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Bayes Theorem and Joseph Smith’s Seer Stone: Extraordinary Claims Require Extraordinary Evidence
Mormons have discovered Bayes! And there are rumors that more and more articles, and potential books are going to start being written using it. This paper is meant to give them a hand at it. Bayes is perfect in helping us grasp that extraordinary claims require extraordinary evidence. There are more than enough in Mormonism to use Bayes as a help.
The power of Bayes is that it provides us with an orderly and solid way of making sure our thinking is more accurate in regards to what we think we know, how evidence (good and bad) affects what we know, and how we can use probability to remain rational in our choices, or at least be helped in making belief choices.
Joseph Smith claimed to have used a seer stone to translate at least part of the Book of Mormon with. By placing it in a hat, and then seeing the translation from an unknown language into English in glowing letters according to one report. We can use Bayes to check on the probability that he was telling the truth.
We use what we know of the world, how things work, as well as looking into as much evidence(s) as we can to justify what we claim we know. The more evidence, the stronger our belief can be justified. It’s really quite straight forward.
So… how likely is the hypothesis Joseph Smith presented concerning a seer stone with God’s power, which helped him translate the Book of Mormon? Whatever evidence there is, how likely is it on the basis of the claim? Is it the kind of evidence we would expect if Smith was being truthful? This is more the Bayesian way of asking questions about our knowledge and updating our knowledge when new evidences come in or are discovered through our researching. Some basics facts, not exhaustive, enough to guide us for now in a short introductory paper.
Joseph Smith was very tight lipped about the entire translation process. This was not a magic show for entertaining the public, but it appears to be more manipulative which does affect people’s lives, the choices they make - hiding evidence that does matter and make a difference in people’s choices they make in life - actions they may do and the consequences. This appears more like a charlatan’s method than a teller of truth. Is this the kind of truth we expect to find on his claim? His reticence appears odd. This does not seem to be in line with reality.
One of the witnesses said glowing letters appeared on the rock, and when Joseph was finished speaking them out loud, and the scribe wrote them out, the rock knew when to let them fade and bring more letters from the unknown language no one in the world could possibly read or know.Is this a property of stones generally? How many such stones like this do we know in history? Revelations in this manner in general anywhere in the world or time? Specifically from one language into another? Glowing words? What is our background knowledge of stones in the world? This appears to be telling against Joseph Smith’s claims.
No church leaders have ever claimed to have used a seer stone, yet they proclaim they are seers in line with Joseph Smith. They also have proven they own one of his seer stones. They hid it for decades and decades, never using it for anything, not even a paperweight as it has lain hidden in a secure vault. It has never been used to translate anything, or help with any unknown languages from antiquity the world has found, faced and confronted. Shouldn’t they use it for the benefit of the church, let alone the world like Joseph Smith did? This seems problematical. Is this the kind of evidence we would expect from the church if they had a genuine direct access source to God’s knowledge, that could be used for good and for learning truth, whether ancient or modern? Not at all.
Joseph encouraged others to get their own seer stones, and some did so. Yet when their revelations began contradicting his (God apparently not keeping his stories straight?) Joseph Smith put the squelch on it immediately, destroyed other’s stones and downplayed and refuted the revelations, as if he had that right. All revelations were to conform to his, since they are the standard of truth, or so he said. Is this the kind of evidence we would have if Joseph Smith’s story held water? Or was he trying to cover his tracks so as to not expose something far more mundane that he had simply invented the whole thing, and others’ taking his literally could cause real problems?
Joseph Smith very rarely ever quoted from or used the Book of Mormon in any of his hundreds of sermons. It was described as “The most correct book” ever given to mankind, yet went entirely unused by him for any pedagogical purposes. A book translated through a stone by God, which leads closer to God than any other, and yet ignored? Is this the kind of evidence we would have if his claim of the stone was true? No.
Martin Harris once during the translation process switched stones doing a test on Joseph Smith when he wasn't looking. When they got back to it after a break, Joseph just sat there and finally looked up and asked what was wrong? The stone was as dark as Egypt! Harris confessed and said he wanted to shut up the mouth of critics. Assuming (not wise, but for now we will let it slide without analysis) it would shut up the mouth of critics Joseph Smith was glad it wasn’t a problem he had caused such as making God angry. So, is this the kind of evidence we would see on the Smith claim? Actually this is in his favor.
