrier is not here looking for the (posterior) probability of the hypothesis. He is looking for the _conditional probability_ of the evidence, _given_ the hypothesis. This is just one step of a Bayesian calculation. He will also need to calculate (or assign) the conditional probability of the evidence given the negation of the hypothesis, as well as the prior probabilities of both, before coming to the question of the posterior probability that he calculates for the hypothesis.

To use the UFO example, let's take a stab at the question of whether there are aliens with FTL travel visiting the planet at least once a year.

Background probabilities? Let's say that aliens anywhere in the universe have 50/50 probability but that the FTL visitation bit (largely contrary to modern physics but with a generous assumption that these physics are incomplete and may be controverted) tips it down to a 25% background probability for the hypothesis that aliens with FTL travel are visiting the planet at least once a year.

Then we come to the particular evidence. Let us say that there are two kinds, the STORIES and the PHYSICAL.

The story evidence is that people report abductions and sightings. This has a very high probability under the hypothesis of alien visitation (let's say 95%) and a moderate probability under the hypothesis of no alien visitation (let's say 60%).

The physical evidence is that there are crop circles. But the physical evidence is also that there are no alien specimens, no parts of alien craft, no clearly visible aliens walking around on dashcams in Russia or security cameras anywhere, no very reliable recordings at all even. So let's say that this physical evidence and its extent has a low probability under the hypothesis of alien visitation (let's say 25%) but a high probability under the hypothesis of no alien visitation (let's say 90%).

Now we need to know the conditional probability of all the evidence given the hypothesis of alien visitation. We're assuming that the two conditional probabilities are independent, and we arrive at the intersection (the probability that all of the evidence obtains, not just some of it, given alien visitation) as 0.95 times 0.25, which is 0.2375.

Likewise we need to know the conditional probability of the evidence given the hypothesis of no alien visitation. Similarly, 0.6 times 0.9, which is 0.54.

(These two different conditional probabilities don't add up to 100% and aren't supposed to. Rather, they complement the conditional probability that _some of the evidence does not obtain_ under the hypothesis and again under the negation of the hypothesis, respectively. It is with those figures that they add up to 100%. That is, they complement P( ~e | h ) and P( ~e | ~h ), respectively.)

What we want to know is P( h | e ). The reason we got the above numbers is because Bayes' formula lets us get from one to the other:

Bayes theorem (corollary, i.e., "Alternative Form")



We want to know P( h | e ), the posterior probability of the hypothesis, and the above formula does it:

e = evidence, h = alien visitation every year

P( h | e ) = P( e | h ) * P( h ) / ( P( e | h ) * P( h ) + P( e | ~h ) * P( ~h ) )
P ( e | h ) = 0.2375
P ( e | ~h ) = 0.54
P ( h ) = 0.25
P ( ~h ) = 0.75

P( h | e ) = 0.2375 * 0.25 / ( 0.2375 * 0.25 + 0.54 * 0.75 ) = 0.12786
= 12.8% probability of alien visitation every year on these numbers

P( ~h | e ) = 0.54 * 0.74 / ( 0.54 * 0.75 + 0.2375 * 0.25 ) = 0.87213
= 87.2% probability of no alien visitation every year on these numbers

I don't enjoy arguing math with a hostile pupil, but I do hope that a worked example (based on real mathematical formulae, not muddling) is helpful.

[Peter Kirby]

As GakuseiDon suggests, there are in fact some problems or issues with Carrier's math.

First, the issue that Carrier treats as binary what is not binary. There are other possibilities besides minimal historicity and minimal mythicism. While historicity and non-historicity (mythicism?) can be considered a hypothesis and its negation, these other things aren't.

Second, the issue of whether the individual pieces of evidence and the conditional probabilities assigned to them can be considered independent. What might it mean if they are not? Well, for example, let's say that instead of having a category for "Gospels," instead Carrier had a category for "the Gospel of Mark," "the Gospel of Luke," "the Gospel of Matthew," "the Gospel of John," "the Gospel of Thomas," and "the Gospel of Peter." It's easy to see why he doesn't! Because, while we'd get very different results if we did this, we'd have foundered on the fact that these probabilities are in no way independent. In the case that the evidence of Mark does not obtain (in some other universe), it is less likely that the rest (Matthew and Luke) obtain, etc. But to use straight multiplication requires the assumption that each piece has its own and the same probability of obtaining, whether or not any of the others have obtained (ie mathematical independence).

But you could say this is a bit unfair because Carrier's math is basically an aid to honesty. The probabilities themselves are nothing more than "reasonable assumptions," so it is only left for Carrier to make more "reasonable assumptions" regarding the independence of the probabilities used and the negligible status of hypotheses other than the two under examination (which he spends some time on).

