Bayesian evaluation for the likelihood of Christ's resurrection (Part 50)

We will consider some more miracles from other religions, but the conclusions here are not difficult to reach. A full-blown analysis is not necessary, as none of them reach anywhere near the level of evidence in Jesus's resurrection. We can just draw parallels to our previous analysis.

So, for example, there's a story of Ichadon, an ancient Korean Buddhist monk. It's said that miraculous signs accompanying his death resulted in the adoption of Buddhism as the state religion. This story is recorded in the "Lives of Eminent Korean Monks" - about 700 years after the fact. As we've mentioned before, this kind of time gap makes any kind of personal testimony impossible, and the level of evidence here only reaches the "some people say..." level, which falls far short of overcoming the small prior against a genuine miracle.

In Islam, the Quran itself is said to be Muhammad's chief miracle - but it is difficult to evaluate this claim. We would need to dig into specific passages in the Quran and interpret it, which would easily end up leading down a rabbit hole. Otherwise, it's hard to say that the mere existence of the text is miraculous. Rather, we want a clear miracle - a miracle that can be recognized as such by anyone, like a resurrection. That is the point of a sign, after all. What good is a sign if it can't be clearly recognized?

The best known Islamic miracle of that kind - a miracle which is clearly a miracle - is Muhammad's splitting of the moon. But even this miracle is highly controversial. There are even certain interpretations - Islamic ones - which deny that this took place at all. They say that it is rather a prophecy that's suppose to take place at some future time.

The clearest case for the splitting of the moon being a literal miracle come from certain hadiths. Now, the sheer number of such citations and the authority they claim is indeed impressive - if one were to stop their investigation there.

But if you actually read these hadiths, you immediately notice how sparse they are in detail. For example, one of the more detailed hadiths on the subject reads:
We were along with God's Messenger at Mina, that moon was split up into two. One of its parts was behind the mountain and the other one was on this side of the mountain. God's Messenger said to us: Bear witness to this.
And... that's it. Clearly, such a claim doesn't score particularly high on the "earnest and insistent" scale. Compare that to, say, the Holy Week narrative in the Gospel of John. The difference in the level of detail is incomparable.

It is furthermore worth noting that these hadiths were generally written down over two hundred years after Muhammad's death. They claim a chain of transmission going back to some contemporaries of Muhammad, but this chain often exceeds five or six people in length. They are therefore distinctly inferior to any of the testimonies in the New Testament in terms of their authenticity, just on this point alone.

Considering these facts, it's hard to see how these sparse testimonies add up to overcome the small prior odds, against a miracle as remarkable as the moon splitting. A rough estimate would assign a Bayes' factor of ~ 1e1-1e3 for each of the 5 or so companions that were suppose to have originated some of the hadiths, giving an overall Bayes' factor of around 1e10, before taking dependence factors into account. Another way to think about this is to consider these hadiths to be parallel to the testimony of the group of apostles mentioned in 1 Corinthians 15, if we only had very sparse records of the apostles from over two hundred years after the fact. This again brings us to numbers far less than 1e10.

On the other hand, a miracle like splitting the moon is indeed highly remarkable - so it should at least be given a prior odds on par with the resurrection, of 1e-11. So the math just doesn't work out. The evidence just doesn't add up. A Bayes' factor of less than 1e10 doesn't overcome a prior odds of 1e-11. In addition, we must consider that there are no credible non-Islamic records of this highly visible and remarkable astronomical event, and the fact that there are good Islamic reasons to disbelieve that this ever happened. At the end of it all, we can be fairly confident that this did not actually happen as a miracle.

So, overall, we can say that our methodology does correctly reject non-Christian miracles. This validates the methodology for the skeptic's test cases, and therefore compels them to accept the results when the same methodology says that Jesus definitively rose from the dead.

Ah - but what about the other miracles in Christianity? Sure, the resurrection might be well-attested, but what about the numerous other miracles in the Bible which has barely any evidence behind it? For example, only Matthew mentions the resurrection of other people at the time of Jesus's death. He only mentions it briefly, in passing. Many of the miracles during the Exodus are also mentioned only in that book. How could a Christian believe in such things, if such level evidence is inadequate according to our methodology?

