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

Questions
1

Hi, I’m familiar with Bayesian updates using discrete data - but I’m confused on how to do the same thing for continuous data, and someone recommended PyMC3. Here’s my example:

My somewhat informative prior distribution of outcomes is this:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm

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])
prior.sort()
plt.plot(prior, norm.pdf(prior, prior.mean(), prior.std()))

figure

And my observed evidence is:

evidence = np.array([27, 20.75, 24.5]

How would I update this very specific prior distribution, given 3 samples of evidence, using PyMC3? I would expect it to lower the density of outcomes below 20 and above 30 and increase the concentration between 20 and 30.

Thanks so much!

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()
1 Like
3

Another option is to use pm.Interpolated, there is a bit more info in Updating priors — PyMC3 3.11.4 documentation

1 Like