[Bernard Muller]

to Gakuseidon,

By using the wrong example. Polling uses averages. But we are not looking for an average here. Think of it being a horse race. In Race A, a horse has 1/5 odds. In Race B, another horse has 1/5 odds. If you heard that both horses had won their respective races, how would you calculate the odds of that happening? Would you use an average of those odds, or multiply them together?

I understand but in order to achieve the conclusion you are stating (the two horses winning their respective race) you depend on two factors: horse C has to win race A, while horse D has to win race B. In that case, I fully agree with you.
When a hypothesis A is TRUE at 1 out of 5 times, but is fully dependent of another hypothesis (TRUE at also 1 out of 5 times) being TRUE (for (hypothesis A being TRUE), then the overall result will be 1/5 x 1/5 = 1/25. Agreed.

However, Extra odds are fully independent from Acts odds. Extra odds do not depend on Acts odds in order to be TRUE.
And in my example about UFOs and aliens, the same thing.

So I would change your example to make it similar to what I propose:
Think of it being a horse race. In Race A, a horse has 1/5 odds. In Race B, another horse has 1/5 odds. If you heard that one horse (out of the two) had won its race, how would you calculate the odds of that happening?
(1/5 odds for race A horse) + (4/5 x 1/5 = 0.8/5 odds for race B horse) = 1.8/5 = odds ratio of 0.36 = converted in probability: 26.5 %
For three horses in three races, with same 1/5 odds:
(1/5 odds for race A horse) + (4/5 x 1/5 = 0.8/5 odds for race B horse) = 1.8/5 + (3.2/5 x 1/5 = 0.64/5 odds for race C horse)= odds ratio of 0.49 = converted in probability: 33.0 %

If the odds on Extra are 1 to five to be TRUE (TRUE = supporting MM) and the odds on Acts are 1 to five to be TRUE, then the combined odds of Extra & Acts will be 1.8 to 5 to be TRUE, with odds ratio of 0.36, converted in probability: 26.5 %

Use two dices: assign TRUE for even numbers (odd 1 to 1) => 1/1. How many times one of the two dices are going to show TRUE (one is enough). You need one dice to show TRUE.
Throw the dices together many times. Note when one dice at least will show TRUE.
I am saying one of any of the two dices will show TRUE three times out of four times.

Carrier has been proposing that in order for the final result to be TRUE, all four components (Extra, Acts, Gospels & Epistles) have to be TRUE. No way. If only one is TRUE, the others do not matter.

If in a court of Law, one piece of evidence is so obvious, clear & direct (such as the suspect saying he/she is guilty as charged) in order to incriminate the suspect of a crime, and the defense cannot think of anything in order to raise doubt about that piece of evidence, then who would need weaker pieces of evidence against the suspect?

Here is the issue.

Cordially, Bernard

[Peter Kirby]

The probability P(e | h) is not stating the odds that a piece of evidence secures the conclusion of the hypothesis, which is how you are understanding it.

Such a concept is not used anywhere by Carrier. It is not the way Bayesian arguments work.

The probability P(e | h) is stating the conditional probability that "e" would occur given that a hypothesis "h" happened.

For example, let's say that the "e" is "the sidewalk is wet now" and that the "h" is "it rained considerably on the sidewalk five minutes ago."

P(e | h) = 0.999

If it rained considerably on the sidewalk five minutes ago, there's 999 out of 1000 odds that we would find the sidewalk wet now.

We would be completely retarded if we then use that to say that there's a 99.9% chance that it rained considerably on the sidewalk five minutes ago, even though that is exactly what you are doing when you misunderstand the meaning and structure of the mathematical argument.

[Bernard Muller]

Is Carrier not making math mistake in OHJ?
Let's again look at:
OHJ extract 1

In PRIOR PROBABILITIES, Carrier has for P(h), ODDS 1/2 with PROBABILITY 33% (1/3)
In PRIOR PROBABILITIES, Carrier has for P(-h), ODDS 2/1 with PROBABILITY 67% (2/3)

No problem here, he has the probabilities calculated correctly.

However, right below, in CONSEQUENT PROBABILITY ON MINIMAL HISTORICITY (h)
Carrier has for GOSPELS 1/1 for ODDS & 100% for PROBABILITY.
That does not make sense and not according to the equation he used above.
The probability for 1/1 should be between the ones for 1/2 and 2/1, and with a calculated value of 50%.
The same error for 46.08%": should be 31 %.
The same error for 72%": should be 42 %.
The same error for 100%" relative to EPISTLES: should be 74 %.
The same error for 10%": should be 9.1 %.
etc.

It looks to me that Carrier was not too careful about his math he used before he got to the Bayes theorem and if he made errors here, he could also make errors there.

I'll take the day off tomorrow. I have a backlog of things to do other than arguing about Carrier's math.

