Probability about Jesus (Christ) existence on earth

[Peter Kirby]

to Peter,

If we use the naive but wrong approach of assuming independence of events for no reason other than sheer bloody-mindedness:

0.99^26 * 0.95^63 = 0.030416

A word is not an event.

Says who?

According to the input data, the probability the whole TF if genuine is:
1 - [(1-0.99)^1 * (1-0.95)^1] = 1 - 0.9405 = 0.0595 = 5.95%

You're confused. That's supposed to be the 'probability that it is not the case that the whole TF is interpolated', by what you're attempting to do there. This includes three particular scenarios: G1 & G2, G1 & ~G2, and ~G1 & G2, where G1 and G2 stand for genuine of the first and second part. (That is, according to what you are attempting to do, and assuming 'genuine' is the same as 'not interpolated'.)

1st event: words discarded by Meier as interpolation (rated 99% probability of interpolation).
2nd event: words kept by Meir as genuine but regardless considered most likely interpolation (rated 95% probability of interpolation).

Then, we have:
0.99^1 * 0.95^1 = 0.9405 = 94.05%
And that result makes lot of sense in view the 99% and 95% implies some small doubts against the possibility of interpolations.

What's the probability that only Meier's special words are genuine, but that the rest are not? That is:

P(Special Part and NOT The Rest)

If you were consistent with preferring to stay with some kind of naive and ignorant approach, it would be (per the quote from you, above):

(1 - 0.99) * 0.95 = 0.0095

But it's not very reasonable to even allow 0.95% chance of that event. These are not independent events.

Apparently, when you are given instructive examples, you choose to try to stay within your comfort zone of ignorance. You're working harder not to understand this subject (by taking instructional material and then trying to argue with it and beat things down to the level with which you are comfortable) than it would actually take to start to understand it.

Just ridiculous. There's no point in arguing with you here. You are ignorant. You can either choose to learn, or not. You've chosen not to learn. So be it.

[Peter Kirby]

I'm sorry. This just goes back to the fact that you consider me so unreliable as a source of information, that you consider your own intuition to be a better source of information, in general, on the subject of mathematical probability. In other words, you can't really learn from me efficiently. You need to find a book or someone knowledgeable that you trust, to try to learn from. Otherwise you're just going to get bogged down in arguing over everything you don't completely understand yet ... never actually getting to the complete understanding part of it.

[Bernard Muller]

to Peter,

Says who?

I said it.

You're confused. That's supposed to be the 'probability that it is not the case that the whole TF is interpolated', by what you're attempting to do there. This includes three particular scenarios: G1 & G2, G1 & ~G2, and ~G1 & G2, where G1 and G2 stand for genuine of the first and second part. (That is, according to what you are attempting to do, and assuming 'genuine' is the same as 'not interpolated'.)

I still think this is the probability (5.95%) which represents the probability the TF (as a whole) being not interpolated, when working from the 99% & 95%. Can you prove it should be 0% with your math?

What's the probability that only Meier's special words are genuine, but that the rest are not? That is:
P(Special Part and NOT The Rest)

I do not care much about that. But:

If you were consistent with preferring to stay with some kind of naive and ignorant approach, it would be (per the quote from you, above):
(1 - 0.99) * 0.95 = 0.0095

Well 0.95 % is certainly acceptable, according to the input data. Of course, this 0.95% can be argued to be practically 0%.

But it's not very reasonable to even allow 0.95% chance of that event. These are not independent events.

I do not see any dependence here, except if the 26 words are interpolated, the 63 words may be also (but that would be disputed by Meier and others), as long as we have strong arguments for that.
However, that seems to be taken care by the 99% & 95%.

You're working harder not to understand this subject (by taking instructional material and then trying to argue with it and beat things down to the level with which you are comfortable) than it would actually take to start to understand it.

Actually, I have been working hard to understand your math, more of it each day.

I have one question for you: do you have something like what you call the "Obfuscator" in your examples with 10 & 8 events?
I think you do: by calling "evidence of interpolation": ~N (which is apparent), and "interpolation": X (which is not apparent). But I want to make sure.

Please don't get frustrated. I am still digesting your math.

Cordially, Bernard

[Peter Kirby]

Please don't get frustrated. I am still digesting your math.

