Bayesian evaluation for the likelihood of Christ's resurrection (Part 37)

Now, what kind of data do we have to determine the shape parameter?

We have the historical data, of course. We have some number of people who are said to have been resurrected in some sense, and each of these people has some amount of evidence associated with their resurrection claim.

We essentially want to "fit" these evidence data into a generalized Pareto distribution, and read off the shape parameter. However, this will be somewhat tricky. We do not have the complete data for all 1e9 reportable deaths throughout human history. We can reasonably assume that the vast majority of them would have essentially zero evidence for a resurrection, but the complete data set would be pretty much impossible to obtain. We don't even have the complete data set just for the outliers - cases like Apollonius or Zalmoxis, where there is a distinctly non-zero level of evidence for a resurrection. Furthermore, the precision on determining the level of evidence is rather poor. All this means that the usual "fit a curve through some kind of x-y scatterplot" approach would not work very well.

However, given that we already know we'll be fitting a generalized Pareto distribution, this is not necessary. We're just looking for the shape parameter, and for that, we merely need to count the number of outliers near the maximum value. Consider the following graph:

This is the same graph as before, in the sense that it just shows the generalized Pareto distribution, scaled so that the probability of x > 1 is 1e-9. Once again, this means that the maximum evidence from 1e9 reportable deaths is likely to appear around x = 1.

However, we now want to focus on how to fit the data. And since the data will have x values less than the maximum, this graph is scaled so that we're focusing to the left of the x = 1 line, instead of the tail to the right.

In particular, note the vast differences in the area under the curve for different shape parameters. The shaded regions represent the probability of finding an "outlier" - a non-Christian resurrection report with at least 20% of the evidence of the maximum report. For instance, the reports of the resurrection of Puhua or Apollonius would be considered an "outlier".

So, let's look at the green curve, with a shape parameter of 20, and a tiny area under the curve. If this were the skeptic's distribution, you'd expect essentially no other outliers. The maximum value would stand by itself, with no other outliers coming anywhere near its value.

Similarly, if the shape parameter is 2, you'd expect perhaps one outlier out of 1e9 samples - one other resurrection report would have at least 20% of the evidence of the maximum.

Lastly, if the shape parameter is 0.2, you'd expect many, many outliers. The probability distribution grows very rapidly as it goes backward from x = 1, and therefore you expect to find many other resurrection reports with a similar level of evidence as the maximum.

So by counting the number of outliers, we can make a determination about the shape parameters.

But... wait a minute. Having more outliers is associated with smaller shape parameters? But didn't smaller shape parameters correspond to a faster-decaying function, and therefore a lower probability for the "skeptic's distribution" generating a Jesus-level of evidence? Wouldn't this lead to the "skeptic's distribution" being less able to explain the evidence for Jesus's resurrection, and therefore make the resurrection more likely?

Are we saying that having MORE non-Christian resurrections reports (like Apollonius or Zalmoxis) make Jesus's resurrection MORE likely?

That is precisely what we are saying. The following analogy may help understand how this could be.
Alice accuses Bob of theft. Bob is known to have come into a sudden possession of $100,000. He is also known to be a gambler. He claims that his sudden fortune came from a lucky night at the card table, but Alice believes that he stole the money - she claims that $100,000 is far too large a sum for Bob to have naturally won through gambling. 
Carol takes on this investigation. She looks into Bob's past gambling history, to see it's realistic for him to have won $100,000 in a single night. She finds that, among Bob's past verifiable winnings, there were two nights where Bob won $5,000 and $3000. These are his most remarkable winnings on record, and Carol cannot find any other instances where he won more than $1000 on a single night. 
Carol concludes that she does not really have enough information. It could be that Bob plays a card game with an erratic payout scheme, where winning 20 or 30 times more money is not that unusual. Maybe it has some kind of "let it ride" or "double or nothing" mechanism which makes such returns plausible. Or maybe Bob himself is an erratic gambler, and decided to bet a lot more money that one night to win the $100,000. Based on all this, Carol decides to be skeptical of Alice's claim that Bob stole the money. Her own "skeptic's distribution" for how much money Bob can win does not decay quickly enough. There are relatively few outliers near his maximum winnings of $5000, and this suggests that it decays very slowly - meaning that the $5000 cannot be established as a limit to what Bob can win. His theoretical winnings can possibly stretch quite far into the higher values, making it impossible to rule out a $100,000 winning. 
But then, Carol has a breakthrough in her investigation. She finds extensive, previously undiscovered records of Bob's gambling winnings, and it shows that Bob has won more than a $1000 on dozens of nights. The maximum that he's won is still $5000, but he's also regularly won thousands of dollars in a single night. 
Carol takes this new information into account, and adjust her "skeptic's distribution" for how much Bob can win in a single night. Clearly, Bob's winnings are not erratic; he regularly wins up to about $5000. But this also establishes, with the weight of those repeated winnings, that this is close to the likely upper limit for what he can win in one night. 
Carol therefore decides to believe Alice. Her "skeptic's distribution" cannot explain how Bob would naturally win $100,000 in a single night, because it goes against his established pattern of regularly winning up to $5000. She pursues the case further, and eventually convicts Bob of theft.
This is not just a story; it can be mathematically established, and it will be in the future posts. For now, this story just provides the intuitive backing for the mathematical results to come.

