Bayesian evaluation for the likelihood of Christ's resurrection (Part 46)

So then, what would count as replicating the evidence for Christ's resurrection?

It's simple. The replication would be a new religious movement based on a "resurrection", which must match or exceed all of the essential components of the original evidence for Christ's resurrection. These components are merely what we've been discussing throughout this series of posts. They consist of the following:

First, the replicated "resurrection" must have sufficient evidence, enough to match the evidence for Jesus's resurrection. Again going back to the summary in 1 Corinthians 15, this must consist of at least the following:

To match Peter, James, and Paul's testimonies, we will require earnest, insistent, and personal testimonies to the replicated resurrection by three specifically named individuals. These individuals must be public figures, prominent enough for there to be a good amount of readily accessible information about them. They must open themselves up to the public, in regular speaking engagements involving cross-examinations - just as Christ's disciples did. 

To match the testimonies of the other apostles, we will require an earnest, insistent, and personal testimony by a group of people, of at least 10 individuals. They must be public figures with known names. Many (but not necessarily all) members of this group must be sufficiently accessible, to allow for public speaking engagements and cross-examinations. They need not all attain to the same level of prominence as the first group, as long as we're certain of who they are and what their testimony is. 

To match the testimonies of the group of 500 disciples, we will require testimonies from a large, specific group of people. They don't have to be named, and they don't have to be insistent in their testimony. They may be mostly private individuals. But they cannot be merely "some people". They should be sufficiently well-defined that many of them are personally known to a public figure, and their testimony should be about a specific, public event.

That's the set of people you'll need to testify concerning this replicated "resurrection". This is merely a repeat of the same conditions that we've previously used in this series. This covers the raw amount of evidence you need.

Second, this evidence for the replicated "resurrection" must have the certain qualities which make conspiracies and other crackpot theories unlikely. Again, this is only what we've covered before - but here it takes on added importance, since we're talking about artificially replicating the evidence.

One of the public, prominent witnesses must be someone who was publically known for being strongly opposed to this new "resurrection" movement from the beginning. This person must have done real, material, public harm to that movement, prior to their change of heart. That change of heart must come from a conviction that this "resurrection" really happened. 

There must not be an obvious prior connection or common cause among the public, prominent witnesses. They must be reasonably independent. 

Within, say, 30 years of its beginning, this "resurrection" movement must cover a wide geographical area with great cultural and linguistic diversity - an appropriate region might be "the Middle East", "the Mediterranean", or "Southeast Asia". Its followers, too, must reflect this diversity. As a corollary, the movement cannot be entirely directed by a central authority, and different parts of it must be in severe contention with one another. 

That 30 years will also mark how long the major witnesses must be "insistent" for. They must unwaveringly testify to the replicated "resurrection" for at least that long.

Material wealth or political power cannot be a tangible, or even likely, reward for joining up. Escape from poverty due to charity is acceptable, but only to a point where the new convert would no longer go hungry and naked.

Third, there are some further implicit factors which now need to be spelled out.

This movement cannot be built on Christianity. Otherwise, the strong dependency factors would ruin the experiment. It must achieve everything from scratch, without a preexisting foundation guiding how things ought be or ought to turn out - just as Christianity itself did. It's okay for the movement to get started in a "Christian nation", it just can't be directly based on or inspired by Christianity. 

The entire replication must be plausible. If you successfully start such resurrection-based movement, but it requires circumstances which occur once in a trillion years, that would not be considered plausible. For example, let's say you find a way to reproducibly convince people of a "resurrection". But it only works on quintuplets who were struck by a ball lightning at the moment of their conception, and it can furthermore only take place when twelve comets brighter than Venus simultaneously show up in the night sky. Such an explanation for the original, Christian resurrection is not plausible, even if it may be naturalistic. Whatever mechanism you use to generate your replication must be likely enough to have had a decent chance of actually occurring in history. 

Lastly, you may not brute force the problem with an overwhelming amount of resources. Recall that Christianity started with Jesus and a handful of his followers, with no great wealth, political power, or specialized scientific knowledge. Your effort must start with similarly humble circumstances. You cannot, for instance, enlist a billionaire to pay off the population of a whole city in some poverty-stricken country, to get them to act out a "resurrection" for your first set of witnesses. You cannot become the dictator of a country and force people to comply with your lies. You cannot impress some primitive, hidden tribe with modern science to get them to believe. You must play fair - using the same means that were available to the early Christians, if indeed their movement actually started naturalistically.

