A Critical Response to Mayer’s Defense of Miracles

In his article Hume’s Fallacy: Miracles, Probability, and Frequency, Paul Mayer attempts to rehabilitate belief in miracles by critiquing David Hume’s well-known argument against them. Mayer’s central move is to accuse Hume of conflating frequentist and Bayesian notions of probability. He argues that, under a Bayesian framework, belief in a miracle can be rational if the testimonial evidence is strong enough. But Mayer’s position fails at a more fundamental level: he does not distinguish between the mere possibility of a miracle and the rational expectation of one. This failure undermines the practical applicability of his argument.

1. Rational Belief Tracks Rational Expectation

Rational belief, especially when governed by Bayesian inference, is not about what is merely possible—it is about what is probable. Bayesian updating demands that belief be scaled to evidence, and this includes prior probabilities derived from our inductive understanding of the world.

Miracles, by definition, are violations of the uniform patterns observed in nature. And it is from those regularities that we derive our inductive expectations. These regularities are not assumptions; they are the result of overwhelming empirical consistency. The prior probability of a miracle is thus extremely low—not arbitrarily, but for good evidential reasons.

2. Low Priors, Heavy Lifting

While Bayesianism permits posterior belief to grow with strong evidence, it does so only in proportion to the prior probability. This is where Mayer’s argument begins to collapse. The miracle hypothesis begins with a vanishingly small prior. That means any confirming evidence must be strong enough to compensate for this starting disadvantage.

The relevant Bayesian update is:

The low value of heavily biases the equation unless is nearly one and is vanishingly small. But human testimony is never that reliable—especially when filtered through layers of belief, culture, and psychological distortion.

3. The Testimony Trap

Mayer places undue epistemic weight on testimony. But human testimony is demonstrably unreliable even in mundane matters. When applied to the extraordinary domain of miracle claims, the weaknesses of testimony compound. False positives, confabulation, motivated reasoning, and confirmation bias all increase the likelihood that a reported miracle is mistaken or fabricated.

Indeed, our inductive knowledge of the unreliability of testimony should itself reduce our confidence in it. So even if the testimony were emotionally compelling or socially reinforced, Bayesian reasoning still prohibits belief unless the testimonial evidence overwhelms the improbability of the claim.

4. Expectation Constrains Belief

Mayer’s defense collapses if one accepts a crucial principle: if you cannot rationally expect a type of event to occur in the future, you cannot rationally believe that it occurred in the past—unless the evidence clears an enormous evidential hurdle.

If a miracle is the kind of thing you do not expect given your epistemic commitments and experience, then you are not rationally entitled to believe that one has occurred based merely on testimony. To accept it as a past event is to accept that your inductive expectations were unjustified, or that testimony has somehow provided greater evidential force than direct physical observation. Both moves are epistemically reckless.

5. Rationality Requires Inductive Coherence

Belief must be tethered to what we reasonably expect, not merely what we cannot logically rule out. Mayer’s argument, by failing to respect the inductive constraint on rational expectation, invites an epistemology where any rare and exotic claim could be deemed rational if bolstered by sufficient narrative support. That is not a defense of rationality—it is a recipe for credulity.

It is true that miracles are not logically impossible. But that is irrelevant. In the absence of strong, reproducible, and independently verified evidence, the rational agent does not believe. Expectation, rooted in past experience and statistical frequency, is the constraint that protects us from unjustified belief. Until that constraint is met—and Mayer offers no mechanism by which it is—rational agents are not justified in affirming miracle claims, no matter how emotionally or culturally persuasive the testimony may be.

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Assigning Priors to Inexplicable Phenomena: A Bayesian Perspective

Introduction

Consider a remote tribe, isolated from modern civilization, encountering a laser pointer for the first time. The bright, focused beam of light defies their understanding of fire, sunlight, or any natural phenomenon they know. To them, this could appear miraculous, perhaps the work of spirits or gods. This scenario mirrors how individuals throughout history have interpreted unexplained events as miracles, often attributing them to supernatural causes. Yet, we know lasers are products of advanced technology, fully explainable by natural laws such as optics and quantum mechanics (Laser Basics). This comparison provides a compelling framework for exploring how to assign initial probabilities, or priors, to hypotheses about phenomena for which no causal chains are available, particularly for minds with limited understanding of the natural world. For those familiar with Bayes Theorem, this article offers a rigorous examination of how to set priors in such cases, using a thought experiment to illuminate Bayesian reasoning.

