First, we import some modules:

In [1]:

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

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 min(
dist.isf(top_percentiles.min()),
np.finfo(float).max / 100) # prevent inf
```

We then write a function to calculate the Bayes factor. The methodology is the same as before: we compare the skeptic's and Christian's likelihood of generating a Jesus-level resurrection report. The various parameters fed into the function determines the specific form of the "skeptic's distribution", whereas the "Christian's distribution" remains fixed with a pdf of 0.5 / 12 for values up to x = 12.

In [3]:

```
def norm_redraw(mu, sigma):
draw = int(round(norm(mu, scale=sigma).rvs(1)[0]))
if draw <= -1:
return norm_redraw(mu, sigma)
else:
return draw
def calculate_bf(
dist_type, #genpareto or lognorm
shape_params_dist, #np.geomspace or np.linspace
sample_size, #1e9 to 1e10+
greater_by, #6 to 10+
outlier_interval, # (0.5, 1) to (0.5, 0.9)
n_historical_outliers, #60 to 300+
n_max_draws=10000,
):
if dist_type == genpareto:
shape_limits = [0.01, 2.0]
elif dist_type == lognorm:
shape_limits = [0.1, 10.0]
shape_params_list = shape_params_dist(
shape_limits[0], shape_limits[1], 200)
shape_params = []
pdf_greater_by = []
n_outliers = []
for shape_param in shape_params_list:
dist = dist_type(shape_param, scale=1, loc=0)
for i in range(n_max_draws):
max_val = max_out_of_n_from_dist(dist, sample_size)
scaled_dist = dist_type(
shape_param, scale=1 / max_val, loc=0)
# approximate binom with norm
p_outlier = (
scaled_dist.sf(outlier_interval[0])
- scaled_dist.sf(outlier_interval[1])
) / scaled_dist.cdf(1)
mu = (sample_size - 1) * p_outlier
sigma = (
(sample_size - 1)
* p_outlier * (1 - p_outlier)) ** 0.5
n_outlier = norm_redraw(mu, sigma) + 1
if n_outlier == n_historical_outliers:
n_outliers.append(n_outlier)
shape_params.append(shape_param)
pdf_greater_by.append(scaled_dist.pdf(greater_by))
result_df = pd.DataFrame({
"shape_params":shape_params,
"pdf_greater_by":pdf_greater_by,
"n_outliers":n_outliers,
})
if result_df.shape[0] < 50:
print("warning: result_df.shape = ", result_df.shape)
if result_df["shape_params"].max() == shape_params_list.max():
print("warning: maxed out shape_param")
bf = (0.5 / 12) / result_df["pdf_greater_by"].mean()
print("Bayes factor: {:.2e}".format(bf))
```

Now, let us explore some of the different possible forms of the "skeptic's distribution", and calculate the Bayes factor for these possibilities.

Here's one we looked at before. It uses the most pro-skeptical assumptions possible to generate the smallest Bayes factor.

In [4]:

```
calculate_bf(
dist_type=genpareto,
shape_params_dist=np.linspace,
sample_size=int(1e9),
greater_by=6,
outlier_interval=(0.5, 1),
n_historical_outliers=60,
)
```

Bayes factor: 1.06e+11

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

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

The multiplicative factor between the level of evidence in 1 Corinthians 15 and the "once in non-Christian world history" level: from 6 to 7

The number of "n_historical_outliers" (That is, the number of reports of a "resurrection", with at least a "some people say..." level of evidence): from 60 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 the Bayes factor.

In [5]:

```
calculate_bf(
dist_type=genpareto,
shape_params_dist=np.linspace,
sample_size=1e10,
greater_by=7,
outlier_interval=(0.5, 1),
n_historical_outliers=200,
)
```

Bayes factor: 1.12e+14

Here we've changed a few more parameters. The distribution type has been changed to lognormal, the shape parameters are now uniform in logs, we've increased the factor by which the Jesus-level of evidence exceeds the maximum, and the number of "outliers" has been increased. The Bayes factor calculated here may perhaps be called "somewhat conservative".

In [6]:

```
calculate_bf(
dist_type=lognorm,
shape_params_dist=np.geomspace,
sample_size=int(1e10),
greater_by=8,
outlier_interval=(0.5, 1),
n_historical_outliers=300,
)
```

Bayes factor: 8.41e+15

Here are few more combinations of parameters which range from "very conservative" to "somewhat conservative".

In [7]:

```
calculate_bf(
dist_type=lognorm,
shape_params_dist=np.linspace,
sample_size=int(1e10),
greater_by=8,
outlier_interval=(0.5, 0.9),
n_historical_outliers=200,
)
```

Bayes factor: 3.80e+14

In [8]:

```
calculate_bf(
dist_type=genpareto,
shape_params_dist=np.linspace,
sample_size=int(2e10),
greater_by=10,
outlier_interval=(0.5, 0.95),
n_historical_outliers=250,
)
```

Bayes factor: 6.12e+15

In [9]:

```
calculate_bf(
dist_type=lognorm,
shape_params_dist=np.geomspace,
sample_size=int(2e10),
greater_by=10,
outlier_interval=(0.5, 1),
n_historical_outliers=200,
)
```

Bayes factor: 1.14e+16