Without getting mathematical, I want to show how Mormons and non-Mormons disagree on things and why a lot of the differences can be illustrated by applying a result of probability theory known as Bayes Theorem.

First a couple of examples of Bayes Theorem in action:

(A) Suppose you have two coins in your pocket, a fair coin and a two-headed coin. You reach in your pocket and grab a coin and flip it. It lands heads. Now, what is the probability that it is the two-headed coin? Well there are FOUR possibile outcomes to the experiment: 1. fair coin heads, 2. fair coin tails, 3. two-headed coin heads and 4. two-headed coin the OTHER heads. If it lands heads we've ruled out option 2. So of the remaining equally likely options two of them are with the two-headed coin and one of them is with the fair coin. So the probability that you've chosen the two-headed coin is 2/3.

OK. you flip it again and it, again comes up heads. Now what is the probability it's the two-headed coin? Now there are FOUR ways to get two heads in a row with the two-headed coin (two choices for the first flip times two choices for the second flip) and only one way to get two heads in a row for the fair coin. So now the probablity it is the two-headed coin has risen to 4/5.

You flip it again and it comes up heads again. By this time you are becoming sure it's the two-headed coin, and the probability is 8/9. A fourth heads would make the probability 16/17 and a fifth heads in a row would make it 32/33.

We'll come back to the coins in a while. Here's a second applilcation of Bayes Theorem which is more practical

(B) There is a test for a certain disease which, if it says you have the disease is correct 95% of the time but gives a "false positive" 5% of the time. You take the test and it says you have the disease. Now, what is the probability you have the disease?

Answer: You have no idea. Not enough information has been given to answer the question (shockingly even many doctors do not get this one right)

Think of this. What if the disease is VERY rare, What if only one in a million people actually have the disease. That means that if the test gives a false positive 5% of the time then it will test positive on 50,000 of the million people who do not have the disease. So the question is, which is more likely, are you the one-in-a million who has the disease or one of the 50,000 who doesn't have the disease but tested positive anyway?

Going back to the coins, I started the problem with two coins. What if I change it to 1001 coins: 1000 fair coins and one two-headed coin? Now you reach in your pocket and randomly grab a coin. Three flips in a row give heads. There are 1000 ways this can be done with a fair coin (1000 coins to choose from each showing heads on 3 successive flips) and 8 ways it can be done with the two-headed coin. Now the probability it is the two-headed coin is 8/1008--MUCH less.

The difference between skeptics and apologists has to do with the initial "estimate" of how many "coins" are in the pocket or what percent of people "have the disease."

Let's apply this reasoning to the famous "NHM" found in Yemen. Mormon apologists are saying "we've found Nahom!" It's called the best evidence yet for the Book of Mormon.

If you start by ASSUMING the Book of Mormon is probably true, then the NHM would be a good choice for the BOM Nahom. If you give a 50% likelyhood to the Book of Mormon being true then finding NHM would up the chances, much like when there were only two coins and one flip came up heads. Or the likelyhood of actually having the disease when the test says you do, if the disease is a very common disease.

But if you start with the assumption that books given by angels are VERY rare, then your situation is more like the scenario with 1001 coins or when only one in a million have the disease. What is more likely? That the book was given by and angel and you've found NHM on a stone in Yemen, or that the book was not given by and angel and you've found NHM on a stone in Yemen? Clearly, if it is assumed that books given by angels are very rare then the second interpretation is more likely.

This is why Mormon apologists shake their heads at critics when they say "coincidence." "Oh," they say, "you're just saying that because you don't like the result." No, this is not why we're just saying that. We're saying that because coincidence in such investigations is the null hypothesis. It's ALWAYS possible to find coincidences. They crop up all the time. It is the duty of the person making the claim to prove that their find is NOT a coincidence, not the duty of the rest of the world to prove it is. As we saw with the 1001 coins, the "coincidence" of 8 heads in a row would be MUCH more likely than having grabbed the two-headed coin.

