Possiblism and Bayes, Part 1 – http://jturnonmormonism.wordpress.com/2014/02/07/possibilism-and-bayes/
There is another way to think about distorted reasoning in the context of Bayes’s rule. This is a consequences of a person starting off with an exceptionally high prior probability based on an epistemology that leans heavily on trust in authority as sufficient grounds for accepting a truth claim, or counting as evidence some sense of direct revelation (subjective spiritual experience). Let’s take it through the basic formulation again in the context of the Book of Mormon.
- H = the hypothesis that the Book of Mormon (BoM) is True
- H* = the hypothesis that the BoM is False
- B = the Pre-DNA background evidence
- E = the new DNA evidence (containing no Israelite markers)
Second, the relevant prior probabilities (with LDS-friendly assignments). Now we represent the person who has repeatedly testified in public, “I know the Book of Mormon is True.”
- P(H|B) = probability BoM is true given pre-DNA evidence = 0.99
- P(H*|B) = probability BoM is false given pre-DNA evidence = 0.01
Next, the relevant updated probabilities are
- P(E|H,B) = probability of the DNA evidence given the BoM is True = 0.1
- P(E|H*,B) = probability of the DNA evidence given the BoM is false = 0.9
Note that this gives an LDS-friendly 0.10 (or 10%) chance that no Israelite DNA would show up in the Americas even if Lehi, Ishmael, etc. had as many offspring as the text suggests. This 10% forces the other to be 90%, since they must add to 1oo%
This leads us to what we are after, the probability the BoM is true after updating the prior probability (99%) with the new DNA evidence
- P(H|E,B) = updated probability that BoM based on new DNA evidence
This final probability is determine completely by the previous ones through Bayes theorem. It is the only rational conclusion that can follow that honors evidence.
. P(H|B) x P(E|H,B)
P(H|E,B) = ______________________________
. P(H|B) x P(E|H,B) + P(H*|B) x P(E|H*,B)
. 0.99 x 0.1
. = _____________ = 0.92
. 0.99 x 0.1 + 0.01 x 0.9
Here we see that even strong contradictory evidence drops the belief probability from 99% to 92%.
If the person felt 99.9% certain, the posterior probability would only drop to 99.1% certain.
If a person is deeply committed to an idea based on a faith-based epistemology, it will take an extraordinary amount of contradictory evidence to make them doubt, or even reach the 50:50 level.
Of course, ours minds do not tend to Bayesian assessments – we ignore the right term in the denominator. Indeed, for a person to hold beliefs with a “provisionality” on par with how a scientist trusts a thermometer over her sense of touch when her hand that’s been steeping in ice water, the person would have to accept that our brains are “steeping” in “ice water.” It’s tough to arrive at that level of detachment – and it might not be a healthy state to achieve, except, perhaps, temporarily in critical circumstances.