This is where the "license plate effect" comes into play. Recall that human testimony can have absurdly high Bayes' factors, such as when you choose to believe a particular record of a chess game. There are over 1e120 possible chess games, so if you believe a particular game record, you're giving that game record a Bayes' factor of something like 1e120. How could a human testimony be so powerful?

It's because, even if the true game did not in fact go as recorded, there's no reason to believe that the mistake or the lie would result in that particular game record. Let's go through this step by step: P(game record|true game went according to the record) is high, because most games do in fact get recorded accurately. But P(game record|true game different from the record) is absurdly low - around 1e-120 - because even if the true game was actually different, there's no reason for the game record to end up as that particular incorrect record. But the ratio between these two probabilities is the Bayes' factor. In other words, the Bayes' factor absolutely explodes for human testimony, when there is no particular reason to pick that particular testimony. Or as Aron Wall put it, "this is a magical aspect of testimony, that it can cancel out any amount of low prior probability so long as it's merely due to there being large amount of detail".

Now, our Bayes' factor of genuine, sincere, insistent human testimony - 1e8 - was for cases where there already was a particular reason for a particular testimony. The collection of such testimonies in the New Testament was amply sufficient to cover the small prior odds against a resurrection, but it is not enough to cover the odds for all the miracles in the Bible.

But, that 1e8 is a minimum value. It can increase almost indefinitely, to values like 1e120 or more, once you accept the resurrection. Because once you accept that someone rose from the dead, it becomes a mere detail that the same person also healed the sick. It's just more details that his death triggered remarkable events. There's no reason for a lie or a mistake to record that particular miracle, if you're willing to accept that this is about the one who conquered the grave. In a similar vein, such a person is probably trustworthy when they vouch for the miraculous stories in Exodus. That is how all the other miracles in the Bible can be believed.

Let's go back to the chess game example. Let's say that a friend of yours claims to have beaten Magnus Carlsen, the current world chess champion. You initially say, "no way. You're nowhere good enough to beat the world champion". But many people, including large, independent, trustworthy groups, all publically, sincerely, and insistently state that your friend did in fact beat the world champion. Convinced by the overwhelming evidence, you eventually come to believe that this chess game actually took place, and that your friend actually won. For this decision, you should count the testimony of each individual as having a Bayes' factor of around 1e8, and the total evidence here must have been enough to overcome the small prior odds of your friend winning.

However, let's say your friend then gives you the record of this remarkable game. Should you believe this game record? Absolutely. You are now giving your friend an enormous Bayes' factor, of something like 1e120. The details - the exact record of the game - can be covered by human testimony, because there is no reason to doubt the record given that you've already accepted the unlikely results of the game. They are mere details, compared to your friend's remarkable victory. There is no reason for a lie or a mistake to result in this particular game record. This is what allows the Bayes' factor for the game record - 1e120 - to be far greater than the initial 1e8 that we gave to our friend's testimony. That is the "license plate effect", and it is what allows us to believe all the other miraculous stories in the Bible, once we accept Christ.

So, all that covers the numerous ways to test our methodology. It has passed them all. In everything there is perfect logical consistency and harmony. We believe all the things that ought to be believed, and reject all the things that ought to be rejected. And this methodology, which passes all the tests of the skeptics and the other religions, clearly concludes that Jesus Christ almost certainly rose from the dead.

The next post will be the summary of the whole series.

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Another post, from the table of contents

Bayesian evaluation for the likelihood of Christ's resurrection (Part 49)

Let us now consider some miraculous stories from the works of Josephus.

Josephus was a Jewish historian who was active in the latter half of the first century. His works include The Jewish War and Antiquities of the Jews. They deal with the contemporary and the ancient history of the Jews, from the perspective of the a Jew living after the Siege of Jerusalem and the destruction of the Second Temple in 70 AD.