Let's extend this idea to two other aspects of "evidence." First, let's look at the idea that EXTRAORDINARY CLAIMS REQUIRE EXTRAORDINARY EVIDENCE. Two-headed coins are Very rare. If you have a coin which you've flipped four times in a row and has come up heads each of the four times, how justified would you be in concluding it's a two-headed coin? Given that fair coins outnumber two-headed coins by millions-to-one, you'd not be justified at all. It's MUCH more likely that it's an ordinary coin defying the odds (even though that's a one-in-sixteen chance with a fair coin) than it's a two-headed coin. To conclude it's a two-headed coin you'd have to get more like 25 heads in a row (or, more simply, look at both sides).

Another claim that we can look at is the idea that "ABSENCE OF EVIDENCE IS NOT EVIDENCE OF ABSENCE." This can be valid in some situations and invalid in others. Let's consider the question of life in the universe. Are we the only planet with life? The fact that we've found no life on any other planets would be an "absence of evidence." But we've looked at only a few planets. And we've not searched those exhaustively. Results in astronomy in the past quarter century have shown that planetary systems around stars are not at all uncommon. There could easily be trillions of planets in the universe. The absence of evidence after looking at a handful is not evidence that none of the others have life. We have to ask the following question: What is the probablity that if there is life out there, on, say, a million different planets that our efforts so far could have missed it? Even with a million life-supporting planets we can expect to not have found one yet so the absence of evidence so far doesn't mean much.

However let's look at a different situation: the absence of archaeological evidence for the Book of Mormon. What is the probability that IF there were this advanced civilization which spoke and wrote a hybrid of Hebrew and Egyptian, used horses and chariots, cultivated wheat and barley, smelted iron and steel, had a Judeo-Christian religion and "filled the land," growing into the millions etc. that it wouldn't have been discovered?

If you were to have asked this question in 1830 there could be a good case such a civilization could had not yet been detected. However in 2015 after teams of archaeologists have combed every square mile of the Americas, using infra-red and ground-penetrating radar, and other modern technologies, and having searched for every bit of archaeological information they could find, then it stretches credulity to the breaking point to claim that they just haven't looked in the right place yet. In such a situation it is proper to say that the absence of evidence is strong evidence of absence.



Edited 2 time(s). Last edit at 08/02/2015 04:05PM by baura.

Thanks baura.

Wonderful post!!!

(This is the first time I have "sort of" understood this...so truly muchly appreciated. :D )

NHM wouldn't even be 50%. NHM is one of the most common phrases of letters and there are 36 instances of NHM that date back to the time of Lehi. NHM refers to the tribe Nihm and its something that has to do with ancestry. It's not a place name at all. They point to the fact that it's near a burial ground but the thing is that it has nothing to do with the burial ground and it's no surprise that the Nihm tribe is near at least one considering that there are so many.

Amazing post Buara

"NHM refers to the tribe Nihm and its something that has to do with ancestry. It's not a place."

Exactly. We know what the letters mean and it's not "Nahom". So, it's not even close to 50%...it's 0%. That's the best evidence apologists have for the BoM...nothing.

Suppose a large, previously unknown species was found in the Americas that lived 2,600 years ago and died out some time after that. Apologists claiming that they were cureloms or cumoms, although a stupid argument, would be a better argument than the NHM argument.

Speaking of coincidence...

NT, KJV, James 1:5

"If any of you lack wisdom, let him ask of God, that giveth to all men liberally, and upbraideth not; and it shall be given him."


Over the centuries, how many devout (or thinking themselves devout) christians have read James 1:5 and attempted to log in with ghawd in order to browse ghawdipidea.org?

There are three 18th century American religions that claim to have or have had an direct, open line with ghawd's office:

1. los mormones 15.3 million
2. 7th Day Adventists 18.1 million
3. Jehovah's Alibis (Jesus couldn't have robbed the bank! He was here with us the whole day!!) 7.9 million

In terms of accuracy, the connoisseur knows that the mormon number is extremely suspect and that in terms of active members, it is in third place.