Bayes Theorem: A Refresher

Bayes Theorem is a cornerstone of probabilistic reasoning, allowing us to update our beliefs about a hypothesis () given new evidence (). It is expressed as:

Here, is the posterior probability of the hypothesis given the evidence, is the likelihood of observing the evidence if the hypothesis is true, is the prior probability of the hypothesis, and is the total probability of the evidence, calculated as:

The prior represents our initial belief in the hypothesis before considering the evidence. Setting appropriate priors is critical, especially when dealing with phenomena we cannot explain due to limited knowledge, as priors heavily influence the posterior probabilities.

The Thought Experiment: A Tribe Encounters a Laser

Imagine a remote tribe observing a laser pointer’s beam creating a dot on a wall. Lacking knowledge of modern technology, they might formulate three hypotheses to explain this phenomenon:

• H1: Known Natural Causes – The laser is explained by natural laws they already understand (e.g., fire or sunlight).
• H2: Unknown Natural Causes – The laser is explained by natural laws they do not yet know.
• H3: Supernatural Causes – The laser is caused by supernatural forces (e.g., spirits or gods).

Given their limited understanding, H1 is unlikely, as the laser does not resemble any known natural phenomenon in their experience. The decision thus hinges on assigning priors to H2 and H3. This scenario parallels historical cases where phenomena like comets were deemed supernatural omens but later explained as celestial orbits (Comets in History). The challenge is to determine how the tribe—or any mind with insufficient knowledge—should rationally assign priors to these hypotheses.

The Challenge of Setting Priors

Assigning priors to H2 and H3 is complex when causal chains are unavailable. The tribe’s cultural beliefs might incline them toward H3, especially if their worldview includes supernatural entities, as seen in historical attributions of diseases to divine punishment (Epilepsy History). However, a rational approach requires considering the broader context of human knowledge and the limits of one’s understanding. Bayes Theorem emphasizes that priors should reflect our best estimate of probability based on available information, but what happens when that information is severely limited?

Historical Perspective: The Success of Natural Explanations

History provides a compelling guide. Many phenomena once considered supernatural have been explained naturally as scientific knowledge advanced:

• Comets: Once omens of divine will, now understood as icy bodies orbiting the sun.
• Diseases: Previously attributed to evil spirits, now explained by microbiology and neurology.
• Eclipses: Thought to be divine interventions, now predictable through astronomy.

This pattern in history suggests that unexplained phenomena are likely to have natural explanations, even if those explanations are not yet known. In Bayesian terms, the base rate—the frequency with which past unexplained phenomena were resolved naturally—supports assigning a higher prior to H2 over H3. While the tribe may lack this historical perspective, they can still apply a principle of humility: recognizing that their knowledge is limited and that many things they do not understand may still operate within the natural world.

Rational Approach: Embracing Ignorance

Even without extensive historical knowledge, the tribe can reason about their own experiences. For example, they might recall phenomena that once seemed mysterious—such as thunder or the rising sun—but later became accepted as part of the natural order, even if not fully understood. This introspection can lead to a higher prior for H2, acknowledging that the laser might be a natural phenomenon beyond their current comprehension. This approach aligns with Cromwell’s rule, which advises against assigning zero probability to any hypothesis unless it is logically impossible (Cromwell’s Rule).

Moreover, assigning a higher prior to H2 is pragmatically beneficial. It encourages investigation and learning, potentially leading to new discoveries, whereas H3 might prompt acceptance of the phenomenon as inexplicable, halting further inquiry. This mirrors the scientific method’s success, which assumes natural explanations and has consistently expanded human understanding (Methodological Naturalism).

Bayesian Calculation: The Impact of Priors

To illustrate the impact of priors, consider a Bayesian calculation for the tribe’s hypotheses. Suppose they observe the laser (evidence ) and assign the following priors based on their initial beliefs:

• (low, as the laser doesn’t fit known natural laws)
• (moderate, allowing for unknown natural causes)
• (high, reflecting cultural belief in supernatural forces)

Assume the likelihoods are:

• (very low, as known laws don’t explain the laser)
• (high, as unknown natural laws could account for it)
• (high, as supernatural causes can explain anything)

Calculate the total probability of the evidence:

Now, compute the posterior probabilities:

Here, the supernatural hypothesis (H3) is favored, largely due to its higher prior.

Now, suppose the tribe adopts a more rational approach, recognizing their ignorance and assigning a higher prior to unknown natural causes:

Recalculate:

Now, the unknown natural causes hypothesis (H2) is favored, demonstrating how priors significantly influence posterior beliefs. This highlights the importance of setting priors thoughtfully, especially in the absence of clear causal chains.