So, that is the challenge. You think you can fake a movement appearing to meet all of the above conditions? Go ahead and try. If you succeed, I will change my mind about the resurrection.

Next week, we will consider still more tests for the validity of my method and the veracity of its conclusion.

---

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Christianity and falsifiability
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Another post, from the table of contents

Bayesian evaluation for the likelihood of Christ's resurrection (Part 45)

Is anyone still skeptical of the fact that Jesus rose from the dead? Well then, here is one more test, straight from a hallmark of the scientific method:

If you think that the evidence for Christ's resurrection was naturalistically produced, then replicate the result.

We have seen thus far that history, in its natural course, has utterly failed to reproduce a Jesus-level of evidence for a resurrection. It has not even come remotely close. And this has not been for a lack of trying, either - we've cited multiple cases where someone tried to produce a resurrection story, only to fall pathetically short of the level of evidence for Christ's resurrection.

But perhaps you might succeed! And really, there isn't any fundamental reason why you can't, if Christianity started naturalistically. In fact, if you are not convinced by the previous arguments, a scientific mindset demands that you give it a try.

So, do you think that there was a massive conspiracy among the disciples to steal Jesus's body and start a new religion? Well, try to start a similar conspiracy of your own! See how well it holds up over the years when people rightly accuse you of being liars, and rightly threaten your reputation and wealth - perhaps even your life and limb!

You think that the Christian resurrection stories started through a mass hallucination, caused by eating a psychoactive plant native to Jerusalem? Well, go find that plant, feed it to a bunch of people, and see if they have the exact same hallucinations about the resurrection of one person!

You think that some gullible religious people couldn't learn to cope with the death of their charismatic leader, and therefore made up the resurrection story? Well, start such a religion yourself, pretend to die, and see what happens!

Don't complain about the scope of the problem, or the amount of people, time, or money you need. Christianity started with Jesus and a handful of his disciples. You and your circle of friends could easily out-scope this group. This is not an experiment that's too big to be attempted. In fact, real-world, large-scale studies on health or sociology regularly out-scope the humble beginning of Christianity.

Don't complain about the right combination of social circumstances needed for Christianity to take off. If you can read this, you probably have access to world-wide travel and possible exposure to an enormous variety of the world's cultures - an advantage that Jesus's disciples did not have. You think that the right set of circumstances only exist in one particular tribal group in Papua New Guinea, or in a specific small town near the outskirt suburbs of Kyoto? You can actually travel to these places, and access the right social and cultural circumstances.

I am being serious here. This is not some cheap taunts against skeptics. If you've read my other posts - if you've even just read the other posts in this series - you know that I welcome the testing of my ideas, and that I'm ready to change my beliefs as a result. If you really do come up with a plausible, naturalistic, reproducible way for Jesus's resurrection reports to have been generated, I will change my mind.

But remember that this works the other way too. We've already seen that the failures of the non-Christian resurrection stories have only made Jesus's resurrection more certain. In the same way, failure in an attempt to replicate Jesus's resurrection reports must, of logical necessity, change your mind. You must become more certain of Christ's resurrection.

Of course, abject failure is in fact the most likely outcome of such an experiment. The experiment will produce something - but that something is not likely to be any better than the many other examples in world history, where a "resurrection" was said to have occurred. It will fall pathetically short of the level of evidence established by Christ's resurrection.

And that is why I, personally, won't conduct this experiment: I think the result will be negative - that it will not really add anything new to the data we already have. Furthermore, I have already done my due diligence, and am already well convinced that Jesus rose from the dead. This only cements my expectation of a negative result. I therefore have little reason to conduct this experiment, no more than I have a reason to reproduce the Michelson-Morley experiment to search for the luminiferous aether - I would rather believe in special relativity.

But the situation is exactly the opposite for a skeptic: they should expect a positive result, that there actually is a way to naturalistically reproduce Jesus's resurrection reports. This would, furthermore, be a new result with high impact, which overturns all the historical accounts thus far. Furthermore, they may be well convinced that Jesus did not really rise from the dead, which would only increase their expectation of success. They therefore have every reason to conduct this experiment - just as Michelson and Morley did for their famous experiment.

So, that is the challenge: if you are a skeptic, you have every reason - including scientific obligation - to try to replicate Jesus's resurrection reports, to achieve the same level of evidence. Refusing the challenge will have its own consequences, concerning your rationality or your actual beliefs.

The next post will outline the specifics of this challenge.

