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Bayesian calibration of a computer code¶

In this example we are going to compute the parameters of a computer model thanks to Bayesian estimation.

Let us denote $(y_1, \dots, y_n)$ the observation sample, $(\vect z_1, \ldots, \vect z_n) = (f(x_1|\vect\theta), \ldots, f(x_n|\vect\theta))$ the model prediction, $p(y |\vect z)$ the density function of observation $y$ conditional on model prediction $\vect z$ , and $\vect\theta \in \mathbb{R}^p$ the calibration parameters we wish to estimate.

The posterior distribution is given by Bayes theorem:

where $\propto$ means “proportional to”, regarded as a function of $\vect\theta$ .

The posterior distribution is approximated here by the empirical distribution of the sample $\vect\theta^1, \ldots, \vect\theta^N$ generated by the Metropolis-Hastings algorithm. This means that any quantity characteristic of the posterior distribution (mean, variance, quantile, …) is approximated by its empirical counterpart.

Our model (i.e. the compute code to calibrate) is a standard normal linear regression, where

$y_i = \theta_1 + x_i \theta_2 + x_i^2 \theta_3 + \varepsilon_i$

where $\varepsilon_i \stackrel{i.i.d.}{\sim} \mathcal N(0, 1)$ .

The “true” value of $\theta$ is:

$\vect \theta_{true} = (-4.5,4.8,2.2)^T.$

We use a normal prior on $\vect\theta$ :

$\pi(\vect\theta) = \mathcal N(\vect{\mu}_\vect{\theta}, \mat{\Sigma}_\vect{\theta})$

where

$\vect{\mu}_\vect{\theta} = \begin{pmatrix} -3 \\ 4 \\ 1 \end{pmatrix}$

is the mean of the prior and

$\mat{\Sigma}_\vect{\theta} = \begin{pmatrix} \sigma_{\theta_1}^2 & 0 & 0 \\ 0 & \sigma_{\theta_2}^2 & 0 \\ 0 & 0 & \sigma_{\theta_3}^2 \end{pmatrix}$

is the prior covariance matrix with

$\sigma_{\theta_1} = 2, \qquad \sigma_{\theta_2} = 1, \qquad \sigma_{\theta_3} = 1.5.$

The following objects need to be defined in order to perform Bayesian calibration:

The conditional density $p(y|\vect z)$ must be defined as a probability distribution.
The computer model must be implemented thanks to the ParametricFunction class. This takes a value of $\vect\theta$ as input, and outputs the vector of model predictions $\vect z$ , as defined above (the vector of covariates $\vect x = (x_1, \ldots, x_n)$ is treated as a known constant). When doing that, we have to keep in mind that $\vect z$ will be used as the vector of parameters corresponding to the distribution specified for $p(y |\vect z)$ . For instance, if $p(y|\vect z)$ is normal, this means that $\vect z$ must be a vector containing the mean and standard deviation of $y$ .
The prior density $\pi(\vect\theta)$ encoding the set of possible values for the calibration parameters, each value being weighted by its a priori probability, reflecting the beliefs about the possible values of $\vect\theta$ before consideration of the experimental data. Again, this is implemented as a probability distribution.
Metropolis-Hastings algorithm(s), possibly used in tandem with a Gibbs algorithm in order to sample from the posterior distribution of the calibration parameters.

import pylab as pl
import openturns as ot
import openturns.viewer as viewer
from matplotlib import pylab as plt

ot.Log.Show(ot.Log.NONE)

Dimension of the vector of parameters to calibrate

paramDim = 3
# The number of obesrvations
obsSize = 10

Define the observed inputs $x_i$ .

xmin = -2.0
xmax = 3.0
step = (xmax - xmin) / (obsSize - 1)
rg = ot.RegularGrid(xmin, step, obsSize)
x_obs = rg.getVertices()