In this short paper (for online message boards) I cannot do an exhaustive analysis, the really good news being as more evidence both pro and con show up, we can always update our Bayesian analysis, one of its sterling virtues of all systems available to us. Let's put this extraordinary seer stone claim to a Bayesian analysis at least to give us the direction and an example of how Bayes works, and see why extraordinary claims really do need extraordinary evidence. First our daunting equation that is truly not at all difficult once it is explained.
xxxxxxxxxxx P(h丨b) x P(e丨h.b)
P(h丨e.b) = -------------------------------------------
xxxxxxxxxxx [P(h丨b) x P(e丨h.b)] + [P(~h丨b) x P(e丨~h.b)]
This just reads the probability of the hypothesis [Joseph Smith used a seer stone to translate the Book of Mormon] given the evidence and our background knowledge of how the world works. That is the meaning of the first term P(h丨e.b).
On the top the first term is P(h丨b) x P(e丨h.b) where the P(h丨b) means the probability of the hypothesis given the background - our everyday knowledge of the world and what we know about how it works in our lives. This is multiplied by the next term on the numerator P(e丨h.b) meaning the probability of the evidence given the hypothesis given our background understanding of reality we live in.
In the denominator, the lower part of the fraction, we repeat the expression from the top, and now we add that to this second part of the expression after the +sign - this part - [P(~h丨b) x P(e丨~h.b)]. The tilda in front of the h - this thing ~h, means that we have established a ratio here. This second part is the other hypothesis which we contrast and compare to ours, so, we have the claim (hypothesis) that Joseph Smith used a seer stone to translate the Book of Mormon. The second expression says this is the hypothesis that Joseph Smith’s claim is not right, hence the tilda negates or is opposite of our claim, however we define that - in this case, made up the story, this the competing hypothesis, for contrast purposes. So we are comparing two claims and the probability of the first claim which ideally includes all evidences and our actual common sense knowledge of reality in this life and about how the world works. OK, enough explaining, read it through a couple times if you don’t at first see it. It is not difficult, I promise.
The term P(h丨b) is a prior probability. This means what do you think about the claim based on what you know right now before seeing any kind of evidence whatever. You hear a claim, and then what do you think after you hear it? That is your prior. There is a prior in your hypothesis and a prior in the hypothesis you are comparing your hypothesis with. The full probability space of the two priors has to come to 1. In probability and percentages, the full space available for use is 1. We express the probability as .10 for 10% - move the dot over two (the meaning of per / cent = percent) places to the right. And .20 = 20%, .30 = 30%, up to .90 = 90% and finally 100% is just 1. Move the dot over 2 places and it gives you two 0’s behind the 1 - wala! 100%! Now there is truly never certainty in probability. We can come close, but nothing is ever really totally 0 or totally 100%. There is always some chance we are wrong.
So, now we know that in our world there are 76 rivers over 1000 miles long and the rivers in the U.S. have 2.5 million miles of rivers. Those rivers have a lot of rocks along them that are smooth and somewhat egg shaped and palm sized like Joseph Smith dug from the well he got his from (it was Sally Chase’s which he borrowed and never returned though promising to, and he stole it from her and then translated the Book of Mormon with it, interestingly enough). And none of them are seer stones. Stones don’t translate anything. That’s our background knowledge. Smith was known to have later given up using seer stones because he said he was used to the Spirit and didn’t need a prop, seems like a convenient way to eliminate bothersome questions and prodding into it all on the face of it.
We also know some around him claimed he had 5 seer stones in all, odd if he didn’t need to use them. And he definitely would not let anyone challenge his own revelation from seer stones. Among all these truly billions and billions of stones, only Joseph’s was a seer stone that was used and known about publicly, and the details he definitely did not want to let out, a good way to keep the ruse going on the face of it, based on all we know about stone’s properties and behavior (none have ever glowed letters from unknown languages that we are aware of). The Middle Ages magician John Dee’s “Shew-stone” or seer stone was actually a blackened mirror of some sort, and wasn’t used in any manner similar to how Joseph claimed he used his. Others in Smith’s day used some stones to find buried treasure (Joseph never succeeded until he dug up the golden plates - but was forbidden by the angel to lust after their financial value), but only Smith had the claim so weird to translate unknown languages.