Cordially, Bernard

[Peter Kirby]

However, right below, in CONSEQUENT PROBABILITY ON MINIMAL HISTORICITY (h)
Carrier has for GOSPELS 1/1 for ODDS & 100% for PROBABILITY.
That does not make sense and not according to the equation he used above.
The probability for 1/1 should be between the ones for 1/2 and 2/1, and with a calculated value of 50%.
The same error for 46.08%": should be 31 %.
The same error for 72%": should be 42 %.
The same error for 100%" relative to EPISTLES: should be 74 %.
The same error for 10%": should be 9.1 %.

No, not at all. For whatever reason, he chose to give the amount both as a fraction and in decimal.

And in the section above that, on prior probabilities, he chose to give it in relative odds form, like 2:1 (which he writes "2/1"), which means 2/3 to 1/3.

Perhaps he should have used a colon instead of a forward slash in the section on prior probabilities, for clarity, but that (and the confusing headings that might be read to suggest that "odds" and "probability" are not the same thing, when they are; or that the numbers under prior probabilities' "odds" are to be read the same way as the numbers under the consequent probabilities' "odds" are, which they are not) would be his only crime there.

You might indeed be making an argument for avoiding difficult math in history, but only incidentally, in that others will not be able to follow it (or do their own work when they are publishing) due to their lack of expertise.

It looks to me that Carrier was not too careful about his math he used before he got to the Bayes theorem and if he made errors here, he could also make errors there.

It looks that way because you are looking for it, and, again, because you are incompetent regarding the math.

I'll take the day off tomorrow. I have a backlog of things to do other than arguing about Carrier's math.

Some more productive (or enjoyable!) use of your time is obviously available, other than coming back to this again, but suit yourself.

If you insist, perhaps you can try learning the math first this time?

[Bernard Muller]

to Peter,

The probability P(e | h) is not stating the odds that a piece of evidence secures the conclusion of the hypothesis, which is how you are understanding it.
Such a concept is not used anywhere by Carrier. It is not the way Bayesian arguments work.

Maybe he should have used that concept instead of torturing his input data in order to fit his Bayes theorem.
Actually, the final Bayes theorem is an anti-climax. Just a ratio of odds reduced by the multiplication of a Prior. So far, I did not find anything wrong with that, I mean the math, even if I object about the input values.
With my concept, I can also put a ratio of odds into that Bayes theorem.

We would be completely retarded if we then use that to say that there's a 99.9% chance that it rained considerably on the sidewalk five minutes ago, even though that is exactly what you are doing when you misunderstand the meaning and structure of the mathematical argument.

I do not see where I fit into that.

Cordially, Bernard

[Peter Kirby]

Maybe he should have used that concept

Maybe, but he doesn't, so you cannot use it to criticize his argument.

Is your idea such a wonderful one, anyway?

The Bayesian approach allows us to make a consistent estimate, given all the facts and assumptions taken into consideration, of the probability that the hypothesis is true or false.

Your concept regards a probability that the hypothesis is demonstrated to be true.

It's not really clear how we are to form intuitions regarding "a probability of proof," when that proof is not actually proof but falls short of proof. Especially when we also have to consider competing evidence regarding a probability of disproof.

Suppose that I have a fingerprint from you, Bernard Muller, on the weapon used to kill a man--and nobody else's. A ha! We have it! It is a 90% proof of guilt! But suppose that we also have you on tape with a million witnesses because you were on Dancing with the Stars on the night of the murder. A ha! A bulletproof alibi. It is a 99% proof of non-guilt!

So we have a 90% proof of guilt and a 99% proof of non-guilt. But you can't be both guilty and not guilty. Your "concept" is half-hatched and doesn't really offer a superior, or even consistent, method of deriving the answer we seek.

Carrier uses the Bayesian approach because it is a valid, established, well-understood scientific and mathematical method. There's nothing wrong with that. And you're criticizing him because he doesn't go with Bernard Mullerian mathematics, whatever that is. Wonderful.

If your concept is such a great one, please, by all means, work it out and publish a paper on it. If it is already published, point to it. Kkthx.

[Bernard Muller]

To Peter,

No, not at all. For whatever reason, he chose to give the amount both as a fraction and in decimal.

And in the section above that, on prior probabilities, he chose to give it in relative odds form, like 2:1 (which he writes "2/1"), which means 2/3 to 1/3.

You did not understand. In PRIOR "section", he used the correct equation to calculate the probabilities from the Odd ratio, 2 to 1 and 1 to 2.
But in the CONSEQUENT section, he used a wrong equation in order to calculate the probabilities.
The right equation is p = ODD1 / (ODD1 + ODD2).
Carrier does not do that in the CONSEQUENT section. The equation he used is p = ODD1 / ODD2 . Wrong equation!

Cordially, Bernard