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Bayesian evaluation for the likelihood of Christ's resurrection (Part 38)

So then, here is the summary of the basic idea:

We assume that the "skeptic's distribution" will take the form of a generalized Pareto distribution.

We will determine the shape parameter of the distribution by looking at how many "outliers" it has.

A person's resurrection report is considered an "outlier" if it has at least 20% of the evidence compared to the non-Christian resurrection report with the maximum evidence.

The "non-Christian resurrection report with the maximum evidence" is taken to be Krishna or Aristeas, with  Apollonius not too far behind. These are taken as having 1/24 of the evidence for the resurrection of Jesus.

Recall that the "some people say... " level of evidence - as per PuhuaOsirisZalmoxis, etc. - corresponded to 1/60th of the evidence for Christ's resurrection. This corresponds to 40% of 1/24, meaning that anyone else with the "some people say..." level of evidence would pass the 20% threshold for an "outlier".

We can therefore calculate the relationship between the number of outliers and the likely shape parameters, and thereby calculate the probability for the "skeptic's distribution" naturally generating a Jesus-level of evidence for a resurrection.

Here is the plan for the program:

We will consider shape parameters from 0.02 to 2.1, in increasing intervals of 0.02. That is to say, we will consider shape parameters 0.02, 0.04, 0.06, etc. all the way up to 2.1. Our region of interest will lie in this range.

We will create a generalized Pareto distribution with that shape parameter, then simulate drawing the maximum value of 1e9 samples from that distribution. We will then estimate the number of outliers in that distribution, and the probability of that distribution generating a sample more than 24 times larger than the maximum.

We will do this 10000 times for each value of the shape parameter. This gives us a table of more than a million rows, with each row containing the shape parameter, the number of outliers, and the probability of generating a Jesus-level of evidence.

If we assume equal prior weights for each of the shape parameters, we can consider the final distribution of the shape parameters to be just its distribution from the subset of the table where the number of outliers is equal to the actual, historical value. That is to say, we just have to look at where the theory fits the data, and consider only those theories. This satisfies Bayes' theorem.

Likewise, the probability of the "skeptic's distribution" achieving a Jesus-level of evidence for a resurrection will just be the mean value of that probability from the same subset.

Every assumption and choice made above favors the skeptic's case. Therefore, the probability obtained at the end will be a maximum probability; the skeptic cannot hope for more than that in explaining Jesus's resurrection "naturally".

We will go over the actual computer program in the next post.

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How to think about the future (Part 2)