So how do we calculate all this? First we find our prior probability. This is not the final probability, though it can influence it. This is not rocket science, and many are dismayed at the subjective nature of it, yet, we all subjectively use priors everyday about claims when they are made, we just don’t formalize them. Bayes is invaluable to us for doing that.
What is my (your) first impression of seer stones translating unknown languages with glowing letters, and knowing when a sentence has been finished and puts up another glowing sentence of words? Be honest. That is just too ludicrous for words. No stone does this. No one else has ever done this with any kind of stone, not even those in possession of the very stone that is said to have done this, today’s church leaders! The prior probability in our mind is what kind of ridiculous claim is this? No one in their right mind is going to believe this (why do you think the church has downplayed it for decades, and hidden the stone until just very recently?!)
In other words, the probability you will believe this is far below 50%. That’s even odds, and there is no way it can possibly by any stretch of the imagination be that high at the start! It is a no-brainer. It’s why missionaries do not teach about it when talking about the Book of Mormon. Yeah it was translated by a stone with glowing letters on it from a hat! What investigator wouldn’t throw the kids out on that one? So let's be realistic, yes we don’t have the exact number down to 10 decimals, but we don’t need that kind of precision to get the accurate enough probability. Would you give this bizarre claim 20% of being real? Too high, entirely, obviously too high still! 2 out of 10? Not a chance! Now, since I am illustrating Bayes here, I will go absolutely out of my mind with charity and very kindly grant the claim - entirely unjustifiably - the high 10% probability, which is astonishingly generous.
Now that is .1 for the prior term in the top part of the equation - P(h丨b).
For P(e丨h.b) meaning the probability of the evidence given the hypothesis given our background understanding of reality we live in, what is the evidence and is it the kind we would expect if the claim is true? Well, we have the book. And it’s a big book! Joseph Smith saId he translated out a book and sure enough, he gave us the book. That is exactly the kind of evidence we would expect if the claim was true, so lets give him a dandy .95 = 95% on this part.
So, the top line looks like P(h丨e.b) = 0.1 x .95 The P(h丨b) = 0.1 x P(e丨h.b) = .95
Then we repeat this on the bottom line of the fraction, P(h丨b) x P(e丨h.b) with those same values. Notice the prior = 0.1, that is, 10%. In the next term on this line, that means the prior has to be ,9 since the priors have to equal 1. So on the claim it has a realism of 10%, and on its opposite, it’s prior that the claim is wrong is 90% - which is too low, but we have to make it equal 1, and we are being generous to Joseph Smith for illustration sake. His actual prior is far closer to 0.0001 - 1 in ten thousand, but let's see what happens the way we have set this up. So for the P(~h丨b) x P(e丨~h.b), we have to set the P(~h丨b to .90. The term we multiply this by, the P(e丨~h.b - the evidence which shows his claim is ludicrous, all that I have described above as problematical we can set safely to .85 = 85% because we haven’t used all the evidence, and evidence either way can change this posterior probability, so 85% says the evidence against Smith’s claim is reasonable and highly probable as being the kind of evidence we would expect to find if Smith was just telling a story. So lets plug all this in and work out the probability.
P(h丨e.b) = 0.1 x .95 = .095 / [.095 - repeating the top] + [.90 x .85] = 0.86 ; so .095/.86 = 0.1104 = 11% (move the dot two places to the right). It is 11% probability that Joseph Smith was telling the truth about the seer stone translating the Book of Mormon.
Now, before anyone freaks out, let's explore something. Let's give Joseph Smith’s claim 99% evidence. Lets bump it from 95% to 99%. And then let’s work it out again mathematically. So we have upped the evidence in favor of Joseph Smith here.
P(h丨e.b) = 0.1 x .99 = .099 / [.099] +[.90 x .85] = .1145 = 11.45% probability his claim is true. Notice when we give his claim a decent jump in evidence it is still not near enough to change much of anything of this extraordinary claim.
