If we assign the existence of God a very low prior belief, then we must also assign a very low prior belief to the interpretation of the Bible as the word of God. In that case, seeing the Bible will not do much to elevate our belief in the claim that God exists, if there are more likely hypotheses to be found.
Then I worked out that the likelihood ratio P(S|H) / P(S|¬H) = ( P(S|A)P(A|H) + P(S|¬A)P(¬A|H) ) / ( P(S|A)P(A|¬H) + P(S|¬A)P(¬A|¬H) ) depends only on our conditional probabilities, not on our prior probabilities. (Here S = “We observe the Bible”, H = “God exists”, and A = “The Bible is the word of God”, as in Hahn & Oaksford.)
So the existence of the Bible can be strong evidence for the existence of God if we use likelihood ratio as a measure of strength of evidence. On the other hand, if we start with a very low prior for God, then even somewhat strong evidence will not be enough to convince us of His existence.
Put another way, the Bible can shift log(odds ratio) by quite a bit, independently of our prior for God; but if we have a sufficiently low prior for God, our posterior credence in God won’t be much higher.
The conditional probabilities are doing a lot of work here, and it seems that in many cases our estimates of them are strongly dependent on our priors.
What are our estimates for P(S|A) or P(S|notA) and how do we work them out? clearly P(S|A) is high since “The Bible is the word of God” directly implies that the bible exists, so it is at least possible to observe. If our prior for A is very low, then that implies that our estimate of P(S|notA) must be also be high, given that we do in fact observe the bible (or we must have separately a well founded explanation of the truth of S despite it’s low probability).
Since having P(S|A) = P(S|notA) in your formula cancels the right side out to 1⁄1, P(S|H) = P(S|notH). We find as S as evidence for or against A weakens, so does S as evidence for or against H by this argument.
So the problem with the circular argument is apparent in Bayesian terms. In the absence of some information that is outside the circular argument, the lower the prior probability, the weaker the argument. That’s not the way an evidential argument is supposed to work.
Even in the case where our prior is higher, the argument isn’t actually doing any work, it is what our prior does to our estimate of those conditionals that makes the likelihood ratio higher. If we’ve estimated those conditionals in a way which causes a fully circular argument to move the estimate away from our prior, then we have to be doing something wrong, because we don’t have any new information.
If we have independent estimates of those various conditionals, then we would be able to make a non-circular argument. OTOH We can make a circular argument for anything no matter what is going on in reality, that’s why a circular argument is a true and complete fallacy: it provides no evidence whatsoever for or against a premise.
I was confused by this:
Then I worked out that the likelihood ratio P(S|H) / P(S|¬H) = ( P(S|A)P(A|H) + P(S|¬A)P(¬A|H) ) / ( P(S|A)P(A|¬H) + P(S|¬A)P(¬A|¬H) ) depends only on our conditional probabilities, not on our prior probabilities. (Here S = “We observe the Bible”, H = “God exists”, and A = “The Bible is the word of God”, as in Hahn & Oaksford.)
So the existence of the Bible can be strong evidence for the existence of God if we use likelihood ratio as a measure of strength of evidence. On the other hand, if we start with a very low prior for God, then even somewhat strong evidence will not be enough to convince us of His existence.
Put another way, the Bible can shift log(odds ratio) by quite a bit, independently of our prior for God; but if we have a sufficiently low prior for God, our posterior credence in God won’t be much higher.
The conditional probabilities are doing a lot of work here, and it seems that in many cases our estimates of them are strongly dependent on our priors.
What are our estimates for P(S|A) or P(S|notA) and how do we work them out? clearly P(S|A) is high since “The Bible is the word of God” directly implies that the bible exists, so it is at least possible to observe. If our prior for A is very low, then that implies that our estimate of P(S|notA) must be also be high, given that we do in fact observe the bible (or we must have separately a well founded explanation of the truth of S despite it’s low probability).
Since having P(S|A) = P(S|notA) in your formula cancels the right side out to 1⁄1, P(S|H) = P(S|notH). We find as S as evidence for or against A weakens, so does S as evidence for or against H by this argument.
So the problem with the circular argument is apparent in Bayesian terms. In the absence of some information that is outside the circular argument, the lower the prior probability, the weaker the argument. That’s not the way an evidential argument is supposed to work.
Even in the case where our prior is higher, the argument isn’t actually doing any work, it is what our prior does to our estimate of those conditionals that makes the likelihood ratio higher. If we’ve estimated those conditionals in a way which causes a fully circular argument to move the estimate away from our prior, then we have to be doing something wrong, because we don’t have any new information.
If we have independent estimates of those various conditionals, then we would be able to make a non-circular argument. OTOH We can make a circular argument for anything no matter what is going on in reality, that’s why a circular argument is a true and complete fallacy: it provides no evidence whatsoever for or against a premise.
What? That’s an argument for P(S|¬A∧S) being high, not an argument for P(S|¬A) being high.