Can Bayes' theorem be used non-fallaciously to argue for miracles?

Question

NOTE: Originally this was a sub-question asked within Did Thomas Bayes truly develop Bayes' theorem in an effort to rebut David Hume's arguments against miracles?, but some commenters correctly pointed out that I should split them into two separate questions.

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So the motivating context is this article:

Bayes’ theorem began as a defense of Christianity

Jordana Cepelewicz, Nautilus, December 20, 2016, on the Christian roots of Bayesian statistics:

Presbyterian reverend Thomas Bayes had no reason to suspect he’d make any lasting contribution to humankind. Born in England at the beginning of the 18th century, Bayes was a quiet and questioning man. … Yet an argument he wrote before his death in 1761 would shape the course of history. It would help Alan Turing decode the German Enigma cipher, the United States Navy locate Soviet subs, and statisticians determine the authorship of the Federalist Papers. Today it has helped unlock the secrets of the brain.

It all began in 1748, when the philosopher David Hume published An Enquiry Concerning Human Understanding, calling into question, among other things, the existence of miracles. According to Hume, the probability of people inaccurately claiming that they’d seen Jesus’ resurrection far outweighed the probability that the event had occurred in the first place. This did not sit well with the reverend.

Inspired to prove Hume wrong, Bayes tried to quantify the probability of an event. He came up with a simple fictional scenario to start: Consider a ball thrown onto a flat table behind your back. You can make a guess as to where it landed, but there’s no way to know for certain how accurate you were, at least not without looking. Then, he says, have a colleague throw another ball onto the table and tell you whether it landed to the right or left of the first ball. If it landed to the right, for example, the first ball is more likely to be on the left side of the table (such an assumption leaves more space to the ball’s right for the second ball to land). With each new ball your colleague throws, you can update your guess to better model the location of the original ball. In a similar fashion, Bayes thought, the various testimonials to Christ’s resurrection suggested the event couldn’t be discounted the way Hume asserted.

In 1767, Richard Price, Bayes’ friend, published “On the Importance of Christianity, its Evidences, and the Objections which have been made to it,” which used Bayes’ ideas to mount a challenge to Hume’s argument. “The basic probabilistic point” of Price’s article, says statistician and historian Stephen Stigler, “was that Hume underestimated the impact of there being a number of independent witnesses to a miracle, and that Bayes’ results showed how the multiplication of even fallible evidence could overwhelm the great improbability of an event and establish it as fact.”

Unfortunately, the article does not explicitly provide a formal mathematical formulation of the argument. However, the intuition is captured in the final bolded quote: multiple independent witnesses can make events that seemed highly improbable a priori become highly probable a posteriori. Is this reasoning valid?

Answer 1

If we assume that the probability of a witness testifying to an event is higher if the event actually happened (a reasonable assumption), then it is true that observing that testimony makes the event more likely. How much more likely it makes it depends on how likely the testimony is in the absence of the event. And of course, if the prior probability is low enough, the increase might not be enough to make it likely (e.g., if the witnesses make the event 1000x more likely, but the prior probability is 0.000042, it's still a pretty safe bet that it didn't happen.)

Multiple independent witnesses might help overcome those sort of bad prior odds: they increase the probability very rapidly as you add witnesses. But this requires that they be genuinely independent not only of each other, but also of any confounding variables that might result in a false observation of the event. And that's the real catch, because a lot of potential confounding factors wouldn't be unique to a given witness.

Consider a less controversial example: UFO sightings. If 100 witnesses say they saw a UFO, that might seem convincing...but if it turns out that the first witness actually saw Venus, the rest of the sightings suddenly become a lot less convincing; the witnesses weren't actually independent, because they were looking at the same night sky and were exposed to the same confounding variable. It's not enough to ask how likely they are to report an alien spaceship given that they actually saw one (P(report|event)). You have to also ask, "What else might have caused them to make that report?" (P(report) = P(report|event)*P(event) + P(report|~event)*(~event)). And it can be really hard to rule out all other potential causes.