As such, he is a valuable resource in understanding the background of early Christianity, and his works are quite compatible with the New Testament. The miracles in his works that we're about study are likewise Jewish in origin, and compatible with Christianity. It would do Christianity no harm if these miracles really took place. They do not meet our earlier criteria of "miracles that expressly support a non-Christian worldview". In fact, Christianity could quite happily accommodate Josephus's stories about the signs surrounding the sacking of Jerusalem (Jewish War, 6.5.3).

After all, Christianity started out as a Jewish sect, and acknowledges the fundamental truth in Judaism. Jesus himself was Jewish, and famously predicted the sacking of Jerusalem. The New Testament acknowledges the existence of miracle workers and exorcists outside the immediate circle of New Testament Christians, of varying degrees of legitimacy, including one that Jesus himself was okay with.

However, even all that doesn't mean that we ought to uncritically accept the miraculous stories in Josephus. We will evaluate some of them, according to the methodology we've used thus far. In doing so, we will also evaluate the methodology itself.

First, Let us consider the story of Honi the Circle-drawer, also known as Onias. Josephus tells his story in the Antiquities of the Jews, 14.2.1-2, briefly mentioning that he once called on God to bring down rain.
Now there was one, whose name was Onias, a righteous man he was, and beloved of God, who, in a certain drought, had prayed to God to put an end to the intense heat, and whose prayers God had heard, and had sent them rain.
The rest of the story is about how a warring faction attempted to get him to use his miraculous powers against their enemies - so apparently his feat was publically known. He is also mentioned in some other Jewish works. But at the end of the day, there just isn't much written about him.

Now, what are we to make of this story? Unfortunately Josephus doesn't mention any specific sources in relating this story. He simply tells it as a story, more than a hundred years after the fact. That means we don't have anything like any of the direct witnesses we have for Christ's resurrection. The best we can do here is infer that Josephus and others must have heard the story from someone, presumably from a group of people. But given that we don't know anything else, again the most we can do is credit this account with the "some people say..." level of evidence. This is nowhere near enough evidence to overcome the small prior against a supernatural event, and we can be fairly certain that this event did not actually occur as a miracle.

Next, let us consider the story of Eleazar the exorcist, as told in the Antiquities of the Jews, 8.2.5:
I have seen a certain man of my own country, whose name was Eleazar, releasing people that were demoniacal in the presence of Vespasian, and his sons, and his captains, and the whole multitude of his soldiers. The manner of the cure was this: He put a ring that had a Foot of one of those sorts mentioned by Solomon to the nostrils of the demoniac, after which he drew out the demon through his nostrils; and when the man fell down immediately, he abjured him to return into him no more, making still mention of Solomon, and reciting the incantations which he composed. And when Eleazar would persuade and demonstrate to the spectators that he had such a power, he set a little way off a cup or basin full of water, and commanded the demon, as he went out of the man, to overturn it, and thereby to let the spectators know that he had left the man; and when this was done, the skill and wisdom of Solomon was shown very manifestly
Well, now this sounds pretty impressive! Josephus names himself as an eyewitness! And also his Roman Patron, the emperor Vespasian, whom Josephus would not invoke lightly! And a great number of Vespasian's associates! Let's see what we can make of this.

First, Josephus and Vespasian are both very well known historical characters, easily on par with the named individual witnesses in 1 Corinthian 15. Josephus furthermore mentions a crowd of other people - Vespasian's sons, captains, and soldiers. This crowd is specific enough to match the crowd of 500 in 1 Corinthians 15. The only thing missing from the roster in 1 Corinthians 15 is the group of apostles, and a third major historical character. So that's nearly as much evidence as there is for Christ's resurrection, right?

But we have not yet considered the full qualifications to match the 1 Corinthian 15 witnesses. First, Josephus's testimony here is nowhere as earnest or insistent as the testimonies of someone like apostle Paul. The above passage is the only thing that's known about this Eleazar. Josephus mentions him this one time, and that's it. Compare that to the twelve disciples, who completely oriented their entire lives around the resurrection of Jesus Christ. Compare also to the apostle Paul, whose every writing can be traced back to his testimony that Jesus rose from the dead. There can really be no comparison.