Now what if we did the unprecedented thing and give him a prior that his bizarre claim had a 20% prior probability of being true, which is wildly too high! But let’s do it for the illustration. Now we calculate P(h丨e.b) = 0.2 x .99 = .198 Now remember, we changed the first prior from .1 enlarging it to .2 (from 10% to 20%), so we have to adjust the negative conclusion prior down from .90 to .80 - 90% to the lower 80%, because the two priors have to equal 1, the full probability space available. So this also helps out Joseph Smith. So the bottom now becomes [.198 - repeating the top] + [.8 x .85] = .198 + .8 x .85 = .878. So dividing the top .198/.878 = .2255 or 22.55% (moving the point two places to the right). So this does help a little, but notice doubling the strength of the claim in favor of Joseph Smith did not do much, demonstrating the power of the notion about extraordinary claims! Double the claims strength and adding to the evidence didn’t accomplish anything to crow about.
OVERTIME: OK, lets go all out ridiculous and charitable and insanely give the prior to Joseph Smith’s claim at 50%! That means we are saying there is a 50% chance that Joseph Smith’s claim is real and believable. Again, the illustration is quite remarkable! So, remember we change the first prior to a whopping 50% from the meager 20%, and it forces the second prior down to 50% from 80% because they both add together to equal 1, which is 100% of the probability space for the two different claims. We shall keep the evidence probabilities the same in both so that we arrive at the equation:
P(h丨e.b) = 0.5 x .99 = .495 for the top.
The bottom = [.495 - repeating from the top] + [.5 x .85] = .425, now add .495 = .92, so we have the final step of .495/.92 = .5380 = 54% rounding up (move point over two places to the right to get 53.80%, rounding up = 54%. The claim of Joseph Smith still after insanely enormous charitable additions to both the prior and the evidence - utterly impossible to agree with in reality of what we know - still gives the perfectly dismal, discouraging, and far too large, thanks to our doled out unjustifiable grace toward him, but not anywhere near large enough to even suggest he was not telling stories is a mere 54%. And having 60% is completely, unacceptably too small for Joseph Smith’s claim to be considered anywhere approaching true, even after giving him the kitchen sink and mansion on the mountain of charity, we can’t even get to 60%.
A powerful illustration that Carl Sagan had it precisely correct - “Extraordinary claims need EXTRA-ordinary evidence.” (my emphasis)
Mormons have discovered Bayes! And there are rumors that more and more articles, and potential books are going to start being written using it. This paper is meant to give them a hand at it. Bayes is perfect in helping us grasp that extraordinary claims require extraordinary evidence. There are more than enough in Mormonism to use Bayes as a help.
The power of Bayes is that it provides us with an orderly and solid way of making sure our thinking is more accurate in regards to what we think we know, how evidence (good and bad) affects what we know, and how we can use probability to remain rational in our choices, or at least be helped in making belief choices.
Joseph Smith claimed to have used a seer stone to translate at least part of the Book of Mormon with. By placing it in a hat, and then seeing the translation from an unknown language into English in glowing letters according to one report. We can use Bayes to check on the probability that he was telling the truth.
We use what we know of the world, how things work, as well as looking into as much evidence(s) as we can to justify what we claim we know. The more evidence, the stronger our belief can be justified. It’s really quite straight forward.
So… how likely is the hypothesis Joseph Smith presented concerning a seer stone with God’s power, which helped him translate the Book of Mormon? Whatever evidence there is, how likely is it on the basis of the claim? Is it the kind of evidence we would expect if Smith was being truthful? This is more the Bayesian way of asking questions about our knowledge and updating our knowledge when new evidences come in or are discovered through our researching. Some basics facts, not exhaustive, enough to guide us for now in a short introductory paper.
Joseph Smith was very tight lipped about the entire translation process. This was not a magic show for entertaining the public, but it appears to be more manipulative which does affect people’s lives, the choices they make - hiding evidence that does matter and make a difference in people’s choices they make in life - actions they may do and the consequences. This appears more like a charlatan’s method than a teller of truth. Is this the kind of truth we expect to find on his claim? His reticence appears odd. This does not seem to be in line with reality.
One of the witnesses said glowing letters appeared on the rock, and when Joseph was finished speaking them out loud, and the scribe wrote them out, the rock knew when to let them fade and bring more letters from the unknown language no one in the world could possibly read or know.Is this a property of stones generally? How many such stones like this do we know in history? Revelations in this manner in general anywhere in the world or time? Specifically from one language into another? Glowing words? What is our background knowledge of stones in the world? This appears to be telling against Joseph Smith’s claims.