Answer 2

There's an interesting anecdotal example of this. Since you're looking for an intuitive answer, I think a compelling anecdote is meaningful. This example centers around this famous picture:

This picture was the first imaging of a black hole. It required doing lots of processing on gobs and gobs of data. Katie Bourman, one of the lead algorithm designers, has a TEDx talk on it and a paper from which I will paraphrase grossly.

The measurements gathered were not in the usual imaging space. For the technically literate, it was a bunch of relative phase measurements in RF signals. If the measurements were continuously gathered perfectly, reconstructing the image would be straight forward. It would just be a convolution, which is a well understood mathematical tool from the 1700's. However, due to the realities of the telescopes used, we only got part of the data. There was quite literally an infinite number of possible reconstructions of this partial data. Finding the "correct" reconstruction would be a reasonable approximation of a miracle for this discussion. The algorithm Bouman and the rest of the team developed selected a "most probable" reconstruction, based on what our simulations of black holes say they should look like. The result is the image above.

This algorithm is comparable to a witness. It is witnessing the image of the black hole. But, as Bouman points out in her talk, its an untrustworthy witness. It's already been primed to see what we want it to see, primed with images of our best simulations of what a black hole should look like. So its hard to trust its a priori testimony.

The solution they developed was to create several other "witnesses." They trained the algorithms using different sets. For instance, they trained it with pictures of other celestial objects. Then they asked this "witness" to provide testimony, in the form of a "most probable" image. If the result looked like a comet or a nebula, that would be a good sign that the witnesses were unreliable. As it turns out, the images were remarkably similar.

They then developed another witness, trained by pictures of natural objects: people, mountains, elephants, etc. It too made a most probable image which was remarkably similar to the original.

So after receiving these additional "testimony" from the different witnesses, Bouman and the team could conclude that it was reasonable to assume the reconstruction was a good reconstruction. It wasn't being tainted by seeing what it wanted to see. We could trust the results a posteriori.

I find this example useful because while algorithms have a lot of similarities to witnesses, they are also simply reproducible programs. The intuition about multiple witnesses can be turned into quantitative measurements of fits, as Bouman does in her paper. So it's both an intuitive argument regarding miracles and witnesses and also a mathematically precise argument about algorithms and reconstructing images from VLBI data from radio telescopes.

Answer 3

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)))

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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.

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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.

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Final Thoughts

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.

Answer 4

Probability vs. Bayesian belief
Firstly, Bayes theorem describes not the probability of the actual event (in its usual frequentest understanding - as likelihood of an event actually happening given the laws of physics), but our belief about the event likelihood, i.e., our estimate of this probability. Thus, if I believe that Jesus resurrection is highly unlikely, but many witnesses claim that this did happen, I have to reasonably modify my beliefs.

Confounding factors/purity of the experiment
The problem is that there are many reasons why people may claim that they have seen Jesus resurrecting, i.e., many other parameters of the problem that the argument above doesn't take into account.

In fact, if we to rely on the biblical accounts, not many people had seen Jesus dead and buried, and few had seen him alive afterwards - see Overview of resurrection appearances in the Gospels and Paul. The only event that occurred in presence of many witnesses was opening of the tomb, which turned out to be empty. Thus, the testimony of each of the witnesses has to be conditioned on factors such as:

• this person has actually seen Jesus dead
• they have seen him afterwards alive
• they have a proof (rather than belief) that it was the same person
• they are telling truth

This significantly diminishes the posterior probability. This gets even worse, if we incorporate in our analysis a likewise Bayesian estimate that a witness violates one of the above conditions - since most people who make the claims about Jesus resurrection actually were not there to see, the credibility of a typical witness is very low.