Furthermore, it's worth noting that Josephus mentions this story in a larger narrative about king Solomon. Yes, that Solomon - the son of king David, builder of the first temple, well known for his wisdom. What's a story involving Vespasian doing in a narrative about king Solomon, who lived a thousand years earlier? Well, it turns out that this whole story is actually an aside, an anecdote that Josephus tells to demonstrate how wise Solomon was. He was saying that Solomon was so wise that his wisdom was used to exorcise demons even after all this time. In other words, the whole story is a parenthetical remark to the main point he was trying to make. In fact, for Josephus's main point, it doesn't even really matter if this exorcist was genuinely supernatural. A fake exorcist invoking Solomon is still evidence for Solomon's renown. This erodes the testimony further on the "earnest" and "insistent" scales, for everyone involved.

In addition, there's significant dependency factors at work here. Josephus clearly has ulterior motives in mentioning Eleazar and his feat. He wants to impress his Roman audience with the wisdom of Solomon, as we just mentioned. This is plainly written out in the text itself.

Furthermore, given that he was writing to a Roman audience about Solomon - one of the most famous Jewish kings - it makes sense that he would invoke Vespasian and his associates, even if might mean falsely implying that they were more impressed than they were in reality. Because, of course, we're not told what Vespasian and his associates thought of the whole affair. Clearly they cannot count as being more impressed than Josephus, as there are no records of any testimony that they, or anyone else, gave concerning this event.

And this complete lack of any other mention of this story makes the dependency factors far worse. The testimonies in 1 Corinthians 15 are of course all attested to in multiple other places in the New Testament, and corroborated by multiple non-biblical sources. We have no doubt that their testimonies are accurately summarized in 1 Corinthians 15. Even Vespasian's healing miracles had multiple attestations. But with this story, it's Josephus and only Josephus, who is himself one of the named witnesses. Everything depends on his testimony, on that small passage he wrote - including his claim that there were other testimonies. This dramatically increases the chance of near-complete dependency among the witnesses.

Lastly, it's again worth noting that a resurrection is nearly impossible to fake, whereas in Josephus's story the narrative itself suggests that nothing remarkable happened. He specifies very simple, physical things involved in the exorcism: a ring was put up to the demoniac's nostril. The man fell down. There were incantations mentioning Solomon. A container of water was knocked over. All these are simple, ordinary, non-supernatural events. There is no mention of whether the man was actually restored, in the sense of being in his right mind, free of demonic influence. Taken altogether, it again looks like nothing much happened, and this is far less remarkable than the resurrection of a man who was confirmed to be dead multiple times, who then came back walking and talking.

So we see that while the roster of witnesses is pretty impressive for Eleazar's exorcism, their testimony is actually very weak in comparison to their parallels in 1 Corinthian 15, according to the previously established rules for matching testimonies.

So, let's consider two separate versions of what happened, as we did for Vespasian's healing miracles. First, did something remarkable happen with Eleazar performing in front of a crowd? And second, was it an actual, supernatural exorcism of an actual demon?

As before, let's give the first, non-supernatural "something happened" version of the event a prior odds of 1e-6. On the evidence side, after taking everything above into account, I'd give Josephus's testimony for himself a Bayes' factor of 1e4 - about half as strong as a full-blown, earnest, insistent testimony. Vespasian's testimony must be significantly less than this, because of the very strong dependency factors involved. It is entirely dependent on Josephus's testimony, and Josephus has a strong motivation to mention Vespasian in appealing to a Roman audience in telling a Jewish story. I'd give Vespasian a Bayes' factor of around 1e2, and his associates about the same number. That all comes to a combined Bayes' factor of 1e8. Or, another way of thinking about this is to say that we really only have Josephus's testimony, but the fact that he's willing to involve Vespasian in the story shows that he's really serious, and that upgrades his testimony to make it earnest. In the end, this very rough calculation comes out to about a final odds of about 1e2. We can be fairly certain that "something happened" here.

