How would I take this prior distribution and evidence and plug it into pymc3?

Questions
2

There are many choices one could make in modeling this data with this prior. But here’s a pretty straightforward implementation of what you have described:

import pymc3 as pm
import numpy as np
import arviz as az
import matplotlib.pyplot as plt

prior = np.array([20.5,15.25,5.0,29.0,11.75,8.5,8.25,14.5,14.25,
 23.25,31.75,44.5,9.75,2.75, 14.25, 7.0])

# we'll throw away this data and instead use a normal 
# prior with mean of np.mean(prior)=16.265625
#  and a SD of np.std(prior)=10.784634885306735

data = np.array([27, 20.75, 24.5])

with pm.Model() as model:
    mean = pm.Normal('mean',
                     mu=prior.mean(),
                     sigma=.5*prior.std())
    sigma = pm.Gamma('sigma',
                     alpha=prior.std(),
                     beta=1)
    likelihood = pm.Normal('likelihood',
                           mu=sigma,
                           sigma=sigma,
                           observed=data)

    trace = pm.sample(return_inferencedata=True)

az.plot_trace(trace)
plt.show()
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