Posted: Dec 18, 2011 3:55 am
Hi Paul -- Please let me know if the following explanation helps. Also, for readers who have a background in this sort of thing, keep in mind I am using the epistemic interpretation of probability, not the frequency or logical interpretations.
Regards,
Jeffery Jay Lowder
Given that inductive arguments (like some versions of the FTA) are shot through with appeals to probability, it is useful to introduce Bayes’s Theorem as a way of clarifying the role of probability in inductive arguments for and against God's existence. Let E be the evidence to be explained, H be our explanatory hypothesis for E, and let B represent our relevant background knowledge. B includes all of our information relevant to H other than evidence E. B includes facts that determine the intrinsic probability of rival explanatory theories and facts that partially determine their explanatory power. E includes unusual facts within the context of this background that need to be explained.
We are now in a position to introduce Bayes’s Theorem, a mathematical formula for representing the effect of new information upon our degree of belief in a hypothesis. In one form, Bayes’s Theorem may be expressed as follows:
PR(H | E & B ) = [Pr(E | H & B) x Pr(H | B)] / Pr(E | B),
where Pr(E | B ) = Pr(E | H & B) x Pr(H | B) + Pr(E | ~H & B) x Pr(~H | B)
Pr(H / B) is the prior probability of H with respect to B—a measure of how likely H is to occur at all, whether or not E is true. Pr(E / H & B) is the predictive power of H—the probability of E, given the truth of both B and H. Pr(E / B) is the prior probability of E with respect to B—how likely E is to be true a priori, whether or not H holds. The explanatory power of H is the ratio of Pr(E / H & B) to Pr(E / B). In other words, explanatory power refers to the ability of a hypothesis to explain (i.e., make probable) an item of evidence. Finally, Pr(H / E & B) is the final probability with respect to the total evidence B and E.
Although it is rarely possible to assign precisely numerical values to each of these components when assessing alleged purported evidence for or against God's existence, Bayes’s Theorem is nevertheless extremely useful because it provides the logical foundation for several of our intuitive beliefs about inductive arguments. Consider, for example, the common-sense principle, “The more implausible the hypothesis, the greater the evidence needed to confirm it.” Bayes’s Theorem captures this principle mathematically by saying that the final probability of H, given E and B, is equal to the product of H’s explanatory power and its prior probability.