Okay. Here's the Khan Academy course on 'dependent probability':

https://www.khanacademy.org/math/probab ... robability

I also linked some videos previously in this thread, that may help.

Says who?

I said it.

Two things, though:

(a) It's not true. We can make sense of the idea of "the probability that a word was interpolated" just as well as (no more or less than) the idea of "the probability that a sentence was interpolated." The sentence being interpolated is literally just the intersection of the events of all the words being interpolated. This might be more clear with some familiarity with the set-theoretic foundations behind math and talking about probability. But I also think it shouldn't be hard to grasp, which leads to...

(b) You're saying this as if it "refutes" the counter-example. But that's pointless. The understanding based on the idea that the special form of the probability rule is true in general... was wrong; this is just a counter-example meant to illustrate it. Someone with infinite patience and time could produce infinite examples. In fact, there have been at least a half dozen throughout the thread so far, using various concepts. So, there's no need to expend effort trying to save the idea. It's just not right.

Well 0.95 % is certainly acceptable, according to the input data.

Ah, but it isn't. See, the "input data" are insufficient, mathematically, to reach a conclusion, because it does not state conditional probability, and conditional probability is required in order to use the multiplication rule for the intersection of events.[1] So... nope!

And it contradicts any informed common sense. The idea that 1% is close to zero is a very ugly way to misunderstand probability. I could accept that one in a billion or one in a trillion is close to zero, in this context. But 1% would mean that we would expect something like this in roughly 1 out of 100 similar examples. However, the idea that a random jumble of nonsense like Meier's extra words just appeared here, is far more exceptional than 1 in 100. I'm disappointed that this was not your intuition also, because it would be easier if you at least understood straightforward cases like this.

I believe that the subject of "interpolations" is super-contentious and especially sensitive in this thread. It's a really bad way to learn the math concepts. I know you asked for "relatable" things, but it really isn't, because there is no shared understanding. Things that only you understand a particular way, which others understand differently, do not facilitate communication or understanding. We're better off with the card game example (say), which received no comment.

I have one question for you: do you have something like what you call the "Obfuscator" in your examples with 10 & 8 events?
I think you do: by calling "evidence of interpolation": ~N (which is apparent), and "interpolation": X (which is not apparent). But I want to make sure.

The "obfuscator" is supposed to represent our uncertainty regarding the truth/falsity of things. In the example with the 10 and the 8 events, it was never certain whether any particular result was true or false. Neither "evidence" or "no evidence" provided full certainty. (The original game with Dr. Q. was slightly different.) However, you were right to point out that the single category does not capture the full range of our beliefs about the strength of evidence (even though those examples didn't treat 'evidence' as proof, there was not any room for levels of 'evidence'). If it really helps, I can possibly do it again with different levels of 'evidence'. But then we'd still be talking about super-contentious examples, and I don't think we're in agreement on general concepts.

I could do more work to try to explain things, but I'm discouraged by the criticism/arguing/etc., so I don't think it will be any better received than the previous attempts from me and from everyone else who's posted in this thread.

[1] When you know that events are independent, you know the conditional probability: you know that it's the same as the unconditioned probability. This is a special case and it's really distorting your understanding because you're treating it as if it's completely natural that P(A | B) = P(A) when that's only one particular possible value for P(A | B) and can't be assumed to be true because it very frequently is not true.

[Bernard Muller]

Peter,
I don't want to bother you.
But I have a question about something which bothers me.
You wrote:

P(N1 | ~X1) = 0.95 - "95% chance of no evidence if not interpolated"
P(~N1 | ~X1) = 0.05 - "5% chance of evidence if not interpolated"
P(N1 | X1) = 0.8 - "80% chance of no evidence if interpolated"
P(~N1 | X1) = 0.2 - "20% chance of evidence if interpolated"

It's OK by me for your two first points.
However, is it realistic that, if interpolated we can have 80% chance of no evidence?
If we have interpolation, we should have a large probability of evidence.
I think:
P(N1 | X1) = 0.2 - "20% chance of no evidence if interpolated"
P(~N1 | X1) = 0.8 - "80% chance of evidence if interpolated"
is a lot more realistic.