So, having more non-Christian reports of a resurrection, with their pathetically low levels of evidence behind them, only make Jesus's resurrection more likely. When skeptics say "don't you know there are numerous other Jesus-like stories of someone dying and resurrecting?", they are only kicking against the goads. The more numerous such cases they come up with, the more firmly it establishes that Jesus really did rise from the dead.

The next post will bring the last several posts together, to fully spec out the program which will compute the complete "skeptic's distribution", from which we can calculate its chances of predicting a Jesus-level of evidence for a resurrection.

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 36)

We've decided on a power law as the general form of the "skeptic's distribution".

The details of the distribution near zero will not particularly matter. We're more concerned about how rapidly it decays at very large values. This allows us quite a bit of leeway in choosing the specific form of the power law distribution, as they all decay similarly as we move along the tail off to the right.

For this reason, I've chosen the generalized Pareto distribution as the specific form of the "skeptic's distribution", guided chiefly by the straightforward interpretation of its parameters. But the choice here will not affect any conclusions. Any other power law distribution would give the same results.

The generalized Pareto distribution is characterized by three parameters: location, scale, and shape. The location parameter determines where the distribution starts. It's where the probability density of the distribution is the largest. As the vast majority of humans have zero evidence suggesting that they rose from the dead, the location parameter should obviously be set at zero.

The scale parameter is irrelevant; it only controls how far the distribution should multiplicatively scale in the horizontal direction, and can be arbitrary changed by changing the unit of evidence we use. As we'll consider all the evidence for our the resurrection reports relative to one another (for example, as a fraction of the amount of evidence for Christ's resurrection), the value for the amount of evidence in some specific units never enters the picture. So we'll just set this parameter to 1, or to whatever is convenient for visualization, and forget about it.

The shape parameter is the interesting one. It's what we really care about. It effectively determines the power in the power law, and controls how quickly the function decays as the amount of evidence increases.

For example, this is what the tail end of the distribution looks like with various shape parameters:

In each case, the distribution has been scaled so that the total probability to the right of the grey line (at x=1) is 1e-9. Essentially, x = 1 is where you would expect the maximum value out of 1e9 samples to appear, corresponding to the level of evidence for the resurrection of a figure like Apollonius or Aristeas.

Note the different rates decay. With the shape parameter at 0.2, the probability density drops to practically zero as we move to larger x values. There is essentially nothing left by the time we've moved to x = 24, even if we integrate out to infinity. Therefore, if this were the final form of the "skeptic's distribution", the probability of generating a Jesus-level of evidence for a resurrection would be essentially zero.

However, with the shape parameter at 2, we see that the decay rate is much slower, and there is a good amount of probability even out at x = 24 and beyond. If this were the "skeptic's distribution", it would have a good chance of generating a Jesus-level of evidence for a resurrection, even if that level were 24 times higher than the runner-up.

A shape parameter of 20 decays more slowly still. It's hardly decaying at all by the time it reaches x = 24. In fact, it decays so slowly that the blue curve with the shape parameter of 2 will eventually move below it. If this were the "skeptic's distribution", it would have a non-negligible chance of generating an event at x values of much higher than 24.

So, it all comes down to the shape parameter. But how shall we decide on its value? Why, by choosing the one that best fits the data according to Bayes' theorem, of course.

