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Estimating key terms in these calculations is complex and heavily reliantdependent on the assumptions made about ~M. If we are open to considering miracles, we might also entertain scenarios involving aliens, advanced technology, yet-to-be-discovered laws of nature, or other extraordinary explanations. TheseSuch alternatives can meaningfully affectsignificantly influence our probability estimates in nuanced ways.

A large number of highly independent, sincere, and consistent witnesses to a purported miracle would be difficult to explain under ordinary, mundane conditions, potentially increasing the likelihood that a miracle actually occurred. However, it would similarly raise the likelihood of alternative, non-trivial waysmiraculous explanations, such as advanced alien technology or a mass hallucination. The probabilistic analysis I provided, based on Bayes' theorem, is not sophisticated enough to distinguish between multiple extraordinary hypotheses that could, in principle, equally explain extraordinary data. As a result, while P(R|M) would increase, thereby raising the overall posterior P(M|R), the effect on P(R|~M) complicates determining an upper bound on how much P(M|R) could ultimately increase.

Estimating key terms in these calculations is complex and heavily reliant on the assumptions made about ~M. If we are open to considering miracles, we might also entertain scenarios involving aliens, advanced technology, or other extraordinary explanations. These alternatives can meaningfully affect our probability estimates in non-trivial ways.

Estimating key terms in these calculations is complex and heavily dependent on the assumptions made about ~M. If we are open to considering miracles, we might also entertain scenarios involving aliens, advanced technology, yet-to-be-discovered laws of nature, or other extraordinary explanations. Such alternatives can significantly influence our probability estimates in nuanced ways.

A large number of highly independent, sincere, and consistent witnesses to a purported miracle would be difficult to explain under ordinary, mundane conditions, potentially increasing the likelihood that a miracle actually occurred. However, it would similarly raise the likelihood of alternative, non-miraculous explanations, such as advanced alien technology or a mass hallucination. The probabilistic analysis I provided, based on Bayes' theorem, is not sophisticated enough to distinguish between multiple extraordinary hypotheses that could, in principle, equally explain extraordinary data. As a result, while P(R|M) would increase, thereby raising the overall posterior P(M|R), the effect on P(R|~M) complicates determining an upper bound on how much P(M|R) could ultimately increase.

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Let:

  • M: a miracle happened
  • ~M: no miracle happened
  • R: reports of a miracle

The posterior probability of a miracle given reports of it is expressed as:

P(M|R) = P(R|M) * P(M) / P(R)

Where:

P(R) = P(R|M) * P(M) + P(R|~M) * P(~M)

Substituting, we can simplify to:

P(M|R) = 1 / (1 + (P(R|~M) * P(~M)) / (P(R|M) * P(M)))

Key Considerations

Prior Probability of a Miracle (P(M))

  • P(M) is arguably very low since most people seldom witness or report miracles, especially of extraordinary types, like a resurrection, which is extremely rare even among miracle claims.
  • Consequently, P(~M) is very high, as it represents the complement of P(M).

Likelihood of Reports Given a Miracle (P(R|M))

  • P(R|M) can reasonably be high if the reports align with realistic, plausible reactions expected from individuals witnessing a genuine miracle.

Likelihood of Reports Given No Miracle (P(R|~M))

  • P(R|~M) depends heavily on the context of R:
    • If ~M holds, R could arise from factors like deception, hallucination, or extraordinary but non-miraculous events (e.g., advanced technology or aliens).
    • For mundane scenarios within ~M, consistent and sincere reports of R would be unexpected unless due to improbable coincidences like group hallucination or coordinated deception.

Thus, while P(R|~M) is plausible, its value depends on how we frame the alternatives under ~M.

Effect of Independent Witnesses

On P(R|M):

  • Adding more independent, consistent witnesses generally increases P(R|M), especially if their reports align with what we would expect to observe if a miracle occurred. This value can approach 1 under these conditions.

On P(R|~M):

  • The effect depends on the specific alternatives within ~M:
    • If ~M includes scenarios like aliens or hidden advanced technology, adding witnesses could make P(R|aliens or technology) more plausible.
    • However, ~M also includes mundane scenarios, where consistent reports of R would remain highly unlikely.
    • Since ~M spans a vast set of possibilities, the fraction of scenarios that make R likely is small compared to those where R is improbable.

Net effect: It’s unclear whether P(R|~M) decreases significantly with more witnesses, as it depends on the weighting of scenarios within ~M.

Final Thoughts

Estimating key terms in these calculations is complex and heavily reliant on the assumptions made about ~M. If we are open to considering miracles, we might also entertain scenarios involving aliens, advanced technology, or other extraordinary explanations. These alternatives can meaningfully affect our probability estimates in non-trivial ways.