So when you stand back and look at ALL the evidence you can gather, why on ghawd's earth would you believe that ghawd picked Joseph Smith, and how long did ghawd sit on his throne, drumming his fingers, waiting for JS to read that verse and act on it?

To bring this back to Baura's excellent post, Joseph Smith is the ultimate double-faced coin; no matter how many times you flip him...

Nicely done as always, baura.
I love explaining Bayes' theorem to beginning statistics students. Once they get it, stats makes a lot more sense. They also start to notice how unsupportable some of the claims they hear in their daily lives are. It's a real eye-opener for them.

I think you may have garbled that false-positive example a bit. "If it says you have the disease [then it] is correct 95% of the time" means just that: if your test is positive, then you are 95% likely to have the disease.

I believe you meant to say that if you really have the disease, then 95% of the time the test will say you have it, but if you really don't have it, then 5% of the time it will still say you have the disease (false positive). This is the scenario where it sounds as though a positive test result is almost surely bad news, when in fact it may be almost surely nothing, if the disease is very rare.

To me, Bayes' Theorem is the reason not to argue with apologists. The theorem says that no amount of evidence ever does anything more than modify the initially-assumed 'prior probability'. If somebody assigns a low enough prior probability to something, then Bayes' Theorem tells you that no amount of evidence will convince them of it. Worse, in fact, Bayes' Theorem provides them with an ironclad justification in logic for their tenacity. All you can say is, we disagree greatly on priors.

Student of Trinity Wrote:
-------------------------------------------------------
> I think you may have garbled that false-positive
> example a bit.

> I believe you meant to say that if you really have
> the disease, then 95% of the time the test will
> say you have it, but if you really don't have it,
> then 5% of the time it will still say you have the
> disease (false positive).

Yes, thank you for the clarification.

Is this statement really true: "If you've got the disease, 95% of the time the test will say so; but if you haven't got the disease, 5% of the time the test will say you do."

The latter statement refers to false positives, which is what you're talking about. The former statement is about false negatives, how often the test misses the disease. These things are independent variables. The probablity of false positive isn't the inverse of false negative. The probablity depends on the test, what it's testing for, and the correlation between the object of the test and the presence of disease--and simply how good the test is at finding its object. In other words, the probability of passing the test and being sick may be different from the probability of failing the test and being well.

A valiant effort at proving Mosiah's Theorem:

http://exmormon.org/phorum/read.php?2,289125,289125#msg-289125

Baura:

I was interested seeing what the Board had to say about Bayes' theorem, and after a search came across this post. Unfortunately I missed it when it was first posted.

Your description here is not accurate as to what Bayes' theorem does. Instead, what you have described is null hypothesis significance testing (NHST) as championed by Ronald Fisher. Bayesian analysis is an alternative to NHST, which is based upon a mathematical consideration of the relationship between conditional probabilities, which is not discussed in your post at all. This confusion is common. The trend in science, and particularly the social sciences, is towards the Bayesian approach rather than NHST.

(See, Kruschke, Doing Bayesian Data Analysis, particularly Chapter 11, where the Bayesian approach is compared with NHST.)

My description here is accurate as to what Bayes theorem says.
As to what it DOES, that depends on what YOU do with it. Bayes
theorem and Bayesian analysis are two different things. Note I
said "Bayes theorem," not "Bayesian analysis."

For the record, Bayes theorem, in its simplest form is

P(A|B) = P(B|A)xP(A)/P(B)

Express P(B) as P(B|A)xP(A) + P(B|~A)xP(~A) and you have the
usual formulation of Bayes Theorem.

Right: Bayes theorem is P(A|B) = P(B|A)xP(A)/P(B)

This formula represents a comparison of conditional probabilities, namely, p(A|B) and p(B|A).