Pragmatic and Epistemic Advantages of Favoring H2

Assigning a higher prior to H2 offers several advantages:

• Encourages Scientific Inquiry: Believing in unknown natural causes motivates exploration, potentially leading to discoveries like optics or quantum mechanics, which explain lasers.
• Avoids Premature Closure: Supernatural explanations can halt inquiry by accepting phenomena as divine mysteries, as seen in historical resistance to studying natural causes of diseases.
• Aligns with Historical Success: The track record of science shows that natural explanations have consistently replaced supernatural ones, from eclipses to electricity.
• Falsifiability and Testability: Natural explanations, even if unknown, are potentially testable, whereas supernatural explanations are often unfalsifiable, offering little explanatory power (Sagan’s Baloney Detection Kit).

These advantages make H2 a more rational choice, even for a tribe with limited knowledge, as it fosters a mindset conducive to learning and progress.

Addressing Cultural Influences

One might argue that the tribe’s cultural beliefs justify a higher prior for H3, as supernatural explanations are deeply ingrained. However, this risks circular reasoning: believing in supernatural causes because of a worldview that assumes them. A rational approach requires stepping back and considering the broader context, including the possibility that cultural beliefs may be incomplete or incorrect. The tribe can reflect on instances where seemingly mysterious phenomena were later understood naturally, even within their own experience, such as seasonal changes or animal behaviors. This introspection can shift priors toward H2, aligning with the principle of Occam’s razor, which favors explanations requiring fewer assumptions. While supernatural hypotheses might assume complex entities like gods, unknown natural causes assume only the existence of undiscovered laws within a consistent natural framework.

Comparative Analysis: Miracles vs. Technology

The comparison between the tribe’s perception of the laser and traditional miracle claims is illuminating. Traditional miracles, such as those in religious contexts, often imply supernatural intervention beyond natural laws, like limbs regenerating (Miracles in Christianity). In contrast, the laser is a natural phenomenon, highlighting that the perception of a miracle is relative to one’s knowledge. This distinction underscores that what seems miraculous may reflect a knowledge gap rather than divine action. In Bayesian terms, assigning priors to miracles requires extraordinary evidence, as per Carl Sagan’s maxim, given the historical rarity of confirmed supernatural events compared to the abundance of natural explanations.

Practical Application: A Framework for Assigning Priors

To formalize the approach, consider the following framework for assigning priors based on the level of understanding:

Level of Understanding	P(H2: Unknown Natural Causes)	P(H3: Supernatural Causes)	Rationale
Limited (e.g., remote tribe)	0.6	0.3	Promotes exploration, avoids premature supernatural conclusions.
Moderate (e.g., early science)	0.8	0.1	Reflects growing success of natural explanations.
Advanced (modern science)	0.95	0.05	Aligns with empirical success, high evidential bar for supernatural claims.

This table illustrates how priors should shift with increasing knowledge, emphasizing natural explanations as understanding grows. For the tribe, starting with a higher prior for H2 encourages a path toward learning, potentially leading to an understanding of lasers through interaction or observation.

Updating Priors with New Evidence

As the tribe gathers more evidence—perhaps observing the laser’s consistent behavior or interacting with those who understand it—they can update their priors. For example, if they learn that the laser is a tool created by humans, would increase, and would decrease, reflecting Bayesian updating. This process mirrors scientific progress, where initial ignorance gives way to understanding through evidence accumulation, as seen in the transition from geocentric to heliocentric models (Heliocentrism).

Conclusion

The thought experiment of a remote tribe encountering a laser illuminates how to assign priors to inexplicable phenomena using Bayesian reasoning. By recognizing the historical success of natural explanations, embracing intellectual humility, and valuing inquiry, we can rationally assign higher priors to unknown natural causes over supernatural ones, even with limited knowledge. This approach not only aligns with the empirical track record of science but also fosters a mindset conducive to discovery and understanding. For the tribe, favoring unknown natural causes opens the door to learning about lasers; for us, it encourages a principled approach to the unknown, grounded in reason and openness to new knowledge.

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The Growing Confidence in Natural Explanations:

Assessing Track Records:

Each generation of humans has faced unexplained phenomena—eclipses, plagues, comets, hallucinations, mysterious lights in the sky. In every era, people were compelled to answer a simple yet pivotal question: What caused this? For most of history, the prevailing answer was supernatural—gods, spirits, curses. Yet over time, these once-enchanted phenomena have been consistently subsumed under natural laws. This cumulative shift in explanatory success is not trivial. It forms the empirical backbone of a rational inductive argument: those living today are more justified than any prior generation in assigning high credence to natural explanations for any new unexplained phenomenon. This essay explores how this track record changes the rational landscape of belief and why appeals to the supernatural have become progressively less defensible.

