Posterior sampling using a PythonDistribution

In this example we are going to show how to do Bayesian inference using the RandomWalkMetropolisHastings algorithm in a statistical model defined through a PythonDistribution.

This method is illustrated on a simple lifetime study test-case, which involves censored data, as described hereafter.

In the following, we assume that the lifetime of an industrial component follows the Weibull distribution , with CDF given by .

Our goal is to estimate the model parameters based on a dataset of recorded failures some of which correspond to actual failures, and the remaining are right-censored. Let represent the nature of each datum, if corresponds to an actual failure, if it is right-censored.

Note that the likelihood of each recorded failure is given by the Weibull density:

On the other hand, the likelihood of each right-censored observation is given by:

Furthermore, assume that the prior information available on is represented by independent prior laws, whose respective densities are denoted by and

The posterior distribution of represents the update of the prior information on given the dataset. Its PDF is known up to a multiplicative constant:

The RandomWalkMetropolisHastings class can be used to sample from the posterior distribution. It relies on the following objects:

• The prior probability density reflects beliefs about the possible values of before the experimental data are considered.

• Initial values of the parameters.

• An proposal distribution used to update parameters.

Additionnaly we want to define the likelihood term defined by these objects:

• The conditional density will be defined as a PythonDistribution.

• The sample of observations acting as the parameters of the conditional density

Set up the PythonDistribution

The censured Weibuill likelihood is outside the usual catalog of probability distributions in OpenTURNS, hence we need to define it using the PythonDistribution class.

import numpy as np
import openturns as ot
from openturns.viewer import View
ot.Log.Show(ot.Log.NONE)
ot.RandomGenerator.SetSeed(123)

The following methods must be defined:

• getRange: required for conversion to the Distribution format

• computeLogPDF: used by RandomWalkMetropolisHastings to evaluate the posterior density

• setParameter used by RandomWalkMetropolisHastings to test new parameter values

Note

We formally define a bivariate distribution on the couple, even though has no distribution (it is simply a covariate). This is not an issue, since the sole purpose of this PythonDistribution object is to pass the likelihood calculation over to RandomWalkMetropolisHastings.

class CensoredWeibull(ot.PythonDistribution):
    """
    Right-censored Weibull log-PDF calculation
    Each data point x is assumed 2D, with:
        x[0]: observed functioning time
        x[1]: nature of x[0]:
            if x[1]=0: x[0] is a censoring time
            if x[1]=1: x[0] is a time-to failure
    """

    def __init__(self, beta=5000.0, alpha=2.0):
        super(CensoredWeibull, self).__init__(2)
        self.beta = beta
        self.alpha = alpha

    def getRange(self):
        return ot.Interval([0, 0], [1, 1], [True]*2, [False, True])

    def computeLogPDF(self, x):
        if not (self.alpha > 0.0 and self.beta > 0.0):
            return -np.inf
        log_pdf = -(x[0] / self.beta)**self.alpha
        log_pdf += (self.alpha - 1) * np.log(x[0] / self.beta) * x[1]
        log_pdf += np.log(self.alpha / self.beta) * x[1]
        return log_pdf

    def setParameter(self, parameter):
        self.beta = parameter[0]
        self.alpha = parameter[1]

    def getParameter(self):
        return [self.beta, self.alpha]

Convert to Distribution

conditional = ot.Distribution(CensoredWeibull())

Observations, prior, initial point and proposal distributions

Define the observations

Tobs = np.array([4380, 1791, 1611, 1291, 6132, 5694, 5296, 4818, 4818, 4380])
fail = np.array([True]*4+[False]*6)
x = ot.Sample(np.vstack((Tobs, fail)).T)

Define a uniform prior distribution for and a Gamma prior distribution for .

alpha_min, alpha_max = 0.5, 3.8
a_beta, b_beta = 2, 2e-4

priorCopula = ot.IndependentCopula(2)  # prior independence
priorMarginals = []  # prior marginals
priorMarginals.append(ot.Gamma(a_beta, b_beta))  # Gamma prior for beta
priorMarginals.append(ot.Uniform(alpha_min, alpha_max)
                      )  # uniform prior for alpha
prior = ot.ComposedDistribution(priorMarginals, priorCopula)
prior.setDescription(['beta', 'alpha'])

We select prior means as the initial point of the Metropolis-Hastings algorithm.

initialState = [a_beta / b_beta, 0.5*(alpha_max - alpha_min)]

For our random walk proposal distributions, we choose normal steps, with standard deviation equal to roughly of the prior range (for the uniform prior) or standard deviation (for the normal prior).

proposal = []
proposal.append(ot.Normal(0., 0.1 * np.sqrt(a_beta / b_beta**2)))
proposal.append(ot.Normal(0., 0.1 * (alpha_max - alpha_min)))
proposal = ot.ComposedDistribution(proposal)

Sample from the posterior distribution

sampler = ot.RandomWalkMetropolisHastings(prior, initialState, proposal)
sampler.setLikelihood(conditional, x)
sampleSize = 10000
sample = sampler.getSample(sampleSize)
# compute acceptance rate
print("Acceptance rate: %s" % (sampler.getAcceptanceRate()))

Out:

Acceptance rate: 0.7194

Plot prior to posterior marginal plots.

kernel = ot.KernelSmoothing()
posterior = kernel.build(sample)
grid = ot.GridLayout(1, 2)
grid.setTitle('Bayesian inference')
for parameter_index in range(2):
    graph = posterior.getMarginal(parameter_index).drawPDF()
    priorGraph = prior.getMarginal(parameter_index).drawPDF()
    graph.add(priorGraph)
    graph.setColors(ot.Drawable.BuildDefaultPalette(2))
    graph.setLegends(['Posterior', 'Prior'])
    grid.setGraph(0, parameter_index, graph)
_ = View(grid)

Define an improper prior

Now, define an improper prior:

logpdf = ot.SymbolicFunction(['beta', 'alpha'], ['-log(beta)'])
support = ot.Interval([0] * 2, [1] * 2)
support.setFiniteUpperBound([False] * 2)

Sample from the posterior distribution

sampler2 = ot.RandomWalkMetropolisHastings(
    logpdf, support, initialState, proposal)
sampler2.setLikelihood(conditional, x)
sample2 = sampler2.getSample(1000)
print("Acceptance rate: %s" % (sampler2.getAcceptanceRate()))

Out:

Acceptance rate: 0.729

Plot posterior marginal plots only as prior cannot be drawn meaningfully.

kernel = ot.KernelSmoothing()
posterior = kernel.build(sample)
grid = ot.GridLayout(1, 2)
grid.setTitle('Bayesian inference (with log-pdf)')
for parameter_index in range(2):
    graph = posterior.getMarginal(parameter_index).drawPDF()
    graph.setColors(ot.Drawable.BuildDefaultPalette(2))
    graph.setLegends(['Posterior'])
    grid.setGraph(0, parameter_index, graph)
_ = View(grid)

View.ShowAll()