This gets even worse, if we take into account that we do not have first-hands accounts of Jesus resurrection, but only those recorded in Gospels. And even among those we rely mostly on those that were selected as canonical by mainstream Christianity centuries(!!) later. Thus some streams the early Christianity, as well as some modern non-mainstream currents do not believe in resurrection all - treating it either in metaphorical sense or even as a ingenious escape from execution, demonstrating Jesus' intellectual superiority. See, e.g., Gnostics and Jesus in Ahmadiyya.

Prior probability
What is missing in the answer above, but has already come up a few times in the comments is that our initial probability is prohibitively low. That is, if we follow Bayes' reasoning in trying to prove that occurrence of a miracle is consistent with scientific findings, we have to estimate a probability of thermodynamically improbable event - - like all the atoms gathering in half of a container or a decomposing human body coming back to life, we have to deal with numbers like P(M)~ 10^{-N_A}, where the Avogadro constant itself is N_A=6 * 10^{23}. The witness testimonies by hundreds or thousands of people (or, if you wish by the totality of the Earth's population ) are not going to change it significantly... unless we make a highly implausible assumption that all these people are truthful and have rigorous medical and scientific training to attest death and life.

For comparison 0.001 (one inn a thousand) is a normally probability accepted threshold in pharmaceutial research, assuming that we can ignore negative effects of a medication. Five sigma standard for rejecting null hypothesis used by CERN is about one in a million - still much larger than the probability that we would get for Jesus' resurrection.

Answer 5

Just to be clear, standard probabilistic methods and Bayesian methods are doing different things. Standard probability works by creating a probabilistic model based on a set of assumptions, and then testing events against that model. In that sense it's a normative methodology. The trick about probabilistic methods is that they are powerful tools if and only if the initial set of assumptions are correct. If the initial set of assumptions are incorrect, in part or whole, then the probabilistic model is inapplicable and the given results are basically worthless. That's why statistics has arrays of different probabilistic models making different assumptions and having different amounts of power.

Bayesian methods, by contrast, make no assumptions except perhaps a centrality assumption (that subsequent events will tend to occur in the direction of the 'center'). Bayesian methods are non-normative, meaning that they are always applicable. But they are also non-modeling, which means that are tests of empirical perceptions not tests of theoretical constructs.

In short, standard probabilistic methods answer the question "Are these observations consistent with the model proposed?", while Bayesian methods answer the question "What unperceived effect do these observations trend towards?".

The problem with Hume's probabilistic approach — and likely what intuitively annoyed Bayes in the first place — is that Hume is making an entirely conjectural model, asserting probabilities for "inaccurate claims" and "miraculous occurrences" without testing or confirmation. Hume is effectively saying:

I think the probability of miracles occurring is really low (i.e., the model I assume is that miracles do not exist), and I think the probability of people being inaccurate or mistaken is reasonably high (the observation quality is questionable), so I'd need a whole lot of such observations to reject the model and accept miracles as true.

But of course, this is just data-waffling: if we choose our asserted probabilities carefully we can get any result we like. Not that I disagree, particularly: if we start with the normative model that miracles don't exist, it would be hard indeed to find sufficient evidence to reject that model. But clearly Bayes does not accept the assumptions behind that model, so from Bayes' perspective the analysis is worthless.

On the other hand, the problem with Bayes' approach is that he isn't actually giving us a likelihood that a miracle occurred; he's giving us the likelihood that people perceived a miracle to occur. The best that Bayesian methods are going to tell us is that it's probable something occurred that a good number of people took as miraculous. Because Bayesian methods don't provide a model, we don't actually do reality testing; we do perception testing. Even if we establish that it's likely people perceived a miracle to occur, we still have to make the extra step to show the perception conforms to the occurrence of an actual miracle. All Bayes' methods can do in this case is support the act of faith, which is arguable all that Bayes (in his role as minister) was aiming for.

Answer 6

I wouldn't say that no evidence of miracles (not even hypothetical evidence) would be sufficient for rational belief. An earlier answer of mine goes a bit more into how we might theoretically verify such a claim with science (along with problems one would run into in trying to do so). The issue is that no miracle or supernatural claim thus far seems to have gotten anywhere close to having such strong evidence.