We will outline this procedure in the next post.

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 35)

Recall that we're constructing a "skeptic's distribution" - the probability distribution of generating a resurrection report with a certain level of evidence. We will construct it from historical, empirical data. This allows us to bypass the mess of trying to compute everything from first principles, and ensures that the this is the correct distribution - a skeptic cannot reject it without rejecting history or empiricism.

So, what form should this "skeptic's distribution" take?

How about using a normal distribution? Well, that would be plainly ridiculous. The data we have thus far has Jesus (very conservatively) having 24 times more evidence for his resurrection than anyone else in history. Our goal is to get the probability for something like this happening.

But if we used a normal distribution for the "skeptic's distribution", this could essentially never happen. Recall that human heights roughly follow a normal distribution. Then, our problem would be analogous to looking for someone 24 times taller than anyone else in history - that is, someone well over 200 feet tall. The probability for something like that is essentially zero. So if we chose the normal distribution, we'd essentially be dooming the skeptic's case from the start.

The same is true for an exponential distribution. An exponential distribution decreases in its probability value by a constant factor for each unit of increase in its domain. As the domain is "level of evidence" in our problem, this means that each piece of evidence would multiply the probability values. That is to say, we'd be treating each piece of evidence independently. And we already saw that even with a reasonable degree of dependence factored in, the probability values already reached numbers like 1e-54, again dooming the skeptic's case.

This is a testament to how quickly these distributions decay as they extend to the right. Their right tails are so "stubby" that the maximum values of their samples are strongly restricted, and getting something 24 times greater than that maximum is essentially impossible. Picking any such distribution would not be taking into account the dependence of the evidence, and would unfairly doom the skeptic's case from the start.

Rather, we need a distribution with a "long tail" - something that has a chance for a new high record to beat the previous record by factors like 24. Something that decays slowly enough that its probability values remain non-negligible as we move further to the right. The distribution should still be realistic and have some justification for being selected, but we want to give the skeptic the best chance.

Taking all that into account, I have chosen a power law function for the "skeptic's distribution". This should not be a surprise - indeed anyone familiar with the statistics of human behavior might have guessed it from just the histogram we're trying to fit:

What makes a power law particularly appropriate? Well, for one, power laws are the quintessential long-tailed distribution. They have one of the longest possible tails, and are fully "capable of  black swan behavior", according to Wikipedia. They can easily have tails so long that the overall distribution has an undefined (that is, infinite) mean. In fact, power laws, as mathematical functions, can decay so slowly that it's not allowed to be a probability density function, because the area under their curve can diverge. One can hardly ask for a more slowly decaying function than that. So this gives the skeptic the best chance at naturally generating a Jesus-level of resurrection evidence.

There exists distributions that decay even more slowly than a power law, but they're rare, obscure, and have no relation to what we're doing. By contrast, power law distributions are ubiquitous in human behavior. They form the basis for the well-known Pareto principle, and they capture the "dependency of evidence" factor we're currently trying to model.

For example, the distribution of income among people follows a power law. A few people, out at the long tail, have a great deal of wealth, because rich get richer - that is, because how rich you get depends on how rich you already are.

The size of cities also follows a power law. There are a few very large cities out at the long tail, because your chances of moving to a city depends on the number of people who already live there.

The number of links to a website follows a power law. There are a few, very popular websites out at the long tail, which have a lot of links to them. This is because a site's chances of getting a link depends on its popularity - that is, on the number of links it already has.

Don't let the specificity of these examples fool you. There are many, many more. Power law distributions are, as I said, ubiquitous in human behavior. They will frequently come up when one human behavior depends on the same kind of behavior, either by others or by the same person.

So it is entirely appropriate that we use a power law to  model the level of evidence for a resurrection report. There will be relatively few reports out at the long tail, like the "resurrection" of Apollonius or Krishna. In the context of things like conspiracy theories, this is because the chances of generating an additional piece of evidence depends on how much evidence it already has.

So there are excellent external reasons and examples to expect that the "skeptic's distribution" will follow a power law. Furthermore, power laws give one of the best possible chances for the skeptic's case, having a very "long tail" and allowing for a "black swan" event like the level of evidence in Jesus's resurrection event.

We will proceed to nail down the specifics of this power law distribution in the next post.

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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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