---

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 44)

Let us summarize the "skeptic's distribution" argument for Christ's resurrection.

We have already seen that any kind of reasonable investigation into Jesus's resurrection accounts would conclusively demonstrate that Jesus did rise from the dead. The only possibility left for the skeptic is to turn to unreasonable hypotheses - that is, to crackpot theories like conspiracies.

The distinguishing feature of these theories is that they postulate a near-total interdependence among the evidence, as if they were all manufactured by a single source - the conspiracy. This allows them to ignore the abundance of evidence for a certain position, and instead attribute it all to a rather unlikely prior.

Now, there are many excellent reasons to reject any kind of conspiracy theory when it comes to Jesus's resurrection. Just the conversion of Apostle Paul would be enough to put it beyond any realistic possibility. But that only scratches the surface of the many anti-conspiracy evidence available to the resurrection. However, we will not quantitatively consider these. We will merely use them to sandbag our conclusion - with the note that they would apply their full, independent force against the skeptic's position, since crackpot theories are all that the skeptic has left at this point.

So then, how do we quantitatively consider the interdependence of evidence, fully taking into account all the different possible crackpot theories?

We construct the "skeptic's distribution" - the probability distribution for achieving a certain level evidence for a resurrection, assuming a skeptical, anti-supernatural worldview. This probability distribution is actually quite accessible, since every single non-Christian resurrection report in world history would be the result of a sample drawn from it. Furthermore, such a distribution would fully take into account the aggregate of all the different types of crackpot theories that actually could have happened in history. The results of all things like conspiracy theories or religious mass delusions would show up in the samples, and the samples can then be extrapolated for things beyond what actually happened in history.

Once we have the "skeptic's distribution", the rest of the calculation is easy. We calculate the "skeptic's probability", which is the probability for the "skeptic's distribution" to generate at least a Jesus-level resurrection report. Since the corresponding "Christian's probability" is of order unity, the Bayes' factor for Jesus's resurrection is essentially the reciprocal of the "skeptic's probability".

We first constructed the "skeptic's distribution" using the most pro-skeptical assumptions possible. Even then, this gave "even odds" of Jesus's resurrection having taken place, under an impossibly favorable set of assumptions for skepticism.

Re-running the calculation with demonstrably more realistic - but still very conservative - assumptions, we saw that the Bayes' factor for Jesus's resurrection was still at least around 1e14 or 1e16. Against a prior of 1e-11 for someone rising from the dead, this puts the final odds for Jesus's resurrection at 1e3 to 1e5 - that is, somewhere between 99.9% to 99.999%. All this is with only part of the evidence for Christianity (those summarized in 1 Corinthians 15), and without considering any evidence that applies specifically against conspiracy theories.

The conclusion is clear: Jesus almost certainly rose from the dead.

We will begin a new line of thinking for the resurrection in the next post.

---

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 43)

This is another Jupyter notebook. It contains python code that generates the probabilities of a "skeptic's distribution" generating a Jesus-level resurrection report.

First, we import some modules:

In [1]:

import numpy as np
import pandas as pd
from scipy.stats import lognorm, genpareto

We then write a function to simulate getting the maximum value out of n samples from a given distribution:

In [2]:

def max_out_of_n_from_dist(dist, out_of_n=1e9):
    manageable_n = 100000
    if out_of_n <= manageable_n:
        return dist.rvs(out_of_n).max()
    else:
        top_percentiles = \
            np.random.rand(manageable_n) * manageable_n / out_of_n
        return dist.isf(top_percentiles.min())

We then write a function to calculate the "skeptic's probability" - the probability of a given "skeptic's distribution" generating a Jesus-level resurrection report. The various parameters fed into the function determines the specific form of the "skeptic's distribution".

In [3]:

def calculate_p_skeptic(
    dist_type, #genpareto or lognorm
    shape_params_dist, #np.logspace or np.geomspace 
    sample_size, #1e9 or 1e10
    greater_by, #24 or 50
    n_outliers, #50 or 250
    n_max_draws=10000,
):
    
    if dist_type == genpareto:
        shape_limits = [0.02, 2,1]
    elif dist_type == lognorm:
        shape_limits = [0.2, 10.0]
        min_shape = 0.2, 
        max_shape = 10.0
    if shape_params_dist == np.logspace:
        shape_limits = [np.log10(x) for x in shape_limits]
    shape_params_list = shape_params_dist(
        shape_limits[0], shape_limits[1], 105)
        