Cordially, Bernard

[Peter Kirby]

Well, it's not a mathematical question, strictly speaking...

From a research standpoint, we would want to have some kind of situation (some data) where we can observe the actual frequency of 'evidence' against the actual frequency of 'interpolation'. This is difficult, to say the least. But, otherwise, we're just making up numbers.

I'm not sure what's "a lot" more realistic. I did not pick those numbers because I thought they were "a lot" less realistic. But, if you really want to see it done again with different numbers, that can be done (as you've noted yourself).

[Bernard Muller]

to Peter,
I redid the calculations for your case with 8 events (viewtopic.php?f=3&t=2753&start=210#p61941) but I changed:

P(N1 | X1) = 0.8 - "80% chance of no evidence if interpolated"
P(~N1 | X1) = 0.2 - "20% chance of evidence if interpolated"

to
P(N1 | X1) = 0.2 - "20% chance of no evidence if interpolated"
P(~N1 | X1) = 0.8 - "80% chance of evidence if interpolated"

I kept the rest of your input data the same than yours.

With your set of input data, you found 99.6% chance of the hypothesis that all the references were interpolated.
However, with my aforementioned change, I found only 11.4% chance of the hypothesis that all the references were interpolated.

Can you check if I made any mistake?
Thanks,

Cordially, Bernard

[Peter Kirby]

P(X1, X2, X3, X4, X5, X6, X7, X8 | ~N1, ~N2, ~N3, ~N4, ~N5, ~N6, ~N7, N8)

For abbreviation, "H" represents "X1, X2, X3, X4, X5, X6, X7, X8" and "E" represents "~N1, ~N2, ~N3, ~N4, ~N5, ~N6, ~N7, N8."

Scenario A Calculations

P(X1 | A) = 0.1
P(~X1 | A) = 1 - P(X1 | A) = 0.9

P(X1 | N1, A) = P(X1, N1, A) / P(N1, A) = (0.02 * P(A)) / (0.875 * P(A)) = 0.02285714285
since
P(X1, N1, A) = P(N1 | X1, A) * P(X1 | A) * P(A) = P(N1 | X1) * P(X1 | A) * P(A) = 0.2 * 0.1 * P(A) = 0.02 * P(A)
P(N1, A) = P(N1 | A) * P(A) = ( P(N1 | X1, A) * P(X1 | A) + P(N1 | ~X1, A) * P(~X1 | A ) * P(A) = (0.2 * 0.1 + 0.95 * 0.9) * P(A) = 0.875 * P(A)

P(X1 | ~N1, A) = P(X1, ~N1, A) / P(~N1, A) = (0.08 * P(A)) / (0.125 * P(A)) = 0.64
since
P(X1, ~N1, A) = P(~N1 | X1, A) * P(X1 | A) * P(A) = P(~N1 | X1) * P(X1 | A) * P(A) = 0.8 * 0.1 * P(A) = 0.08 * P(A)
P(~N1, A) = P(~N1 | A) * P(A) = ( P(~N1 | X1, A) * P(X1 | A) + P(~N1 | ~X1, A) * P(~X1 | A ) * P(A) = (0.8 * 0.1 + 0.05 * 0.9) * P(A) = 0.125 * P(A)

P(N1 | A) = P(N1 | A, X1) * P(X1 | A) + P(N1 | A, ~X1) * P(~X1 | A) = 0.2 * 0.1 + 0.95 * 0.9 = 0.875
P(~N1 | A) = 1 - P(N1 | A) = 0.125

Thus:

P(E | A) = (0.875 ^ 1) * (0.125 ^ 7) = 4.17232513e-7

Scenario B Calculations

P(X1 | B) = 1
P(~X1 | B) = 1 - P(X1 | B) = 0

P(X1 | N1, B) = P(X1 | B) = 1
P(X1 | ~N1, B) = P(X1 | B) = 1

P(N1 | B) = P(N1 | B, X1) * P(X1 | B) + P(N1 | B, ~X1) * P(~X1 | B) = 0.2 * 1 + 0.95 * 0 = 0.2
P(~N1 | B) = 1 - P(N1 | B) = 0.8

Thus:

