Bayesian evaluation for the likelihood of Christ's resurrection (Part 33)

So the diversity of the individuals involved in Christ's resurrection testimonies already make a high degree of interdependence unlikely. One could hardly find a less likely group of people to enter into a world-spanning conspiracy. You would expect disparate parts of such a group to be constantly at odds with each other, destroying the conspiracy almost immediately.

In fact, that's pretty much what happened: the disparate parts of the group were constantly at odds with each other - and yet, the "conspiracy" was preserved.

There were hints of confusion and division even before Jesus's crucifixion, in things like the disciples arguing about who will be the greatest, or who will be sitting by Jesus's side when he establishes his kingdom. Peter even berated Jesus for announcing his upcoming death, and there seems to have been a general confusion about the nature of the movement - are they going to lead an uprising against Rome? Do they need to be armed?

After Christ's ascension, very early in the book of Acts, there was a conflict between the Greek-speaking members and the other Jewish members of the Church, concerning the equitable distribution of food to the widows. This was a big enough deal that the Church instituted a whole new tier of leadership - the first deacons - to address the issue. And yet, the central tenant of the "conspiracy" - the resurrection - was unchanged.

Soon thereafter, an intense persecution befell the church. Several key members were killed, and the church was scattered across the known world. Gentiles were evangelised around this time as well (Cornelius, Ethiopian eunuch) - which in itself caused no small controversy. All of this further fragmented an already very diverse church. The problem was so bad that various evangelists regularly encountered people with very incomplete knowledge about Jesus. There was a group who did not know about the baptism of the Holy Spirit, and Apollos had to have his knowledge completed by Priscilla and Aquila. Still others were only attracted to the power associated with the name of Jesus and wated to misusing it outright, like Simon Magus and the seven sons of Sceva. And despite all this persecution, fragmentation, and confusion, the "conspiracy" held together.

In the middle of all this, Paul - already mentioned as one of the early persecutors of the church - miraculously converted to Christianity, and became one of its foremost evangelists, to the point of becoming one of the named witnesses in 1 Corinthians 15. He then got embroiled in the central controversy of the early Christian Church: how to handle Gentile believers. This controversy got so heated that Paul once had to publicly rebuke Peter for his stance, and James wrote his epistle with a vastly different emphasis from Paul on what it means to truly be a "believer". In other words, this controversy set all three of the named witnesses in 1 Corinthians 15 against one another, to some degree. And yet, the "conspiracy" endured.

And that's not the end to the divisions of the early church - A number of outright heretical groups had to be condemned - Paul pronounced anathema to a group proclaiming "a different gospel", and John pointed out certain "antichrists" at large in the world, and also named the works of the Nicolaitans as the objects of Jesus's hate. And despite all this division, the "conspiracy" remained.

Again, what kind of conspiracy does this? What conspiracy kills off its leader, fragments itself into dozens of different pieces, bitterly fights itself on internal controversies, condemns some parts of itself, and still survives? And all for what purpose? Persecution, controversy, and death? That is all that any insider might have hoped to receive by adhering to their conspiracy. As Paul himself says in 1 Corinthians 15: "If in Christ we have hope in this life only, we are of all people most to be pitied."

If the "conspiracy" is that Jesus really did rise from the dead, and that this was the central truth that held early Christianity together, despite all of its divisions - then all this makes sense. But if you want all this to be the result of some made-up story, then you have to postulate a completely ridiculous conspiracy - one where the leaders somehow concocted the greatest and most effective lie the world had ever seen, despite being an inept, fractious group of people with little control over their followers. Or, you can instead postulate a truly vast conspiracy, one which planned for all this persecution and division and infighting from the beginning. You can postulate whatever you'd like. That's the whole appeal of conspiracy theories. But at the end, the prior probability for any conspiracy you postulate will be absolutely miniscule.

The next post will begin the process of putting all this on a quantitative footing.


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Bayesian evaluation for the likelihood of Christ's resurrection (Part 32)

So, could the resurrection testimonies really have a near-total dependency among them? Could they have been generated by a conspiracy of some sort? There are a multitude of reasons to think they were not.

First, there is the story of apostle Paul - one of the named witnesses in 1 Corinthians 15, and someone who first started out as a zealous persecutor of Christianity. He is then supernaturally converted by literally seeing the light on the road to Damascus, and becomes Christianity's most effective evangelist. How many conspiracies have something like that in their narrative?