That might put me at odds with Hume, but I also have almost 300 years of technological and scientific advancement to work with, which has significantly improved our ability to figure out what actually happened based on some evidence, and which gives us precedented means of documenting things, with most people in much of the modern world carrying a recording device around with them at all times. If Hume wants to debate me on that, he'd be more than welcome, and I'd try to document his resurrection well enough to be sufficient for rational belief, in ways that would far surpass the evidence for miracle claims of old. (I'll use a summoning spell to channel Hume, to argue against this later in this answer. /s) Of course, video evidence is quickly becoming less and less reliable, as deepfakes become more convincing and easier to make, but we might still overcome this by having evidence from enough reliable sources.

What is a miracle?

A miracle could be defined as a violation of a law of nature: something that happens unexpected or unpredictably given the observable events leading up to it, which cannot be explained by events leading up to it, and which is unlike anything that has been well-demonstrated. It may also not leave physical evidence that can be independently discovered and analysed, to reach the conclusion that the miracle happened (if we include this in the definition of a miracle, that would somewhat invalidate what I said above).

It's not a term I'm personally too fond of, because people inevitably end up trying to make pointless semantic arguments, rather than addressing the issue that the claim is far outside or contradicting how we understand nature to work. It's not a categorical difference (few things in this world truly are) - it's really more indicating that the claim is on the far end up of the spectrum of how "extraordinary" a claim is, how unexpected it would be, given all the evidence we have.

Eyewitness testimony?

The arguments in question focus largely on eyewitness testimony. While we have historically put a whole lot of weight into such testimony, we have shown it to be one of the least reliable forms of evidence.

Price/Stigler doesn't necessarily seem to disagree with this, given the "even fallible evidence..." statement.

As it relates to Bayes, Price argument says independent witness accounts may overwhelm the great improbability of a miraculous event. This possibility is largely hypothetical.

In practice, this is going to be a sticking point that undermines their argument for any given miracle claim more often than not. Humans have similar biases, so many people could misinterpret the same event in the same way. This effect has been demonstrated in controlled experiments. This effect is turned up to 11 when one considers people who were taught similar things and who share similar beliefs. Also, memory is malleable. Witnesses often unknowingly align their stories when they share their testimonies with one another. Interrogation can get people to confess to something, and become convinced that they actually did it, despite strong evidence that they didn't do it.

Never mind if we're talking about people witnessing the same event: this is explicitly not independent.

Note: none of this is saying that more witnesses isn't better evidence than fewer witnesses. It's just saying that they're far from independent, and this undermines Price's argument, which requires more independence than there may be.

Regarding Jesus' resurrection, since that is specifically mentioned: The so-called witness testimony for the resurrection is incredibly weak. People say there were 500 independent witnesses, when in fact it's just one guy claiming there were 500 people, and he wasn't even an eyewitness. The gospels were written decades after the events in question by anonymous authors. There's good reason to believe they copied from one another, and that the resurrection was a later addition to Mark. See: Wikipedia. They outright contradict one another in terms of e.g. who was at the tomb. Contemporary authors make no mention of this extraordinary event, nor of the related zombie uprising (Matthew 27:52-53 "And the graves were opened; and many bodies of the saints which slept arose, and came out of the graves after his resurrection, and went into the holy city, and appeared unto many"). At most we have someone mentioning that he heard that people were talking about a dude named Jesus. If there is some amount of eyewitness testimony that can overwhelm the great improbability of a miraculous event, this certainly isn't it.

Assuming a miracle to prove a miracle?

If we want to use Bayes' theorem for determining the probability of a miracle, we need to start from some "prior" probability of that miracle.

A reasonable starting probability would be crucial for an accurate probability estimation, but this is something we just don't have. This ends up being little more than a guess, and people make some questionable assumptions about how events affect the probability. This seems to be little more than presupposing one's conclusion and using motivated reasoning to try to reach one's desired conclusion.

