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#! /usr/bin/env python
# We sample the from the posterior distribution of the parameters of a mixture model.
# This example is drawn from Example 9.2 from *Monte-Carlo Statistical methods* by Robert and Casella (2004).
import openturns as ot
import openturns.testing as ott
import numpy as np
ot.TESTPREAMBLE()
ot.RandomGenerator.SetSeed(100)
# Sample data with :math:`\mu_0 = 0` and :math:`\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))
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.JointDistribution([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(1000)
# Extract the relevant marginals: the first (:math:`mu_0`) and the second (:math:`\mu_1`).
posterior_sample = s[:, 0:2]
mean = posterior_sample.computeMean()
stddev = posterior_sample.computeStandardDeviation()
print(mean, stddev)
ott.assert_almost_equal(mean, [-0.078428, 2.80587])
ott.assert_almost_equal(stddev, [0.0463082, 0.108863])
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