Now, the conspiracy theorist can still say "that was obviously a part of the plan! You've been taken in by their trap!" After all, that's precisely what a conspiracy theory is designed to do. But as I said before, while this allows them to "explain" apostle Paul by keeping the Bayes' factor for his testimony to around 1, it is still a significant blow against a conspiracy theory. For a conspiracy that has planned for such a conversion is far less likely a priori than one that has not. In postulating a more vast, deep, and comprehensive conspiracy, the theorist has postulated a less likely one.

In fact, Paul's conversion is so unlikely that it's probably enough by itself to debunk the most common types of conspiracy theories. Ascribing Paul's actions to a conspiracy is like planning to punch a stranger in the streets in hopes of making money from the ensuing lawsuit, or asking a politician to concede an already won election to their bitter political rival solely out of respect and goodwill. Human beings just don't work that way.

But we're just getting started. Let's look at James - the biological brother of Jesus, and another of the named witnesses in 1 Corinthians 15. Consider his relationship with the rest of the early Christian movement.  Earlier on in his ministry, there is good reason to think that there were some strained relationships between Jesus's family and his disciples. And yet, after the resurrection James is considered one of the chief disciples, and is named as a witness to Christ's resurrection. Could this have been the result of a conspiracy?

Unlikely. Conspiracies are, I think, either family affairs or professional affairs. The relatively rare cases where it involves both are cases where the family member was already in the professional circle from the beginning (e.g. a family of politicians or bankers). I haven't really heard of a conspiracy which starts in a professional setting, then spreads to encompass estranged family members, as key players. Of course, you can postulate that James was in on the plan from the beginning - but you're again just postulating a bigger, and therefore a less likely, conspiracy.

So there is already a great deal of independence for Paul and James from the rest of Jesus's disciples, which include Peter. So our three named witnesses in 1 Corinthians 15 are quite unlikely to be dependent, and therefore their testimonies are unlikely to be the product of a conspiracy.

But the independence of the witnesses don't stop here. The twelve disciples may be thought of as a fairly interdependent group - after all, they were twelve Jewish males who all followed one leader. But looking into their background reveals a good amount of diversity. Some of them were fishermen - but their number also included, at a minimum, a tax collector (working for Rome) and a zealot (revolutionaries working against Rome). It's not easy to come up with three groups that would have gotten along less with each other than a tax collector, a zealot, and a regular Jewish worker, like a fisherman. Could a conspiracy rise from such a group? It's not impossible, but it's also not likely.

The diversity is further magnified in the earliest converts to Christianity, at the Pentecost. According to Acts 2, these people were from all over the known world. Many of them did not even consider Hebrew, Aramaic, or Greek to be their native tongue. Again, could a conspiracy spread out so quickly to such a diverse group, as the very first people to be taken in? It must have been a very flexible and compelling conspiracy indeed - and therefore a very unlikely one.

Lastly on the point of diverstiy, there are of course the women. They go unmentioned in the public declarations of 1 Corinthians 15, because women were not considered reliable witnesses in the 1st century Jewish society. Yet they are featured prominently in the actual narrative in all of the gospels - as the group that did not abandon Jesus at the cross, and the first witnesses to the risen Christ. What kind of conspiracy does this? Why have the first witnesses to the resurrection be a class of people the society considers unreliable? Why include them in the story at all, if you're not going to publically mention them among the chief witnesses?

If it's all true, then all this makes sense. But as a conspiracy theory, each one is a mystery. One can construct a conspiracy theory that fits all this, but such a conspiracy would be a rare one indeed, and highly unlikely a priori.

We will go over more reasons against conspiracies next week.


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Some vague, unoriginal thoughts about the election