As for the "actually supernatural exorcism" case, we'll again start with a prior of 1e-8. As for the Bayes' factors, Josephus himself gets 1e3. He clearly tells the story as an exorcism which he himself witnessed, but his focus on the physical aspects of the story sounds like he has some doubts himself as to whether there was actually anything spiritual going on - so he loses an order of magnitude compared to the "something happened" case above. Vespasian and his associates each get a 1e1. The drop here for them is due to the lack of any testimony concerning what they thought about the event. Overall, the combined Bayes' factor is 1e5. Again, we can think of this as being entirely up to the testimony of Josephus - he starts with a Bayes' factor of 1e8 as in the "something happened" case, but loses 3 orders of magnitude because of the relative ease of faking this kind of exorcism, and his lack of any mention of how Vespasian and his associates reacted. So then, this Bayes' factor of 1e5 is set against a generous prior of 1e-8, resulting in a final odds of 1e-3. Therefore, we can choose to disbelieve that anything actually supernatural happened in the story, with some confidence.

The calculations here are quite rough. The testimonies involved in this story about Eleazar are different from the ones summarized in 1 Corinthians 15, along multiple dimensions. A more accurate calculation would be much more time-consuming. But I'm fairly certain that the results here are good to a couple orders of magnitude, and I'm therefore willing to say that something probably happened, but that it probably wasn't anything supernatural.

Again, does that conclusion sound reasonable? Is that what you would have concluded upon reading this passage from Josephus? Good - then the methodology we used is further validated, and the resurrection of Christ is therefore made more certain.

We will examine more non-Christian miracles next week.

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 48)

The following are the accounts of the healing miracles of the Roman emperor Vespasian.
Vespasian himself healed two persons, one having a withered hand, the other being blind, who had come to him because of a vision seen in dreams; he cured the one by stepping on his hand and the other by spitting upon his eyes.
 - Cassius Dio, Roman History, 65.8 
Vespasian as yet lacked prestige and a certain divinity, so to speak, since he was an unexpected and still new-made emperor; but these also were given him. A man of the people who was blind, and another who was lame, came to him together as he sat on the tribunal, begging for the help for their disorders which Serapis had promised in a dream; for the god declared that Vespasian would restore the eyes, if he would spit upon them, and give strength to the leg, if he would deign to touch it with his heel. Though he had hardly any faith that this could possibly succeed, and therefore shrank even from making the attempt, he was at last prevailed upon by his friends and tried both things in public before a large crowd; and with success. At this same time, by the direction of certain soothsayers, some vases of antique workmanship were dug up in a consecrated spot at Tegea in Arcadia and on them was an image very like Vespasian.
 - Suetonius, The Lives of the Twelve Caesars: Divine Vespasian, 7.2 
During the months while Vespasian was waiting at Alexandria for the regular season of the summer winds and a settled sea, many marvels continued to mark the favour of heaven and a certain partiality of the gods toward him. One of the common people of Alexandria, well known for his loss of sight, threw himself before Vespasian's knees, praying him with groans to cure his blindness, being so directed by the god Serapis, whom this most superstitious of nations worships before all others; and he besought the emperor to deign to moisten his cheeks and eyes with his spittle. Another, whose hand was useless, prompted by the same god, begged Caesar to step and trample on it. Vespasian at first ridiculed these appeals and treated them with scorn; then, when the men persisted, he began at one moment to fear the discredit of failure, at another to be inspired with hopes of success by the appeals of the suppliants and the flattery of his courtiers: finally, he directed the physicians to give their opinion as to whether such blindness and infirmity could be overcome by human aid. Their reply treated the two cases differently: they said that in the first the power of sight had not been completely eaten away and it would return if the obstacles were removed; in the other, the joints had slipped and become displaced, but they could be restored if a healing pressure were applied to them. Such perhaps was the wish of the gods, and it might be that the emperor had been chosen for this divine service; in any case, if a cure were obtained, the glory would be Caesar's, but in the event of failure, ridicule would fall only on the poor suppliants. So Vespasian, believing that his good fortune was capable of anything and that nothing was any longer incredible, with a smiling countenance, and amid intense excitement on the part of the bystanders, did as he was asked to do. The hand was instantly restored to use, and the day again shone for the blind man. Both facts are told by eye-witnesses even now when falsehood brings no reward.
 - Tacitus, Histories, 4.81
So, what are we to make of these accounts?