Using Bayes' theorem from a prior probability is generally also only used for multiple independent events under the same probability, such that one's reasonable prior can be sufficiently updated to get to an accurate value (this is why the independence of witnesses is crucial). A one-time event, unlike anything we've seen before, wouldn't really qualify.

Bayes' theorem is a very useful tool. But using it in this way entails too many assumptions to be considered in any way reliable as a means of determining truth.

This also applies to probabilistic reasoning more broadly: people try to allude to probabilities of things we just don't have reasonable means of determining probabilities for, for which probabilistic reasoning seems to fail altogether. If someone wants to build an extensive probabilistic model of reality and/or epistemology, I'd be all for that (with some doubts about the practical viability of that). But selectively applying such models to singular claims, without concrete and well-supported probabilities - that doesn't really get us anywhere.

Is a miracle claim evidence for that miracle?

In trying to use Bayes to establish miracles, there may be some underlying assumption (or risk of one) that someone saying a miracle happened necessarily best supports the particular thing that they say happened. And that if you just keep stacking more and more such claims on top of one another, eventually you'll meet the threshold of "sufficient evidence".

This is questionable.

I've already addressed the unreliability of eyewitness testimony, and how multiple witnesses to the same event can still be quite unreliable.

But there's a potential false dichotomy here.

This is often presented as if there are just 2 alternatives: (1) the event happened exactly as the testimony says, and that this is explained by the explanation the witnesses give, or (2) the event didn't happen at all, or happened in a vastly different way from described.

Simply put, this seems to be far from the only 2 options.

If we consider, for example, some hypothetical witnesses to a hypothetical resurrected person, that the witnesses say is a god, we might posit any of the following:

• Of course, the event may not just have happened at all, or not remotely as described. This possibility would probably be the go-to explanation for most miracle claims, until sufficient evidence is presented, and this is going to be one of the most difficult possibilities to overcome.
• The resurrected person is actually just a convincing lookalike. This would be less plausible if the witnesses are friends and family of that person, and if they spend significant time together, but those things would make it harder, not impossible. Also, such people may be more easily blinded by intense emotion from the seeming reunion with a lost loved one, such that they may be more inclined to dismiss doubts that the person really is their loved one.
• The event may have been exaggerated or miscommunicated. This could be due to unreliable memory or a game of telephone. The actual events may be more or less miraculous.
• The person appeared to be dead, but wasn't actually dead. This is less of a problem these days, with more modern medical technology, but telling whether someone is dead is not actually as easy as one might think. Some substances could give the appearance of death, and there were significant fears of being buried alive. Death is also a gradual process of organs shutting down and brain activity ceasing - someone is not declared dead when their heart stops, but rather when we can't restart their heart (never mind that we can keep people alive for some time with machines and that heart transplants are a thing). Which is to say: death is not as absolute as people often tend to think, and there are various points from which someone can still come back from. Of course, the viability of this explanation depends on the circumstances of death, as well as what is done with the body.
• For the possibility that the person was actually fully and completely dead, and then stopped being dead:
  • Maybe there's some natural explanation for this that we just don't have yet.
  • Maybe they have some "mystical" ability that enables resurrection, that some people are just randomly, rarely born with.
  • Maybe everyone has this ability, and they were just the only one (so far, which we've documented) to figure out how to use it.
  • Maybe they are the god witnesses claim.
  • Maybe they are some other non-natural entity, that lived in a human body and resurrected for any number of reasons.
  • Maybe they were resurrected by some non-natural entity, for any number of reasons.

Some of these may be more or less plausible than others (and I'm not arguing that these are necessarily plausible alternatives to explain Jesus' resurrection specifically). But the point is that there are far more than just 2 possibilities. Even supposing that a "miracle" happened, we'd still need a lot more supporting evidence to say much more than "this person resurrected".