[...]
Surely the nations are like a drop in a bucket;
they are regarded as dust on the scales;
he weighs the islands as though they were fine dust.
Lebanon is not sufficient for altar fires,
nor its animals enough for burnt offerings.
Before him all the nations are as nothing;
they are regarded by him as worthless
and less than nothing.
[...]
Do you not know?
Have you not heard?
Has it not been told you from the beginning?
Have you not understood since the earth was founded?
He sits enthroned above the circle of the earth,
and its people are like grasshoppers.
He stretches out the heavens like a canopy,
and spreads them out like a tent to live in.
He brings princes to naught
and reduces the rulers of this world to nothing.
No sooner are they planted,
no sooner are they sown,
no sooner do they take root in the ground,
than he blows on them and they wither,
and a whirlwind sweeps them away like chaff.
[...]
Why do you complain, Jacob?
Why do you say, Israel,
“My way is hidden from the Lord;
my cause is disregarded by my God”?
Do you not know?
Have you not heard?
The Lord is the everlasting God,
the Creator of the ends of the earth.
He will not grow tired or weary,
and his understanding no one can fathom.
He gives strength to the weary
and increases the power of the weak.
Even youths grow tired and weary,
and young men stumble and fall;
but those who hope in the Lord
will renew their strength.
They will soar on wings like eagles;
they will run and not grow weary,
they will walk and not be faint.



[...]
Do not put your trust in princes,
in human beings, who cannot save.
When their spirit departs, they return to the ground;
on that very day their plans come to nothing.

Blessed are those whose help is the God of Jacob,
whose hope is in the Lord their God.
He is the Maker of heaven and earth,
the sea, and everything in them—
he remains faithful forever.
He upholds the cause of the oppressed
and gives food to the hungry.
The Lord sets prisoners free,
the Lord gives sight to the blind,
the Lord lifts up those who are bowed down,
the Lord loves the righteous.
The Lord watches over the foreigner
and sustains the fatherless and the widow,
but he frustrates the ways of the wicked.

The Lord reigns forever,
your God, O Zion, for all generations.
Praise the Lord.


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Bayesian evaluation for the likelihood of Christ's resurrection (Part 31)

Let us examine this general class of theories, that postulate a near-total interdependence in the evidence against them. What kind of theories are they? What are their properties? Is it fair to characterize them as "crackpot" theories?

Now, note that such theories requires a conspiracy of some kind, almost by definition. Near-total interdependence means that what appeared to be many pieces of evidence was really just controlled by a singular false entity, which manufactured all the other pieces of evidence. Whether this source was a group of disciples or an elite Roman secret society or some space aliens or whatnot doesn't particularly matter - All such theories share the following traits.

The first thing to note about such theories is that they have very low priors probabilities to begin with. Indeed, among those skeptical of Christ's resurrection, a theory of this type is almost never their first choice. Few people want to be labeled a conspiracy theorist, after all. The skeptics want the resurrection testimonies to have been produced "naturally". They'll invoke known social phenomena such as myth generation over a long time, or religious fervor or delusion. They want such ordinary explanations to be a plausible way to generate the resurrection testimonies. Of course, what we've demonstrated thus far is that such explanations are in fact not plausible - that they're faced against a Bayes' factor of more than 1e54.

Maybe some people will say that they'd rather be a conspiracy theorist than believe in the resurrection. But even so, such people only say this as a backup, while still trying to argue for a more ordinary explanation.

So, conspiracy theories and other similar hypothesis have low prior probabilities, even in the mind of skeptics. This is appropriate, as conspiracies are in fact very rare.

Secondly, these 'near-total interdependence of evidence' theories are designed to ignore the evidence. They are chosen precisely because they allow their adherents to say "but that's exactly what they want you to think!" to any evidence you bring against them. It's important to note that this is not an accidental, fortuitous property of these theories. 'Near-total interdependence of evidence' is the defining feature of such theories, and it's precisely that feature which allows them to dismiss all the evidence which would weigh against more likely theories.

In combination, the above two facts mean that such theories cannot really hope to win the day. Since they start with a low prior, and are designed for ignoring the evidence, they cannot really hope to prevail - they need evidence to increase that low prior probability, but they're designed mostly to ignore evidence.

Note that, when a conspiracy theorist ignores evidence by saying "that's exactly what they want you to think!", this doesn't actually help the theory. It merely turns a piece of evidence against the theory into no evidence. Yes, the conspiracy theory has "explained" the evidence, but only about as well as the rival theory. The Bayes' factor therefore stays around 1, meaning nothing has changed on that front, and the probability for the conspiracy theory remains at its low prior value.

But, such evidence does still hurt they conspiracy theory, because the prior probability itself is now a lower value. A greater conspiracy that explains more - one that is more vast and has planted more evidence and covered it up better - is a priori less likely to have come about than a lesser conspiracy. So a piece of evidence that the conspiracy has to dismiss does still hurt the theory. The hope of the conspiracy theorist is that this harm in the prior probability will be less than the exponential rate of harm that a fully independent piece of evidence would normally cause.