P(E | B) = (0.2 ^ 1) * (0.8 ^ 7) = 0.04194304

Updated Scenario Probabilities

P(A | E) = P(A, E) / P(E) = 3.75509262e-7 / 0.00419464195 = 0.00008952117
P(B | E) = 1 - P(A | E) = 0.99991047883

since

P(A, E) = P(E | A) * P(A) = 4.17232513e-7 * 0.9 = 3.75509262e-7
P(E) = P(E | A) * P(A) + P(E | B) * P(B) = 3.75509262e-7 * 0.9 + 0.04194304 * 0.1 = 0.00419464195

Hypothesis Probability

Recall that:

P(X1 | N1, A) = P(X1, N1, A) / P(N1, A) = (0.02 * P(A)) / (0.875 * P(A)) = 0.02285714285
P(X1 | ~N1, A) = P(X1, ~N1, A) / P(~N1, A) = (0.08 * P(A)) / (0.125 * P(A)) = 0.64
P(X1 | N1, B) = P(X1 | B) = 1
P(X1 | ~N1, B) = P(X1 | B) = 1

Because of conditional independence (and because the equations for X2, X3, etc. are similar to the above), we multiply to get:

P(H | E, A) = (0.02285714285 ^ 1) * (0.64 ^ 7) = 0.00100526777
P(H | E, B) = 1^10 = 1

Finally, we can use the law of total probability to get the result of interest:

P(H | E) = P(H | E, A) * P(A | E) + P(H | E, B) * P(B | E) = 0.00100526777 * 0.00008952117 + 1 * 0.99991047883 = 0.99991056882

99.99% now...

You have effectively made "evidence" into something "more meaningful" for indicating the "truth" of interpolation.

There are better ideas for getting that number down, such as introducing multiple categories of "evidence" and putting some of them into weaker categories (not a stronger one, as you've done here).

[Bernard Muller]

to Peter,
Yes, I made a mistake in my calculation, using a * instead of a + in one place.
The good news is I got the whole thing on a spreadsheet and it worked for all the three cases, except for mine, with 10 events.
But I think you made a mistake on your last calculation: for P(B I E) you have 0.035064. It should be 0.03821.
That changes the overall result slightly from 0.035065 to 0.03827.
When time permitting, I'll experiment with different inputs and "publish" my result on this thread. I am open for testing any set of inputs that you or anyone else see fit.

Cordially, Bernard

[Bernard Muller]

P(A) = 0.9 - "the scenario where interpolations had occurred each with 10% probability" (chance interpolations)
P(B) = 0.1 - "the scenario where interpolations had occurred each with 100% probability" (orchestrated interpolations)
P(N1 | ~X1) = 0.95 - "95% chance of no evidence if not interpolated"
P(~N1 | ~X1) = 0.05 - "5% chance of evidence if not interpolated"
P(N1 | X1) = 0.8 - "80% chance of no evidence if interpolated"
P(~N1 | X1) = 0.2 - "20% chance of evidence if interpolated"

For 8 X's & 7 ~N's: 99.6%
For 8 X's & 6 ~N's: 98.6%
For 8 X's & 5 ~N's: 95.0%
For 8 X's & 4 ~N's: 84.2%
For 8 X's & 3 ~N's: 59.7%
For 8 X's & 2 ~N's: 29.2%
For 8 X's & 1 ~N's: 10.3%
For 8 X's & 0 ~N's: 3.1%

We need to assume at least 3 or 4 items are known to be sure interpolations to say the other items are also interpolations.

Keeping the same inputs, except changing two as such:
P(N1 | X1) = 0.2 - "20% chance of no evidence if interpolated"
P(~N1 | X1) = 0.8 - "80% chance of evidence if interpolated"

For 8 X's & 7 ~N's: 99.99%
For 8 X's & 6 ~N's: 99.74%
For 8 X's & 5 ~N's: 93.44%
For 8 X's & 4 ~N's: 33.72%
For 8 X's & 3 ~N's: 1.784%
For 8 X's & 2 ~N's: 0.064%
For 8 X's & 1 ~N's: 0.000%
For 8 X's & 0 ~N's: 0.000%

We need to assume at least 5 items are known to be sure interpolations to say the other items are also interpolations.

Cordially, Bernard