For the case of Jesus, some might say that there is that supporting evidence, and that they're only trying to establish the resurrection claim itself. But I'd say there are far too many contradictions or inconsistencies, and far too little evidentiary support, for the claims Christians commonly make, of an all-loving all-powerful deity that wants a relationship with us, and that we need to "accept Jesus" to have eternal life or suffer for eternity (or be erased from existence), etc. If I were to accept that a resurrection happened, practically anything else seems more likely than Christianity as it's presented. This is, of course, putting aside the insufficiency of the evidence to establish that Jesus did in fact resurrect.

Answer 7

There is ultimately no objective, canonical way to assign prior probabilities to a hypotheses. And there is ultimately no objective, canonical way to update those prior probabilities after evidence has come forth, given that different people view evidence for a hypotheses in different ways. This is partially because probability is a subjective construct created to model our uncertainty. In reality, hypotheses don’t have “true” probabilities. They are either true or false, and Bayes theorem, along with probability in general, was used as a tool to help navigate through this uncertain world.

So technically, Bayes theorem can be used to justify anything “non fallaciously” as long as the updating system is not self contradictory.

Arguably though, it makes no sense to assign anything but a very low prior to miracles. This is not an assumption but rather a result of the fact that nothing breaking known scientific laws has ever been reliably verified. Yes, there have been reports of miracles, but given the tumultuously long history of most people not observing miracles, it stands to reason that one should prefer sticking with the status quo. In other words, one should think that the lack of observations implies the non existence of miracles rather than to think that the supernatural decided to intervene in such a way where miraculous reports can’t be easily verified.

Answer 8

Yes, Bayes theorem can be used to talk about miracles and priors simultaneously non-fallaciously and in good faith. Or priors and unicorns. Or priors and QAnon. Bayesian thinking does not tell you what to believe in; it only tells you how to modify your beliefs once you have additional experience. Thus, it's just as perfectly acceptable for a Catholic priest who is talking the probability of miracles to use Bayes' work as to use percentages and fractions. But expressing belief in the language of mathematics is not a physical endorsement of its existence as 92.5% of university educated gnomes will assure you. Since Classical Empiricism, it has become fashionable to determine existence, not based on logical arguments, but empirical evidence.

Answer 9

One of the variables of the Bayes Theorem is the a priori probability of an event. To assign a probability to the occurrence of a given miracle, you would have to know in advance, as a premise, whether and how often that miracle should occur in that circumstance. The probability given to the miracle in the conclusion is dependent and proportionate to the probability that was assigned to it in the premises.

This could be used non-fallaciously to determine the occurrence of a given miracle, only given prior knowledge of the probability of that class of miracles (e.g. assuming prior knowledge of the probability of the class of resurrections, you could calculate the probability of a particular resurrection), without taking into consideration the evidence for the miracle. If it makes sense to call an event with a previously known propensity a miracle.

The probability of a given miracle given evidence (e.g. calculating the probability of a given resurrection given a report of a resurrection) is also a necessary variable which requires, as a premise, the knowledge of distinguishing true evidence of a miracle from false evidence of a miracle, which is evidently lacking in the case under consideration. The case under consideration would therefore have to be an unusually obscure case, while the generic case with a general probability assigned would have to be clear-cut cases of miracles where the evidence can be evaluated clearly.

To give a concrete example, to determine the probability of a given resurrection given a report, one would have to know the prior probability of any given dead human being resurrected (or whatever subset of humans is under consideration), the probability of a report of a resurrection, and the probability of a report in the event of a resurrection. This requires a knowledge of 1) whether and how often resurrections occur, independent of reports, 2) a method of determining if a reported resurrection is true, independent of the report, and 3) a method of determining how often resurrections result in reports. The existence, and frequency, of both resurrections and reports of resurrections, as well as a correct method of evaluating other reports of resurrections, is already assumed in the premises. This contrasts with the implicit lack of a method of evaluating the particular resurrection under investigation.