So the most such a theory can realistically hope for is a kind of non-total loss, where they lose less quickly and hope to say "at least it's not impossible!" at the end.

Now, there are very particular kinds of evidence that does help them - the ones that specifically demonstrates a conspiracy. Something like a document from a secret meeting that lays out the nefarious master plan would work. But, of course, for a vast majority of these theories, such evidence does not exist.

So, given all these traits - given that they are highly unlikely theories that are designed to ignore the evidence, with little chance at any positive evidence for them - I think it's fair to call them crackpot theories.

In fact, in the specific case of Christ's resurrection, the situation is even worse for these theories - for there are many factors within the resurrection testimonies that are highly effective in working against them. We will examine these in the next post.


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Bayesian evaluation for the likelihood of Christ's resurrection (Part 30)

Let us recall our purpose in collecting these non-Christian stories about a "resurrection": we wanted to verify our Bayes' factor for the evidence of Christ's resurrection. My claim is that it's at least 1e54.

The first part of our plan was to find the non-Christian resurrection story with the most evidence behind it. If we make the naturalistic assumption about these stories, we can then say that this level of evidence is approximately what corresponds to a Bayes' factor of 1e9. For by the virtue of having the most evidence, such a resurrection story would have narrowed the field down to itself - one case - from the approximately 1e9 reportable deaths in history.

As it turned out, the "resurrection"s of Krishna and Aristeas had the most evidence behind them, amounting to roughly 1/24th of the evidence for Christ's resurrection. According to our program, this must be assigned a Bayes' factor of roughly 1e9. Then 24 times that amount of evidence would correspond to raising the Bayes' factor to the 24th power - meaning, the evidence for Christ's resurrection has a Bayes' factor of... 1e216.

So yes, that does verify that the Bayes' factor is "at least 1e54". It furthermore demonstrates how much of an underestimate that value is. Recall that, in a slightly different context, I mentioned that the full odds for the resurrection would be far in excess of 1e100, and that our values for the Bayes' factors were drastic underestimates. All that is verified by this completely different methodology, of comparing with non-Christian resurrection stories.

But that's not all. This comparison also provides yet another layer of verification, in that it allows us to check the Bayes' factor of 1e8 for a disciple's testimony about Christ's resurrection. You see, among the non-Christian resurrection stories we've seen, there was not a single case of a person making an earnest, insistent testimony about someone rising from the dead. That says something about the strength and rarity of such testimonies. Granted, we have not investigated every existing non-Christian resurrection story - but if such a testimony really has a Bayes' factor of 1e8, there should be about ten such testimonies for us to find. The fact that we have not found a single one puts a lower bound on the Bayes' factor, of just about 1e8. As usual, there's some nitpicking possible, depending on whether you think there are a hundred or thousands of non-Christian resurrection stories. But it's unlikely for any of that to change the value of 1e8 by more than a couple of orders of magnitude. So our estimate about the strength of the disciples' testimony has now also been verified.

We can now be very confident that Jesus rose from the dead. Our previous calculation which first gave us this confidence has now been verified in multiple ways, using completely different methodologies - by double-checking with the historical background of non-Christian resurrection stories. Everything checks out, and all the numbers are in harmony.

But... all this has been computed under the assumption that there isn't any extreme dependence in the disciple's testimonies. We've already accounted for "normal" dependence, like ordinary social pressure or group conformity. But we have not yet accounted for the possibility that the entire set of testimony about Jesus's resurrection might have been been engineered to be in agreement by some unknown force. That is to say, we've been discounting crackpot theories - like a conspiracy by the disciples to steal Jesus's body, or an alien mind-controlling all the witnesses to the resurrection.

Ignoring such theories is fine and good, as long as both sides of the debate are agreed in dismissing them. Most doubters of the resurrection do not subscribe to these extreme theories, so carrying out our calculations in this way up to this point was still productive. However, they're now facing a double-checked Bayes' factor exceeding 1e54 for the resurrection. This makes the posterior probability against the resurrection so tiny, that the small prior probability assigned to crackpot theories now seem much larger in comparison. Someone set on disbelief can no longer ignore these theories. Indeed they have no other choice: they must fully embrace these crackpot theories.

We will begin to address such theories starting next week.


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