    shape_params = []
    p_max_greater_by = []
    n_outliers_estimation = []
    
    for shape_param in shape_params_list:
        dist = dist_type(shape_param, scale=1, loc=0)
        for i in range(n_max_draws):
            shape_params.append(shape_param)
            max_val = max_out_of_n_from_dist(dist, sample_size)
            p_max_greater_by.append(dist.sf(max_val * greater_by))
            p_outlier = \
                (dist.sf(max_val * 0.2) - dist.sf(max_val)) \
                / dist.cdf(max_val)
            n_outliers_estimation.append(
                int(round(p_outlier * sample_size)))

    result_df = pd.DataFrame({
        "shape_params":shape_params, 
        "p_max_greater_by":p_max_greater_by,
        "n_outliers":n_outliers_estimation,
    })
    
    match_df = result_df[result_df["n_outliers"] == n_outliers]

    if match_df.shape[0] < 50:
        print "warning: match_df.shape = ", match_df.shape
    if match_df["shape_params"].max() == shape_params_list.max():
        print "warning: maxed out shape_param"
    
    p_skeptic = match_df["p_max_greater_by"].mean()
    
    return p_skeptic

Now, let us explore some of the different possible forms of the "skeptic's distribution", and calculate their "skeptic's probability".

Here's one we looked at before. It uses the most pro-skeptical assumptions possible to generate the maximum possible "skeptic's probability".

In [4]:

calculate_p_skeptic(
    dist_type=genpareto,
    shape_params_dist=np.linspace,
    sample_size=int(1e9),
    greater_by=24,
    n_outliers=50,
)

Out[4]:

4.139635934580683e-12

Here's another possibility, only changing the most questionable parameters to the edges of their likely values. Here are the changes we're making:

The prior distribution of the shape parameters: from being uniform in linear space to uniform in logarithmic space.

The sample size (that is, the number of reportable deaths in world history): from 1e9 to 1e10.

The number of "outliers" (That is, the number of reports of a "resurrection", with at least a "some people say..." level of evidence): from 50 to 200.

All of these changes are almostly certainly true to at least that extent. The actual truth may be even more extreme - for example, the number of outliers may actually be in the thousands.

This gives a very conservative answer for how small the "skeptic's probability" may be.

In [5]:

calculate_p_skeptic(
    dist_type=genpareto,
    shape_params_dist=np.logspace,
    sample_size=int(1e10),
    greater_by=24,
    n_outliers=200,
)

Out[5]:

1.866911316237698e-14

Here we've changed a few more parameters. The distribution type has been changed to lognormal, we've increased the factor by which the Jesus-level of evidence exceeds the maximum, and the number of "outliers" has been increased. The "skeptic's probability" calculated here may perhaps be called "likely".

In [6]:

calculate_p_skeptic(
    dist_type=lognorm,
    shape_params_dist=np.logspace,
    sample_size=int(1e10),
    greater_by=50,
    n_outliers=300,
)

Out[6]:

3.855529808259304e-16

Here is another combination of parmeters which may be called "likely".

In [7]:

calculate_p_skeptic(
    dist_type=lognorm,
    shape_params_dist=np.logspace,
    sample_size=int(3e10),
    greater_by=70,
    n_outliers=200,
)

Out[7]:

6.095046577090992e-17

So, those are some of the possible values for the "skeptic's probability". Recall that there are other, previously enumerated factors which we have not yet taken into account.

In particular, all the calculations up to this point only take into the amount of testimonies for Christ's resurrection. We have not yet considered the specific kinds of evidence for Christianity that directly counters things like conspiracy theories or mass religious delusions. The ample amounts of such evidence is particularly effective, because such crackpot theories are what makes up most of the skeptic's possibilities at this point. The effect of considering these additional evidence are hard to quantify, but they'd easily be worth another several orders of magnitude.

So we see that the "skeptic's probability" of 4e-12 really was a nearly impossible best-case scenario for skepticism. A very conservative - but not fantastical - value would be more like 1e-14, and a likely scenario gives values like 1e-16. Again, all this considers only part of the evidence for Christianity, and the "skeptic's probability" is bound to be another several orders of magnitude smaller.

The Bayes' factor then is essentially the reciprocal of this value. That is, the Bayes' factor for Jesus's resurrection is at least on the order of 1e14 to 1e16, amply sufficient to overcome a prior odds of 1e-11, with little possible doubt remaining. Jesus almost certainly rose from the dead.

The next post will be a summary of this whole "skeptic's distribution" argument.

---

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