No church leaders have ever claimed to have used a seer stone, yet they proclaim they are seers in line with Joseph Smith. They also have proven they own one of his seer stones. They hid it for decades and decades, never using it for anything, not even a paperweight as it has lain hidden in a secure vault. It has never been used to translate anything, or help with any unknown languages from antiquity the world has found, faced and confronted. Shouldn’t they use it for the benefit of the church, let alone the world like Joseph Smith did? This seems problematical. Is this the kind of evidence we would expect from the church if they had a genuine direct access source to God’s knowledge, that could be used for good and for learning truth, whether ancient or modern? Not at all.
Joseph encouraged others to get their own seer stones, and some did so. Yet when their revelations began contradicting his (God apparently not keeping his stories straight?) Joseph Smith put the squelch on it immediately, destroyed other’s stones and downplayed and refuted the revelations, as if he had that right. All revelations were to conform to his, since they are the standard of truth, or so he said. Is this the kind of evidence we would have if Joseph Smith’s story held water? Or was he trying to cover his tracks so as to not expose something far more mundane that he had simply invented the whole thing, and others’ taking his literally could cause real problems?
Joseph Smith very rarely ever quoted from or used the Book of Mormon in any of his hundreds of sermons. It was described as “The most correct book” ever given to mankind, yet went entirely unused by him for any pedagogical purposes. A book translated through a stone by God, which leads closer to God than any other, and yet ignored? Is this the kind of evidence we would have if his claim of the stone was true? No.
Martin Harris once during the translation process switched stones doing a test on Joseph Smith when he wasn't looking. When they got back to it after a break, Joseph just sat there and finally looked up and asked what was wrong? The stone was as dark as Egypt! Harris confessed and said he wanted to shut up the mouth of critics. Assuming (not wise, but for now we will let it slide without analysis) it would shut up the mouth of critics Joseph Smith was glad it wasn’t a problem he had caused such as making God angry. So, is this the kind of evidence we would see on the Smith claim? Actually this is in his favor.
In this short paper (for online message boards) I cannot do an exhaustive analysis, the really good news being as more evidence both pro and con show up, we can always update our Bayesian analysis, one of its sterling virtues of all systems available to us. Let's put this extraordinary seer stone claim to a Bayesian analysis at least to give us the direction and an example of how Bayes works, and see why extraordinary claims really do need extraordinary evidence. First our daunting equation that is truly not at all difficult once it is explained.
xxxxxxxxxxx P(h丨b) x P(e丨h.b)
P(h丨e.b) = -------------------------------------------
xxxxxxxxxxx [P(h丨b) x P(e丨h.b)] + [P(~h丨b) x P(e丨~h.b)]
This just reads the probability of the hypothesis [Joseph Smith used a seer stone to translate the Book of Mormon] given the evidence and our background knowledge of how the world works. That is the meaning of the first term P(h丨e.b).
On the top the first term is P(h丨b) x P(e丨h.b) where the P(h丨b) means the probability of the hypothesis given the background - our everyday knowledge of the world and what we know about how it works in our lives. This is multiplied by the next term on the numerator P(e丨h.b) meaning the probability of the evidence given the hypothesis given our background understanding of reality we live in.
In the denominator, the lower part of the fraction, we repeat the expression from the top, and now we add that to this second part of the expression after the +sign - this part - [P(~h丨b) x P(e丨~h.b)]. The tilda in front of the h - this thing ~h, means that we have established a ratio here. This second part is the other hypothesis which we contrast and compare to ours, so, we have the claim (hypothesis) that Joseph Smith used a seer stone to translate the Book of Mormon. The second expression says this is the hypothesis that Joseph Smith’s claim is not right, hence the tilda negates or is opposite of our claim, however we define that - in this case, made up the story, this the competing hypothesis, for contrast purposes. So we are comparing two claims and the probability of the first claim which ideally includes all evidences and our actual common sense knowledge of reality in this life and about how the world works. OK, enough explaining, read it through a couple times if you don’t at first see it. It is not difficult, I promise.
