Elder Holland's talk has generated a lot of discussion particularly related to the idea of whether Joseph's and Hyrum's behavior in the last days of their lives was evidence in favor of the truthfulness of the Book of Mormon. Speciifically, Holland sees their attitude towards the Book of Mormon and particularly the fact the they turned to the Book of Mormon for solace and comfort in those last days as supportive of the truthfulness of the Book of Mormon.
I think it is worth being formal about this to show under what conditions Elder Holland is correct. First, in the context I see his use of word evidence as equivalent to saying that when one considers this datum she will now conclude it is more likely that the Book of Mormon is true than before she encountered the datum. Under what conditions is that true?
The first thing to note is that we can apply this to the belief that the Book of Mormom is true rather easily. Bayes rule was developed for evaluating new data in precisely this way.
P(T) = probability of the Book of Mormon is true ... obviously P(F)=1-P(T) is the probability the Book of Mormon is false.
P(D|T) = the probability we observe the behavior pointed out by Elder Holland given the Book of Mormon is true.
P(D|F) = the probability that we observe the behavior given the book of Mormon is false.
Suppose P(T) = 0.6 (nothing sacred about 0.6, change it to whatever you want besisdes 0 or 1 and the argument follows just fine, 0 and 1 have to be dealt with as special cases) and P(D|T) = 0.10 and P(D|F) = 0.05 . Then
P(The book of Mormon is True|The new datum) = P(T|D)
Bayes rule
P(T|D) = P(D|T)*P(T)/(P(D|T)*P(T) + P(D|F)*P(F)
P(T|D) = 0.10*0.6/(0.10*0.06+0.4*0.05) = 0.75
So the person is now more confident the Book of Mormom is true under these conditions. Notice what matters here for this is to be supporting evidence. It just has to be more likely that Joseph Smith would behave this way if the Book of Mormon is true. It doesn't matter that there is positive probability that he would act this way even if the Book of Mormon is fraudalent. In order for it not to effect the probability that it is true positively you have to believe that Joseph is at least equaly likely to behave this way if the book is fraudulant. So for example,
P(D|T) = P(D|F) = 0.10
P(T|D) = 0.10*0.60/(0.10*0.60+0.4*0.10) = 0.6
No change in the probability. If you believe that then it isn't supportive but as long as you think P(D|F) is even ever so slightly less then P(D|T) then the evidence is supportive.
Magnitude also matters here. If you believe that this behavior is really unlikely if the book of Mormon is false then this going to effect your beliefs about the truthfulness of the Book of Mormon a great deal:
P(D|T) = 0.2 P, P(D|F) = 0.02
P(T|D) = 0.20*0.60/(0.20*0.60+0.4*0.02) = 0.9375
Of course, if the gap is small (particularly when the probabilities are big in magnitude as well) then the effect is small but it still matters:
P(D|T) = 0.40, P(D|F) = 0.39
P(T|D) = 0.40*0.60/(0.40*0.60+0.4*0.39) = 0.606
I don't think this is some dramatic piece of evidence in favor of the Book of Mormom but I do think P(D|T) > P(D|F). I don't think the difference is huge or dramatic but I think arguing against the inequality here is a tough road. Good job, Elder Holland.
I think it is worth being formal about this to show under what conditions Elder Holland is correct. First, in the context I see his use of word evidence as equivalent to saying that when one considers this datum she will now conclude it is more likely that the Book of Mormon is true than before she encountered the datum. Under what conditions is that true?
The first thing to note is that we can apply this to the belief that the Book of Mormom is true rather easily. Bayes rule was developed for evaluating new data in precisely this way.
P(T) = probability of the Book of Mormon is true ... obviously P(F)=1-P(T) is the probability the Book of Mormon is false.
P(D|T) = the probability we observe the behavior pointed out by Elder Holland given the Book of Mormon is true.
P(D|F) = the probability that we observe the behavior given the book of Mormon is false.
Suppose P(T) = 0.6 (nothing sacred about 0.6, change it to whatever you want besisdes 0 or 1 and the argument follows just fine, 0 and 1 have to be dealt with as special cases) and P(D|T) = 0.10 and P(D|F) = 0.05 . Then
P(The book of Mormon is True|The new datum) = P(T|D)
Bayes rule
P(T|D) = P(D|T)*P(T)/(P(D|T)*P(T) + P(D|F)*P(F)
P(T|D) = 0.10*0.6/(0.10*0.06+0.4*0.05) = 0.75
So the person is now more confident the Book of Mormom is true under these conditions. Notice what matters here for this is to be supporting evidence. It just has to be more likely that Joseph Smith would behave this way if the Book of Mormon is true. It doesn't matter that there is positive probability that he would act this way even if the Book of Mormon is fraudalent. In order for it not to effect the probability that it is true positively you have to believe that Joseph is at least equaly likely to behave this way if the book is fraudulant. So for example,
P(D|T) = P(D|F) = 0.10
P(T|D) = 0.10*0.60/(0.10*0.60+0.4*0.10) = 0.6
No change in the probability. If you believe that then it isn't supportive but as long as you think P(D|F) is even ever so slightly less then P(D|T) then the evidence is supportive.
Magnitude also matters here. If you believe that this behavior is really unlikely if the book of Mormon is false then this going to effect your beliefs about the truthfulness of the Book of Mormon a great deal:
P(D|T) = 0.2 P, P(D|F) = 0.02
P(T|D) = 0.20*0.60/(0.20*0.60+0.4*0.02) = 0.9375
Of course, if the gap is small (particularly when the probabilities are big in magnitude as well) then the effect is small but it still matters:
P(D|T) = 0.40, P(D|F) = 0.39
P(T|D) = 0.40*0.60/(0.40*0.60+0.4*0.39) = 0.606
I don't think this is some dramatic piece of evidence in favor of the Book of Mormom but I do think P(D|T) > P(D|F). I don't think the difference is huge or dramatic but I think arguing against the inequality here is a tough road. Good job, Elder Holland.
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