The Game of Explanation: Competing Models

Every unexplained phenomenon invites a model. Historically, the explanatory game has pitted two broad categories against each other:

• H2: The event is due to natural causes, whether currently known or unknown.
• H3: The event is due to supernatural causes, involving entities or forces outside natural law.

These are not symmetrical in their evidential performance. Over centuries of testing and observation, the models invoking natural causes have accumulated explanatory wins. Supernatural explanations, by contrast, have suffered an unbroken series of retreats. What was once attributed to gods—lightning, disease, volcanic eruptions—is now accounted for by meteorology, microbiology, and plate tectonics.

This asymmetric track record does more than demote supernatural explanations. It alters the rational structure of prior assignment. If one epistemic player—H2—has won every round of explanatory contestation for centuries, it is rational to increase one’s prior for H2 and diminish it for H3 accordingly. This is not pragmatism. It is inductive rigor.

Induction Over Time: The Compounding Base Rate

Bayesian reasoning is not simply a method for updating beliefs in light of single instances of evidence. It also thrives on patterns of outcomes. With each passing generation, humanity gains more data points in the grand explanatory ledger:

• Demonic possession → epilepsy and schizophrenia.
• Divine wrath → viral and bacterial contagions.
• Spirit apparitions → optical illusions, neurological disorders.
• Miraculous healings → placebo effects, spontaneous remission, misdiagnosis.

This long series of explanatory “corrections” results in a base rate that heavily favors H2. The prior probability of any new phenomenon being best explained by natural causes rises because the total weight of history has shown that such explanations succeed where supernatural ones do not. In other words, the posterior belief from yesterday becomes today’s prior.

This rolling inductive advantage is not speculative. It is cumulative and historical. A person in the 14th century might be justified in thinking plague was supernatural. A person in the 21st century is not. The explanatory track record—available in the public domain of science—has dramatically shifted the burden.

Supernatural Hypotheses: The Problem of Degeneracy

One reason supernatural explanations lose traction over time is their epistemic degeneracy. They are too pliable, capable of explaining anything but predicting nothing. Unlike natural hypotheses, which constrain the kinds of phenomena they allow and thereby risk falsification, supernatural hypotheses are typically unfalsifiable.

When the rain doesn’t fall, one can say the gods are angry. When the rain does fall, one can say the gods are pleased. This degeneracy robs supernatural models of the risk of being wrong—an essential feature of epistemic reliability. In contrast, natural models expose themselves to error and therefore earn trust when they persist unrefuted.

This makes supernatural explanations not just historically underperforming, but structurally inferior as explanatory candidates.

Why Today’s Minds Are Rationally Better Positioned

Modern humans are not epistemically equal to their ancestors—not because they are more intelligent, but because they stand on an evidential mountain built over centuries. When someone today sees a new unexplained phenomenon—say, a mysterious glowing orb hovering over a forest—it would be epistemically irresponsible to assign equal priors to natural and supernatural causes.

The rational response is to recognize that in the entire history of investigated mysterious lights, every confirmed explanation to date has fallen under H2. This does not disprove the supernatural, but it enforces a rational prior heavily skewed toward natural causes. The weight of history is itself data.

Objections and Misplaced Fairness

Some argue that this stance unfairly prejudges supernatural hypotheses. But Bayesianism is not about fairness; it is about credence proportional to precedent and coherence. The prior is not an expression of bias—it is a reflection of past performance. And H3 has performed abysmally.

To assign a high prior to H3 “just in case” is to ignore the empirical record. Worse, it is to place supernatural explanations on undeserved epistemic parity with a naturalist approach that has delivered medicine, space travel, quantum theory, and molecular biology.

Conclusion

We are not the first generation to encounter the unexplained, but we are the first to inherit a near-flawless record of successful naturalistic explanations for what was once deemed miraculous. This historical asymmetry justifies a profound shift in our prior probabilities. Bayesian reasoning compels us not to treat H2 and H3 as equals but to weight them based on performance. Today, that performance makes it vastly more rational to believe that any new unexplained phenomenon is natural in origin—even if its causal chain is not yet known.

The track record of natural explanations is not merely suggestive—it is decisive. And that is why, absent compelling evidence to the contrary, our priors must favor the natural.

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