The term P(h丨b) is a prior probability. This means what do you think about the claim based on what you know right now before seeing any kind of evidence whatever. You hear a claim, and then what do you think after you hear it? That is your prior. There is a prior in your hypothesis and a prior in the hypothesis you are comparing your hypothesis with. The full probability space of the two priors has to come to 1. In probability and percentages, the full space available for use is 1. We express the probability as .10 for 10% - move the dot over two (the meaning of per / cent = percent) places to the right. And .20 = 20%, .30 = 30%, up to .90 = 90% and finally 100% is just 1. Move the dot over 2 places and it gives you two 0’s behind the 1 - wala! 100%! Now there is truly never certainty in probability. We can come close, but nothing is ever really totally 0 or totally 100%. There is always some chance we are wrong.
So, now we know that in our world there are 76 rivers over 1000 miles long and the rivers in the U.S. have 2.5 million miles of rivers. Those rivers have a lot of rocks along them that are smooth and somewhat egg shaped and palm sized like Joseph Smith dug from the well he got his from (it was Sally Chase’s which he borrowed and never returned though promising to, and he stole it from her and then translated the Book of Mormon with it, interestingly enough). And none of them are seer stones. Stones don’t translate anything. That’s our background knowledge. Smith was known to have later given up using seer stones because he said he was used to the Spirit and didn’t need a prop, seems like a convenient way to eliminate bothersome questions and prodding into it all on the face of it.
We also know some around him claimed he had 5 seer stones in all, odd if he didn’t need to use them. And he definitely would not let anyone challenge his own revelation from seer stones. Among all these truly billions and billions of stones, only Joseph’s was a seer stone that was used and known about publicly, and the details he definitely did not want to let out, a good way to keep the ruse going on the face of it, based on all we know about stone’s properties and behavior (none have ever glowed letters from unknown languages that we are aware of). The Middle Ages magician John Dee’s “Shew-stone” or seer stone was actually a blackened mirror of some sort, and wasn’t used in any manner similar to how Joseph claimed he used his. Others in Smith’s day used some stones to find buried treasure (Joseph never succeeded until he dug up the golden plates - but was forbidden by the angel to lust after their financial value), but only Smith had the claim so weird to translate unknown languages.
So how do we calculate all this? First we find our prior probability. This is not the final probability, though it can influence it. This is not rocket science, and many are dismayed at the subjective nature of it, yet, we all subjectively use priors everyday about claims when they are made, we just don’t formalize them. Bayes is invaluable to us for doing that.
What is my (your) first impression of seer stones translating unknown languages with glowing letters, and knowing when a sentence has been finished and puts up another glowing sentence of words? Be honest. That is just too ludicrous for words. No stone does this. No one else has ever done this with any kind of stone, not even those in possession of the very stone that is said to have done this, today’s church leaders! The prior probability in our mind is what kind of ridiculous claim is this? No one in their right mind is going to believe this (why do you think the church has downplayed it for decades, and hidden the stone until just very recently?!)
In other words, the probability you will believe this is far below 50%. That’s even odds, and there is no way it can possibly by any stretch of the imagination be that high at the start! It is a no-brainer. It’s why missionaries do not teach about it when talking about the Book of Mormon. Yeah it was translated by a stone with glowing letters on it from a hat! What investigator wouldn’t throw the kids out on that one? So let's be realistic, yes we don’t have the exact number down to 10 decimals, but we don’t need that kind of precision to get the accurate enough probability. Would you give this bizarre claim 20% of being real? Too high, entirely, obviously too high still! 2 out of 10? Not a chance! Now, since I am illustrating Bayes here, I will go absolutely out of my mind with charity and very kindly grant the claim - entirely unjustifiably - the high 10% probability, which is astonishingly generous.
Now that is .1 for the prior term in the top part of the equation - P(h丨b).
For P(e丨h.b) meaning the probability of the evidence given the hypothesis given our background understanding of reality we live in, what is the evidence and is it the kind we would expect if the claim is true? Well, we have the book. And it’s a big book! Joseph Smith saId he translated out a book and sure enough, he gave us the book. That is exactly the kind of evidence we would expect if the claim was true, so lets give him a dandy .95 = 95% on this part.