We apply the methodology that we've been using all this time. How much evidence is there for these miracles? And is it enough to overcome the small prior?

As before, we first look at the people providing the testimony. Who claimed that this actually happened? We have three accounts by three well-known historians, but they're merely reporting what they heard from others in their research. Now, we didn't count Matthew, Mark, Luke and John as witnesses for merely writing about Christ's resurrection. We only counted those who actually gave personal testimonies. So we can no more count the three historians above as witnesses. Who were their sources? Who were the actual, original individuals that personally witnessed and reported Vespasian's miracles?

None of the accounts give specific names for such individuals. We have some vague characters, such as "people of Alexandria" or "[Vespasian's'] friends" - but there are no named characters, except perhaps emperor Vespasian himself. However, this group of people seem to be well specified: they're better than the "some people" level of evidence that we've seen so much of thus far. The witnesses are the crowd of people who gathered in Alexandria and saw Vespasian heal these two people. Tacitus mentions eye-witnesses, and presumably he could have gotten to these specific individuals if he had to. So, overall, I would say that this testimony is on par with the 500 disciples witnessing Christ's resurrection. The Bayes' factor for such a testimony is in excess of 1e8, according to our previous calculations. It would be greater still if you counted Vespasian himself.

So, that's a pretty big Bayes' factor, right? So this event actually happened?

But now we run into the problem of precisely defining what "this event" was. Did Vespasian "heal" two people in front of a large crowd? The prior odds for such an event is decently large - certainly much larger than someone coming back from the dead. Let's say, for the sake of argument, that it's around 1e-6. This is easily overpowered by the Bayes' factor of the testimonies above, which exceeds 1e8. We can be decently confident that such an event happened.

But was the healing supernatural? Now we're talking about an entirely different kind of event, with different prior odds. A supernatural healing of this type would be almost as unlikely as a resurrection. Since a resurrection had prior odds of 1e-11, let's be generous and assign this a prior odds of 1e-8. But... isn't the Bayes' factor still large enough to overcome that?

No, not at all. For the Bayes' factor itself now changes. For one, we can no longer count on Vespasian's testimony at all. Apart from the massive conflict of interest (which we'll address later), Vespasian himself doesn't believe that he can actually heal these people in the beginning. Tacitus explicitly reports that everything was perfectly achievable through mundane means, and that Vespasian only attempted the healings when he was informed of this possibility.

Therefore, Vespasian himself is certainly not testifying to anything like a supernatural healing here. Neither, for that matter, is the crowd itself, if things went according to what's in Tacitus's account. In fact Tacitus goes to some lengths to provide naturalistic explanations for the "healing" - to such a degree that his account should actually be counted as evidence against a supernatural healing. He calls these people "most superstitious", and nearly explicitly says that anyone in the crowd who actually believed in a supernatural healing would have been deceived. And there is nothing in either of the other two accounts to contradict his.

Tacitus's account is the earliest, most detailed, and most explicit in mentioning witnesses. The other two are mostly just summaries of his. And yet, in this best account, the idea of a supernatural healing is almost explicitly refuted. So what happens to the evidence? Where is the testimony?

There is essentially none left. At best it's reduced to that familiar, unspecific "some people say..." level. This is nowhere enough to overcome a prior odds of 1e-8, and therefore we can be very confident that a supernatural healing did not take place here.

In summary: on the question of whether there was a public spectacle where Vespasian "healed" two people, a prior odds of something like 1e-6 is overcome by a Bayes' factor exceeding 1e8 - therefore we can reasonably hold that this actually took place. But on the question of whether this "healing" was supernatural, a prior odds of 1e-8 is essentially unmoved against a "some people say..." level of evidence. We are therefore very certain that the "healing" was not supernatural.

And all this is without taking into account the enormous potential for deception, conspiracy, political shenanigans, or a publicity stunt. Vespasian was a newly crowned Roman Emperor, after all. Taking that into account would lower the final probabilities even further, for both the supernatural and the mundane versions of the event. In the end, I think our methodology brings us to the point where there's something like even odds for some kind of public spectacle taking place, but the nature of the event was almost certainly not a supernatural healing.

