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Gibbs sampling of the posterior distributionΒΆ

We sample the from the posterior distribution of the parameters of a mixture model.

X \sim 0.7 \mathcal{N}(\mu_0, 1) + 0.3 \mathcal{N}(\mu_1, 1),

where \mu_0 and \mu_1 are unknown parameters. They are a priori i.i.d. with prior distribution \mathcal{N}(0, \sqrt{10}). This example is drawn from Example 9.2 from Monte-Carlo Statistical methods by Robert and Casella (2004).

import openturns as ot
from openturns.viewer import View
import numpy as np

ot.RandomGenerator.SetSeed(100)

Sample data with \mu_0 = 0 and \mu_1 = 2.7.

N = 500
p = 0.3

mu0 = 0.0
mu1 = 2.7
nor0 = ot.Normal(mu0, 1.0)
nor1 = ot.Normal(mu1, 1.0)
true_distribution = ot.Mixture([nor0, nor1], [1 - p, p])
observations = np.array(true_distribution.getSample(500))

Plot the true distribution.

graph = true_distribution.drawPDF()
graph.setTitle("True distribution")
graph.setXTitle("")
graph.setLegends([""])
_ = View(graph)
True distribution

A natural step at this point is to introduce an auxiliary (unobserved) random variable Z telling from which distribution X was sampled.

For any nonnegative integer i, Z_i follows the Bernoulli distribution with p=0.3, and X_i | Z_i \sim \mathcal{N}(\mu_{Z_i}, 1.0).

Let n_0 (resp. n_1) denote the number of indices i such that Z_i=0 (resp. Z_i=1).

Conditionally to all X_i and all Z_i, \mu_0 and \mu_1 are independent: \mu_0 follows \mathcal{N} \left(\sum_{Z_i=0} \frac{X_i}{0.1 + n_0}, \frac{1}{0.1 + n_0} \right) and \mu_1 follows \mathcal{N} \left(\sum_{Z_i=1} \frac{X_i}{0.1 + n_1}, \frac{1}{0.1 + n_1} \right).

For any i, conditionally to X_i, \mu_0 and \mu_1, Z_i is independent from all Z_j (j \neq i) and follows the Bernoulli distribution with parameter

p = \frac{p \exp \left( -(X_i - \mu_1)^2 / 2 \right) }{p \exp \left( -(X_i - \mu_1)^2 / 2 \right) + (1-p) \exp \left( -(X_i - \mu_0)^2 / 2 \right) }

We now sample from the joint distribution of (\mu_0, \mu1, Z_0,...,Z_{N-1}) conditionally to the X_i using the Gibbs algorithm. We define functions that will translate a given state of the Gibbs algorithm into the correct parameters for the distributions of \mu_0, \mu_1, and the Z_i.

def nor0post(pt):
    z = np.array(pt)[2:]
    x0 = observations[z == 0]
    mu0 = x0.sum() / (0.1 + len(x0))
    sigma0 = 1.0 / (0.1 + len(x0))
    return [mu0, sigma0]


def nor1post(pt):
    z = np.array(pt)[2:]
    x1 = observations[z == 1]
    mu1 = x1.sum() / (0.1 + len(x1))
    sigma1 = 1.0 / (0.1 + len(x1))
    return [mu1, sigma1]


def zpost(pt):
    mu0 = pt[0]
    mu1 = pt[1]
    term1 = p * np.exp(-((observations - mu1) ** 2) / 2)
    term0 = (1.0 - p) * np.exp(-((observations - mu0) ** 2) / 2)
    res = term1 / (term1 + term0)
    # output must be a 1d list or array in order to create a PythonFunction
    return res.reshape(-1)


nor0posterior = ot.PythonFunction(2 + N, 2, nor0post)
nor1posterior = ot.PythonFunction(2 + N, 2, nor1post)
zposterior = ot.PythonFunction(2 + N, N, zpost)

We can now construct the Gibbs algorithm

initialState = [0.0] * (N + 2)

sampler0 = ot.RandomVectorMetropolisHastings(
    ot.RandomVector(ot.Normal()), initialState, [0], nor0posterior
)
sampler1 = ot.RandomVectorMetropolisHastings(
    ot.RandomVector(ot.Normal()), initialState, [1], nor1posterior
)

big_bernoulli = ot.ComposedDistribution([ot.Bernoulli()] * N)

sampler2 = ot.RandomVectorMetropolisHastings(
    ot.RandomVector(big_bernoulli), initialState, range(2, N + 2), zposterior
)

gibbs = ot.Gibbs([sampler0, sampler1, sampler2])

Run the Gibbs algorithm

s = gibbs.getSample(10000)

Extract the relevant marginals: the first (mu_0) and the second (\mu_1).

posterior_sample = s[:, 0:2]

Let us plot the posterior density.

ks = ot.KernelSmoothing().build(posterior_sample)
graph = ks.drawPDF()
graph.setTitle("Posterior density")
graph.setLegendPosition("bottomright")
graph.setXTitle(r"$\mu_0$")
graph.setYTitle(r"$\mu_1$")
_ = View(graph)

View.ShowAll()
Posterior density

Total running time of the script: ( 0 minutes 12.724 seconds)