We have established that the resurrection has, at a minimum, even odds of having taken place. Let us retrace our steps and demonstrate that this is, in fact, the minimum.

Looking back, we see that our first decision was to choose a power law distribution as the "skeptic's distribution". As we mentioned when we made the choice, this is the most pro-skeptical choice we can make that fits the facts. Power law distributions have one of the longest possible tails, which can decay very slowly. They're fully capable of a "black swan" event. Furthermore, they're ubiquitous in human behavior, in that they're naturally generated when an increase in a value depends on the value itself. For this reason, the distributions of personal wealth, city sizes, and website popularity all follow a power law distribution. It's therefore appropriate to use it to model the buildup of evidence through possibilities like conspiracy theories or religious mass delusions.

However, there are excellent reasons to believe that the true "skeptic's distribution" will die off more quickly than a power law distribution, especially when we extend it to 24 times the maximum observed value. You see, few power law distributions can actually extend off to infinity - some external factor will intervene to cut off the distribution at very large values.

Consider city sizes, which we just mentioned. The population of cities follows a power law, and this holds up pretty well as long as we consider populations up to tens of millions of people. However, if we try to extend this out to infinity, the distribution no longer holds. We run into external factors which limit city sizes, such as the total population of humanity or the logistics of city growth in a given geography. For example, the largest city in South Korea is Seoul, with about 10 million people. A city 24 times larger than that would have over 200 million people - much larger than the total population of South Korea, which is only 50 million. Such a South Korean city cannot exist - not because its probability would be too small according to the power law distribution, but because it runs into external factors, like the fact that a city cannot be larger than the country to which it belongs. That is to say, the power law distribution for city sizes is limited, or cut off, at the long tail.

You can imagine similar arguments for personal wealth and website popularity. An individual cannot actually have "all the money in the world", and a website cannot be linked from more websites than the number that actually exist. Likewise, naturalistically generated evidence for resurrection stories cannot follow a power law distribution out to infinity. Other, external factors will cut off or strongly attenuate the probability as such resurrection stories gains more momentum.

For this reason, the true "skeptic's distribution" is almost certainly something that looks like a power law over the actually existing samples, but decays more quickly thereafter. A number of distributions - like a log-normal distribution or a power law with an exponential cutoff - follow this behavior. In each case, these other distributions with their "shorter" tails would help the resurrection case. So adopting a generalized Pareto distribution, which is a genuine power law all the way out to infinity, was therefore the most pro-skeptical choice we could have made.

Next, we considered evenly-spaced shape parameters, in intervals of 0.02, for our distribution. That is to say, we chose a uniform distribution over the shape parameter as our prior. Again, this almost certainly unduly favors skepticism. Consider what such a prior distribution means: the true shape parameter would be 1000 times more likely to be between 1000 and 2000 than between 0 and 1. It would be infinitely more likely to be greater than 1 than to be less than 1. Remember that a larger shape parameter favors the skeptic's case, and we have chosen a prior that favors these larger values. It is only through the weight of the evidence that this prior distribution gets reigned in, but choosing such a biased prior still biases the end results.

The more common and reasonable choice of prior in such circumstances is to consider shape parameters which increase linearly in their logarithms. For example, we may consider shape parameters like 0.01, 0.1, 1, 10, 100, and so on. The idea is that we don't know what the order of magnitude of the shape parameter would be, and therefore consider each order of magnitude equally. Of course, such a prior favors the smaller shape parameters compared to the uniform distribution that we actually used, meaning that it helps the case for the resurrection. So once again, our choice of evenly-spaced shape parameters was the most pro-skeptical choice we could have made.

Next, we considered the maximum value of 1e9 samples drawn from our "skeptic's distribution". That value of 1e9 was chosen as the number of "reportable deaths" in world history. That is, this is the number of deaths that had a chance to be witnessed, documented, or told about in a story. It excludes those deaths where nobody could have made a statement about that death, even if a genuine resurrection took place.

But a moment's reflection shows that this number is too small. Only 1e9 - one billion - "reportable deaths" in world history? More people than that have died just in the last century, and virtually all of these deaths have been "reportable" according to the definition above. Surely a more realistic figure would easily be above 1e10.

This is important, because this sets an upper bound on the probability of generating a Jesus-level resurrection report. A report with the most evidence out of 1e9 samples has a probability of about 1e-9 of being generated. The most evidence out of 1e10 samples would correspondingly have a probability around 1e-10. We then calculate the chances of generating a report with 24 times more evidence.

It's clear that the larger the number of samples, the smaller the probability of generating a report with a level of evidence comparable to the maximal sample. The probability of beating that by a factor of 24 is smaller still. So, the more samples we use, the smaller the probability for the "skeptic's distribution" generating a Jesus-level resurrection report. In other words, using 1e9 as the number of "reportable deaths" was a pro-skeptical choice. The true value is definitely much larger - easily above 1e10. And using this true value would only strengthen the case for the resurrection.

In the next post, we will continue going over more reasons why our previous calculation gives the minimum possible probability value for the resurrection.

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Christians, read your Bibles

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