Do you agree with that assessment? Does it seem reasonable to you? Good - then you are compelled to correspondingly increase your faith in the methodology we used, and therefore increase your degree of belief in Christ's resurrection.

But wait! Can a similar type of reasoning be used against Christ's resurrection? Could it be argued that "something" probably happened with a man we now call Jesus, but that it was not anything supernatural?

No, it cannot. The reasons that existed for Vespasian that allowed for such an argument simply does not exist for Christ's resurrection.

For one, a resurrection is nearly impossible to fake. One can fake being healed of blindness, or of a defective limb, without much effort. You can do it right now - you just need a little bit of acting skills. Just a bit of placebo effect or the excited anticipation of the crowd can be enough to get someone to walk around for a few steps, or convince a man with poor vision that he sees better. That is all that is required to generate the above accounts of Vespasian's miracles. But can you imagine making such an argument for a resurrection? "Are you sure that it wasn't just the placebo effect that cured his death? Or sometimes, if everyone in the crowd anticipates it, a corpse can be encouraged enough to get up and walk."

We believe that nothing supernatural took place in Vespasian's case, partly because what he achieved in healing is not all that remarkable. There are many possible naturalistic explanations. But in Christ's case, you need a naturalistic explanation for a man who was confirmed dead multiple times, who then came back walking, talking, eating, teaching, converting skeptics, and giving missions. Good luck getting all that with common naturalistic explanations like "placebo effect" or "crowd anticipation".

But secondly, and far more importantly, the evidence itself points towards a naturalistic explanation for Vespasian, and a supernatural event for Jesus. We are, as ever, evaluating and following the evidence. The evidence itself - in the form of Tacitus's account - spells out that Vespasian's miracles likely had naturalistic causes. In Jesus's case, it's again the evidence itself - in the form of the text of the New Testament - that consistently and repeatedly tells us that Jesus's resurrection was a supernatural event.

In order to draw an equivalence, and say that "something happened, but nothing supernatural" in both cases, you would need the same kind of evidence. You would need something like the Gospel of John explicitly stating that Jesus's disciples stole the body, and spelling out exactly how they did it and why there was nothing supernatural required. If the Gospel of John actually said such things, then you could draw an equivalence between the "miracles" of Vespasian and the resurrection of Jesus.

But, of course, the Gospel of John does not in fact say that. It is no good making up evidence you don't have - we are to only follow the evidence we do actually have. So, merely mentioning that "something happened, but probably nothing supernatural" is worthless. It's wishful thinking about evidence you don't have. Speculations - merely mentioning possibilities - do absolutely nothing against a Bayesian argument. It is evidence, not speculations, that move probabilities.

So in considering all of the above, you see that the methodology is perfectly consistent in concluding that nothing supernatural happened in Vespasian's stories, while also concluding that Jesus rose from the dead.

We will examine more cases next week.

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 47)

A common argument from skeptics is that we cannot accept the miraculous stories about Jesus while simultaneously rejecting them for all non-Christian miracle-workers in world history. But that is nonsense. Of course we can discriminate between these stories. It just comes down to discerning which ones have enough evidence.

So, for instance, we've already shown that the stories about Jesus's resurrection have far more evidence behind them than any other resurrection stories in world history. We've done the math. And that math, with its self-consistent logical rigor, compels us to both accept Jesus's resurrection, and reject the other resurrection accounts. It merely comes down to their respective level of evidence.

But what about other, non-resurrection miraculous stories? Could any such stories of non-Christian origins be true? A Christian must answer "no" for the most part. There may be some allowances for God sending 'rain on the righteous and the unrighteous', but certainly any miracles that expressly support a non-Christian worldview must be false.

And here, both Christians skeptics can find common ground. We both believe that a large majority of non-Christian miracle stories must be false. And if the Bayesian methodology that I've employed thus far is sound, it ought to be able to come to that conclusion. And by doing so, the methodology will demonstrate that soundness - in accordance to Bayes' rule, for both Christians and skeptics.

So, we will begin this extension into non-resurrection miracles, starting next week. Our first case will be the stories of miraculous healing attributed to the Roman Emperor Vespasian.

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