There was no hint of this in the OP. Again, what you described in the OP was NHST. There is a very straightforward difference between the two approaches.

Finally, Bayesian analysis is simply the use of Bayes theorem in the assessment of data to obtain probability conclusions for belief formation.

I once contemplated creating a website with a Bayes calculator that allowed people to calculate the probability that "the Church is true," given their own prior probabilities, but it would be a little bit of work, I'm not facile with web page creation, and I wonder whether the prior probabilities would be such that they would swamp out the evidence anyway.

And there is this calculator:
http://www.lds4u.com/lesson1/bayesian.htm

Hilarious -- and useful!

I put in 0% for a priori belief that the BoM is true (which is how any question should be approached, without pre-conceived bias either way).

I put in 0% chance of answer to prayer that it is or isn't true (since there's no evidence "prayer" is ever answered).

The probability the book is true came out, of course, 0%.

Which is as it should be :)

Of course, if your prior probability is zero, then the result under Bayesian analysis (your posterior probability) must also be zero-no matter what your other assumptions are. So, e.g. if your prior probability assumption of the BoM being true is zero, there can be no evidence for it because under the Bayesian formula there can be no evidence that raises a prior if that prior is predetermined as zero. Intuitively, if there is no chance that a given proposition is true, then there can be no evidence that could possibility support that proposition.

Formally, let x be the BoM is true, and p be prayer confirmation of x, then if p(x) is zero, you have,

p(x|p) = 0 because you have already assumed that there is no probability of x. So, p cannot affect it. Applying the Bayes theorem:

p(x|p)= (p(p|x))(p(x)) / p(p), consider:

Suppose one assigns p(p) to be 99% (Thus, the probability of a prayer confirmation that the BoM is true is 99%) You still get:

First, p(p|x) is 99; Since p(x) is zero, it does not affect p, i.e. the probability of p does not change with x because x cannot occur, i.e. it has zero probability.

But, then 99 X p(x) = 0, since zero times anything is zero. Moreover, zero divided by any number is zero, so 0/99 is 0, and your posterior probability remains at zero.

Thus, assigning a prior of zero is a deal-breaker for Bayesian analysis. It stops the analysis in its tracks.

The moral to this story (I think) is that it may be more useful to assign a very low prior probability rather than zero, because, as indicated, a zero assignment takes the matter out of Bayesian analysis. On the other hand a low probability assignment allows you to compare how even a high probability piece of evidence affects the prior relatively little, not enough to alter beliefs.

In the Mormon context, if the probability of the BoM being true is so low (but not zero) that a Baysian analysis of, say, NHM as evidence, however compelling it might be to some, affects the probability of the falsehood of the BoM relatively little, there is no reason to change one's view that the BoM is false.

HB

Yes, that's correct.
My point was that there IS zero probability (given existing evidence) that the BoM is true. So no amount of "confirmation" can change that. At least not in that construct.

Noted atheist writer and author Richard Carrier uses Bayes Theorem extensively in evaluating the validity of the evidence supporting the existence of Jesus.

One example is his well-received book "On the Historicity of Jesus: Why We Might Have Reason for Doubt".

http://smile.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Daps&field-keywords=richard+carrier&rh=i%3Aaps%2Ck%3Arichard+carrier

Here Bayesian analysis is used to examine multiple points of data derived from information available at the time of Jesus. Carrier doesn't tell readers what conclusion to come to, but provides them an excellent way to evaluate the evidence for themselves. It's not light reading, but well worth the effort.

Still waiting for a historicist to plug their numbers into Carrier's model :( Instead, historicists refuse to apply Bayes Theorum or any other mathematical model for that matter. I still don't know what percentage Ehrman's "Jesus almost certainly existed" is equivalent to, but they are historians who are good at linguistics, not math, so I am not sure why I expect so much of them. I suspect that historians don't even study statistics as part of their degrees.