So, the top line looks like P(h丨e.b) = 0.1 x .95 The P(h丨b) = 0.1 x P(e丨h.b) = .95
Then we repeat this on the bottom line of the fraction, P(h丨b) x P(e丨h.b) with those same values. Notice the prior = 0.1, that is, 10%. In the next term on this line, that means the prior has to be ,9 since the priors have to equal 1. So on the claim it has a realism of 10%, and on its opposite, it’s prior that the claim is wrong is 90% - which is too low, but we have to make it equal 1, and we are being generous to Joseph Smith for illustration sake. His actual prior is far closer to 0.0001 - 1 in ten thousand, but let's see what happens the way we have set this up. So for the P(~h丨b) x P(e丨~h.b), we have to set the P(~h丨b to .90. The term we multiply this by, the P(e丨~h.b - the evidence which shows his claim is ludicrous, all that I have described above as problematical we can set safely to .85 = 85% because we haven’t used all the evidence, and evidence either way can change this posterior probability, so 85% says the evidence against Smith’s claim is reasonable and highly probable as being the kind of evidence we would expect to find if Smith was just telling a story. So lets plug all this in and work out the probability.
P(h丨e.b) = 0.1 x .95 = .095 / [.095 - repeating the top] + [.90 x .85] = 0.86 ; so .095/.86 = 0.1104 = 11% (move the dot two places to the right). It is 11% probability that Joseph Smith was telling the truth about the seer stone translating the Book of Mormon.
Now, before anyone freaks out, let's explore something. Let's give Joseph Smith’s claim 99% evidence. Lets bump it from 95% to 99%. And then let’s work it out again mathematically. So we have upped the evidence in favor of Joseph Smith here.
P(h丨e.b) = 0.1 x .99 = .099 / [.099] +[.90 x .85] = .1145 = 11.45% probability his claim is true. Notice when we give his claim a decent jump in evidence it is still not near enough to change much of anything of this extraordinary claim.
Now what if we did the unprecedented thing and give him a prior that his bizarre claim had a 20% prior probability of being true, which is wildly too high! But let’s do it for the illustration. Now we calculate P(h丨e.b) = 0.2 x .99 = .198 Now remember, we changed the first prior from .1 enlarging it to .2 (from 10% to 20%), so we have to adjust the negative conclusion prior down from .90 to .80 - 90% to the lower 80%, because the two priors have to equal 1, the full probability space available. So this also helps out Joseph Smith. So the bottom now becomes [.198 - repeating the top] + [.8 x .85] = .198 + .8 x .85 = .878. So dividing the top .198/.878 = .2255 or 22.55% (moving the point two places to the right). So this does help a little, but notice doubling the strength of the claim in favor of Joseph Smith did not do much, demonstrating the power of the notion about extraordinary claims! Double the claims strength and adding to the evidence didn’t accomplish anything to crow about.
OVERTIME: OK, lets go all out ridiculous and charitable and insanely give the prior to Joseph Smith’s claim at 50%! That means we are saying there is a 50% chance that Joseph Smith’s claim is real and believable. Again, the illustration is quite remarkable! So, remember we change the first prior to a whopping 50% from the meager 20%, and it forces the second prior down to 50% from 80% because they both add together to equal 1, which is 100% of the probability space for the two different claims. We shall keep the evidence probabilities the same in both so that we arrive at the equation:
P(h丨e.b) = 0.5 x .99 = .495 for the top.
The bottom = [.495 - repeating from the top] + [.5 x .85] = .425, now add .495 = .92, so we have the final step of .495/.92 = .5380 = 54% rounding up (move point over two places to the right to get 53.80%, rounding up = 54%. The claim of Joseph Smith still after insanely enormous charitable additions to both the prior and the evidence - utterly impossible to agree with in reality of what we know - still gives the perfectly dismal, discouraging, and far too large, thanks to our doled out unjustifiable grace toward him, but not anywhere near large enough to even suggest he was not telling stories is a mere 54%. And having 60% is completely, unacceptably too small for Joseph Smith’s claim to be considered anywhere approaching true, even after giving him the kitchen sink and mansion on the mountain of charity, we can’t even get to 60%.
A powerful illustration that Carl Sagan had it precisely correct - “Extraordinary claims need EXTRA-ordinary evidence.” (my emphasis)
