This post displays up close the charts used in this Youtube video, which is a follow-up to this recent stream. In the video, which I won't try to reproduce here, I go into more detail about why it is important to use R and ~R in evaluating the evidence for the resurrection (that is, to compare the theory that the resurrection happened to the theory that it didn't happen) rather than comparing R only to some conjunction of specific, alternative explanations of the evidence (e.g., "Jesus didn't rise and the body was stolen and Peter had a hallucination and the stories in the Gospels were invented or embellished, etc.").
A partition, as I emphasize here and in the stream, is a set of mutually exclusive and jointly exhaustive propositions.
I argue that comparing the explanatory power of R to that of a "best naturalistic alternative" is epistemically uninformative and obscures the real evidential situation, as would be the case if one did something similar in any non-religious historical case. The video contains various perhaps-surprising probabilistic facts such as...Two incompatible theories can both be confirmed by a body of evidence. Just because there is an odds form of Bayes' Theorem without a partition, it doesn't follow that you can calculate the actual posterior probabilities of the two theories involved without using a partition.
Here are the images:
Odds form of Bayes' Theorem with a partition:
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7 comments:
Just a quick not to thank you for your blog and to let you know that people do, in fact, read it. I ran across some of your pages while working on a book on the resurrection of Jesus. We should all be thankful for Al Gore's amazing internet!
Kind regards,
Steve Reed, North Carolina
Thank you Steve, I appreciate it. Even though I don't update it regularly, I keep a lot of content here.
You can see a significant issue with this line of thinking here -- https://bblais.github.io/posts/2026/May/14/how-to-inflate-your-bayes-factor-with-nothing/. Essentially, the increase in the Bayes Factor using the full partition gets canceled in the prior -- making the partition unnecessary, and making the focus on the Bayes Factor alone misleading.
Hi, Brian: There actually is no problem with this analysis, significant or otherwise. You simply prefer, for unknown reasons, to analyze an empirical inference in what to me seems like a confusingly roundabout way, in which one evaluates the *specific evidence set E for* some salient hypothesis H (e.g., testimony to some event, alleged video of that event, or what-not) by comparing the prior probability of H to the prior probability of some highly gerrymandered theory, ~H1, which is a subhypothesis of ~H specifically generated to give probability to the entirety of the set E to its conditional probability on H. It is not even clear that there always *exists* such a ~H1 (unless one simply considers a non-explanatory declaration "~H and E, somehow" to be a hypothesis), and such a procedure is absolutely riddled with opportunities for epistemic confusion. In contrast, the odds form of Bayes' Theorem, so far from being misleading, yields the correct answer even in your "exercise for the reader" and does so with ease. There you create a distribution in which the prior odds are 99/1 against H and the Bayes factor is 39/1 in favor of H from some evidence or set of evidence E. The use of the odds form shows immediately that *of course* E increases the probability of H but is not sufficient to swamp the prior odds. We have there posterior odds of 39/99 (H over ~H), and an extremely simple calculation (39/99 + 39) yields the posterior probability of H as approximately .28.
I have actually seen a professional probability theorist confused by (I would conjecture) not using a partition in evaluating the evidence for miracles, to the point that he kept switching back and forth between a naturalistic hypothesis which was the conjunction of a set of theories specifically evolved to account for the evidence for the resurrection (and frankly, even so it didn't do a very good job!) and the individual conjuncts, or naturalism more broadly. In other words, he treated the statement that a miracle has a low prior probability as equivalent to saying that any naturalistic hypothesis, however convoluted and consisting of however many independent parts conjoined together, evolved to account for the evidence for the miracle, will always have a higher prior probability than that of the miracle. Needless to say, "Naturalism" is not the same thing as "this incredibly specific conjunctive naturalistic hypothesis that we dreamed up in a blatantly ad hoc manner so as to try to account for the evidence for this specific miracle." I believe that the failure to think in terms of partitions is at least partly to blame for this sort of error, which is rife.
(Continued) Finally, I stress that I deem the full partition Bayes factor analysis to be the most natural way to conceive of evidence for *any* specific event for which there is specific evidence. When we say, "The evidence for a moon landing is very strong," a very natural way to think of that is as a reference to the *specific* evidence (videos, interviews, etc.) that man actually landed on the moon, not to more general evidence such as a knowledge of the state of technology at the time and whether it would have been good enough for a moon landing. The latter is relevant to the prior. The former is the specific evidence for the event. I submit that considering a Bayes factor using a partition makes a lot more sense than instead saying, "How does the prior probability of a moon landing compare to the prior probability of no moon landing plus a conspiracy hypothesis specifically gerrymandered to account for this whole set of evidence that we have?" If there actually were a subhypothesis of the negation that gave *identical* probability to the whole of the specific set E to that given by H (and I actually question that), and if one were able to think about it with perfect clarity (also unlikely) and deduct from the prior of the subhypothesis all that needed to be deducted for the various ad hoc embellishments one would need to make, the final result would be the same. But I can't imagine why anyone would try to go about it in such a way, and I submit that if we were considering a variety of mundane events for which we have specific evidence, no one would think of preferring such a method.
One more point: There is a longstanding and lively debate on the best way to measure evidential force. Various methods are suggested, such as measuring the difference between the prior and the posterior of H, another one sometimes called the r measure, which is P(H|E)/P(H), and more candidates. Tim and I favor the likelihood ratio, sometimes called the L measure. Sometimes probability theorists use the log of the L measure. In any event, I don't know of anyone who suggests that we should measure the force of evidence in the round-the-barn fashion that Brian is suggesting: "Take a highly ad hoc version of ~H, if you can find one, that gives precisely the same conditional probability to the set of evidence E that H gives to E, while leaving the probability of E given all the rest of the partition as 0, then somehow or other calculate the prior probability of this subhypothesis ~H1, then calculate the posteriors of H and ~H as constituting a new partition in the same ratio as the priors of H and ~H1. Good luck with that." The whole point of finding a good measure of the force of specific evidence E is to figure out the most intuitive way to measure *what E actually does* qua evidence, not to try as hard as possible to make it look like E isn't doing anything. Part of the reason we favor the L measure (an extremely well-known measure, favored by plenty of people who wouldn't be caught dead being Christians) is because it separates out the power of the specific evidence so cleanly from separate considerations that feed into the priors. In any event, our use of the L measure has precisely nothing to do with weird religious apologists engaging in some sort of probabilistic skulduggery to be misleading.
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