pymc3: hierarchical model with multiple obsesrved variables

Question

I have a simple hierarchical model with lots of individuals for which I have small samples from a normal distribution. The means of these distributions also follow a normal distribution.

import numpy as np

n_individuals = 200
points_per_individual = 10
means = np.random.normal(30, 12, n_individuals)
y = np.random.normal(means, 1, (points_per_individual, n_individuals))

I want to use PyMC3 to compute the model parameters from the sample.

import pymc3 as pm
import matplotlib.pyplot as plt

model = pm.Model()
with model:
    model_means = pm.Normal('model_means', mu=35, sd=15)

    y_obs = pm.Normal('y_obs', mu=model_means, sd=1, shape=n_individuals, observed=y)

    trace = pm.sample(1000)

pm.traceplot(trace[100:], vars=['model_means'])
plt.show()

I was expecting the posterior of model_means to look like my original distribution of means. But it seems to converge to 30 the mean of the means. How do I recover the original standard deviation of the means (12 in my example) from the pymc3 model?

Answer 1

This question was me struggling with the concepts of PyMC3.

I need n_individuals observed random variables to model the y and n_individual stochastic random variables to model the means. These also need priors hyper_mean and hyper_sigma for their parameters. sigmas is the prior for the standard deviation of y.

import matplotlib.pyplot as plt

model = pm.Model()
with model:
    hyper_mean = pm.Normal('hyper_mean', mu=0, sd=100)
    hyper_sigma = pm.HalfNormal('hyper_sigma', sd=3)

    means = pm.Normal('means', mu=hyper_mean, sd=hyper_sigma, shape=n_individuals)
    sigmas = pm.HalfNormal('sigmas', sd=100)

    y = pm.Normal('y', mu=means, sd=sigmas, observed=y)

    trace = pm.sample(10000)

pm.traceplot(trace[100:], vars=['hyper_mean', 'hyper_sigma', 'means', 'sigmas'])
plt.show()