Bayesian evaluation for the likelihood of Christ's resurrection (Part 34)

Can we quantitatively tackle things like conspiracy theories? What do we do about the interdependency of evidence? One can already imagine the objections to any such attempt. Every assumption would be questioned, and every ridiculous possibility brought up demanding a full numerical treatment. Even if a traditional conspiracy were to be fully debunked in a numerical argument, a skeptic would just weasel the argument to be about a "groupthink induced by religious fervor" instead, and when that got debunked, they would just move on to "have you considered aliens?" Indeed, such weaseling is often the point of bringing up things like conspiracy theories in the first place: not to actually advocate for them, but to make the calculation appear intractable.

But I did say in my last post that I will approach this problem quantitatively - and that's exactly what I'm going to do. Furthermore, my argument will take EVERYTHING into account - government conspiracies, religious groupthink, practical jokes by aliens, everything. Every single possibility for every conceivable degree of evidence dependence will be fully considered.

In addition, empirical evidence will be the foundation of my whole argument. That is, in fact, the key that makes it totally comprehensive. Do you remember the following graph?



That is the level of empirical evidence that history has actually recorded for the resurrection of various individuals. It's a partial histogram - note the differing number of people with different amounts of evidence for their resurrection. This suggests a probability distribution.

Of course, the graph above isn't the complete record of everyone - it's a small sampling of some people who have the most evidence for their resurrection. But if we had a complete record, we could get a very accurate model for their underlying probability distribution. What would that probability distribution represent?

If we exclude Jesus and the other Christian resurrection reports, the probability distribution we get would be the EXACT model that an empirical skeptic of Christianity MUST use, in predicting the likelihood of a resurrection report. Essentially, the idea is that we can calculate the probability of getting a certain level of evidence for a resurrection, based on how frequently similar reports have come up in history.

Note that, because the raw data is gathered from empirical reports collected in history, this automatically takes things like conspiracy theories into account. The possible interdependency of the evidence is fully included in this model. So you think that a great deal of evidence can be built up through a conspiracy, because the evidence doesn't have to be independent? The distribution includes all such evidence-manufacturing conspiracies that actually existed in history. You want to switch your argument to a religious mass delusion instead? The result of all such mass delusions are also included, at the level of evidence that they actually generated in history.

How about something that has probably never happened in history at all, like an alien resurrecting someone as a joke? Even these possibilities are included, through at least two mechanisms. For one, there are a great many multitude of such unlikely scenarios - and at least one of them might have actually occurred in history, even if a specific one of them was unlikely. So they would have recorded their evidence in aggregate. And secondly, even if such unlikely scenarios never occurred, they can still be accounted for in the modeling of the probability distribution from the samples we actually have. As an analogy, if you were to model people's heights by sampling a thousand people, you will still deduce that human heights follow a roughly normal distribution, and can thereby figure out that there may be someone out there who's at least 7 feet tall, even if such a person was not in your sample.

So you see, this method does in fact take everything into account. It does generate the exact model that an empirical skeptic of Christianity must use. That's the great thing about arguing from empirical, historical records. You can bypass all the difficult and controversial calculations about the probabilities of conspiracies, or the precise degree of dependence among the evidence. All of that automatically gets incorporated into the historical data at their actually correct historical values, and all we have to do is to read off the final result.

Once we have this "skeptic's distribution", the rest of the calculation is fairly straightforward. We can calculate the probability of generating a Jesus-level of resurrection evidence, from the point of view of the skeptic's hypothesis. From a Christian perspective, this probability is within an order of unity, so the skeptic's probability then essentially becomes our Bayes' factor. We then simply see if this Bayes' factor is enough to overcome the low prior probability for a resurrection.

Now, I will use 1e-11 as the prior probability for a resurrection. This is higher than the 1e-22 that I had used earlier - but recall that I only got 1e-22 by simply giving away an additional factor of 1e11 for no reason. I can't afford to just give that away when faced against something as insidious as conspiracy theories. 1e-11 is still smaller than anything that a skeptic can demand based on empiricism, and the rest of the argument will be constructed so that even the minimum Bayes' factor will exceed 1e11. The actual Bayes' factor may exceed 1e22 by the end - I just can't be as precise in my calculations at that level.

We will start constructing this "skeptic's distribution" in the next post.

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On becoming a good person
The lifetime of evil (part 1)
Another post, from the table of contents

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 wanted 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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How to make a fractal
Another post, from the table of contents

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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Another post, from the table of contents

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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Another post, from the table of contents

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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"Simon, son of John, do you love me?"
The limits of science as evidence for Christianity
Another post, from the table of contents