Miracles, Bayesianism, and Circular Reasoning
Last Thursday Joe concluded his critique of Bayesian antimiraculous arguments by casting doubt on the empirical basis of prior probabilities on the
issue: "This is admittedly subjective, and all one need do is examine it
to see this…. These questions (setting the prior for God) are matters for
theology." That's hard to deny. In cases like these, Bayes' Theorem
involves an inescapable element of subjectivity.
Recall that per Bayes' Theorem, the probability of a hypothesis (H), given evidence E and background knowledge K, equals the conjunction of its explanatory power and prior probability:
Recall that per Bayes' Theorem, the probability of a hypothesis (H), given evidence E and background knowledge K, equals the conjunction of its explanatory power and prior probability:
P(E│H & K) x P(H│K)
P(H│E & K) = -----------------------------
P(E│K)
Posterior probability, P(H│E & K), therefore increases as explanatory power and prior probability increases, all else being equal; while posterior probability decreases with increases in total probability of the evidence P(E│K), all else being equal. To "confirm" the hypothesis and demonstrate that its probability is greater than any competing hypothesis, the resulting posterior probability should be greater than .5. Likewise a posterior probability of < .5 would disconfirm the hypothesis, where explanatory power and prior probability, P(E│H & K) x P(H│K), are too low to clear the probability of the evidence. But in non-statistical contexts (like historical matters), where prior probabilities cannot be calculated, it's difficult to see how Bayesian reasoning can establish a trustworthy posterior probability. How then do atheists and skeptics typically build a Bayesian case against miracles?
It appears the skeptical approach involves toggling between two distinct notions of probabilistic inference, not just Bayesianism but also frequentism. Basically, frequentism is used to establish an extremely low prior probability, P(H│K), of miracles as a class of events that contradict the laws of nature, before being plugged into a Bayesian framework. Bayesian machinery is then employed to show why realistically no amount of evidence will ever overcome that extreme improbability.
But the probability of an individual miracle may diverge quite widely from one situation to the next, as the evidence for some miracles is much stronger than for others. History is not a measurable sample space, after all, and historical events are not random variables. Antecedently improbable events do occur, and when they impact the world in significant ways historians take note of them. The prior probability of the 2001 terror attack on the World Trade Center, for example, was evidently quite low, or at least not high enough to trigger even minimal preventative measures; yet the event certainly happened anyway. This is essentially "the problem of the single case," as Victor Reppert and others have observed.
So what can we expect to happen when an extremely low prior probability is assigned to a miracle claim with an unusually impressive quantity of admissible, relevant evidence cited in support of it? We can expect the evidence to be rejected as insufficient. But arguably this dismissal of evidence is just the sort of thing that makes the prior probability so exceedingly low in the first place. From what I can tell this circular procedure serves to artificially dilute the probabilities of miracles and routinely sweep evidence for them under the rug.
As Joe argued, what seems to be a sophisticated mathematical disconfirmation of miracles is therefore little more than a case of circular reasoning dressed in the rational garb of probability calculus: "There’s never been any proof of miracles before so we can’t accept that there is any now. But that’s only because we keep making the same assumption and thus have always dismissed the evidence that was valid." That is, miracles cannot be confirmed by evidence because their prior probability is too low, and the prior probability of miracles is low as it is because no evidence has confirmed them. Skeptics have been begging the question against miracles for centuries. Evidently they're still doing it today.
[Edited for clarity 3/16 - DM]
Posterior probability, P(H│E & K), therefore increases as explanatory power and prior probability increases, all else being equal; while posterior probability decreases with increases in total probability of the evidence P(E│K), all else being equal. To "confirm" the hypothesis and demonstrate that its probability is greater than any competing hypothesis, the resulting posterior probability should be greater than .5. Likewise a posterior probability of < .5 would disconfirm the hypothesis, where explanatory power and prior probability, P(E│H & K) x P(H│K), are too low to clear the probability of the evidence. But in non-statistical contexts (like historical matters), where prior probabilities cannot be calculated, it's difficult to see how Bayesian reasoning can establish a trustworthy posterior probability. How then do atheists and skeptics typically build a Bayesian case against miracles?
It appears the skeptical approach involves toggling between two distinct notions of probabilistic inference, not just Bayesianism but also frequentism. Basically, frequentism is used to establish an extremely low prior probability, P(H│K), of miracles as a class of events that contradict the laws of nature, before being plugged into a Bayesian framework. Bayesian machinery is then employed to show why realistically no amount of evidence will ever overcome that extreme improbability.
But the probability of an individual miracle may diverge quite widely from one situation to the next, as the evidence for some miracles is much stronger than for others. History is not a measurable sample space, after all, and historical events are not random variables. Antecedently improbable events do occur, and when they impact the world in significant ways historians take note of them. The prior probability of the 2001 terror attack on the World Trade Center, for example, was evidently quite low, or at least not high enough to trigger even minimal preventative measures; yet the event certainly happened anyway. This is essentially "the problem of the single case," as Victor Reppert and others have observed.
So what can we expect to happen when an extremely low prior probability is assigned to a miracle claim with an unusually impressive quantity of admissible, relevant evidence cited in support of it? We can expect the evidence to be rejected as insufficient. But arguably this dismissal of evidence is just the sort of thing that makes the prior probability so exceedingly low in the first place. From what I can tell this circular procedure serves to artificially dilute the probabilities of miracles and routinely sweep evidence for them under the rug.
As Joe argued, what seems to be a sophisticated mathematical disconfirmation of miracles is therefore little more than a case of circular reasoning dressed in the rational garb of probability calculus: "There’s never been any proof of miracles before so we can’t accept that there is any now. But that’s only because we keep making the same assumption and thus have always dismissed the evidence that was valid." That is, miracles cannot be confirmed by evidence because their prior probability is too low, and the prior probability of miracles is low as it is because no evidence has confirmed them. Skeptics have been begging the question against miracles for centuries. Evidently they're still doing it today.
[Edited for clarity 3/16 - DM]
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