Define the parametric model $\vect z = f(x,\vect\theta)$ that associates each observation $x$ and value of $\vect \theta$ to the parameters of the distribution of the corresponding observation $y$ : here $\vect z=(\mu, \sigma)$ where $\mu$ , the first output of the model, is the mean and $\sigma$ , the second output of the model, is the standard deviation.

fullModel = ot.SymbolicFunction(
    ["x", "theta1", "theta2", "theta3"], ["theta1+theta2*x+theta3*x^2", "1.0"]
)

To differentiate between the two classes of inputs ( $x$ and $\vect\theta$ ), we define a ParametricFunction from fullModel and make the first input (the observations $x$ ) its parameter: $f_x(\vect \theta) := f(x, \vect \theta)$ . We set $x = 1$ as a placeholder, but $x$ will actually take the values $x_i$ of the observations when we sample $\vect\theta$ .

linkFunction = ot.ParametricFunction(fullModel, [0], [1.0])
print(linkFunction)

ParametricEvaluation([x,theta1,theta2,theta3]->[theta1+theta2*x+theta3*x^2,1.0], parameters positions=[0], parameters=[x : 1], input positions=[1,2,3])

Define the observation noise $\varepsilon {\sim} \mathcal N(0, 1)$ and create a sample from it.

ot.RandomGenerator.SetSeed(0)
noiseStandardDeviation = 1.0
noise = ot.Normal(0, noiseStandardDeviation)
noiseSample = noise.getSample(obsSize)

Define the vector of observations $y_i$ , here sampled using the “true” value of $\vect \theta$ : $\vect \theta_{true}$ .

thetaTrue = [-4.5, 4.8, 2.2]

y_obs = ot.Sample(obsSize, 1)
for i in range(obsSize):
    linkFunction.setParameter(x_obs[i])
    y_obs[i, 0] = linkFunction(thetaTrue)[0] + noiseSample[i, 0]

Draw the model predictions vs the observations.

functionnalModel = ot.ParametricFunction(fullModel, [1, 2, 3], thetaTrue)
graphModel = functionnalModel.getMarginal(0).draw(xmin, xmax)
observations = ot.Cloud(x_obs, y_obs)
observations = ot.Cloud(x_obs, y_obs)
observations.setColor("red")
graphModel.add(observations)
graphModel.setLegends(["Model", "Observations"])
graphModel.setLegendPosition("topleft")
view = viewer.View(graphModel)

Define the distribution of observations $y | \vect{z}$ conditional on model predictions.

Note that its parameter dimension is the one of $\vect{z}$ , so the model must be adjusted accordingly.

conditional = ot.Normal()

Define the mean $\mu_\theta$ , the covariance matrix $\Sigma_\theta$ , then the prior distribution $\pi(\vect\theta)$ of the parameter $\vect\theta$ .

thetaPriorMean = [-3.0, 4.0, 1.0]

sigma0 = [2.0, 1.0, 1.5]  # standard deviations
thetaPriorCovarianceMatrix = ot.CovarianceMatrix(paramDim)
for i in range(paramDim):
    thetaPriorCovarianceMatrix[i, i] = sigma0[i] ** 2

prior = ot.Normal(thetaPriorMean, thetaPriorCovarianceMatrix)
prior.setDescription(["theta1", "theta2", "theta3"])

The proposed steps for $\theta_1$ , $\theta_2$ and $\theta_3$ will all follow a uniform distribution.

proposal = ot.Uniform(-1.0, 1.0)

Test the Metropolis-Hastings sampler¶

Creation of a single component random walk Metropolis-Hastings (RWMH) sampler. This involves a combination of the RWMH and the Gibbs algorithms.

initialState = thetaPriorMean

We create a RandomWalkMetropolisHastings sampler for each component. Each sampler must be aware of the joint prior distribution. We also use the same proposal distribution, but this is not mandatory.

mh_coll = [
    ot.RandomWalkMetropolisHastings(prior, initialState, proposal, [i])
    for i in range(paramDim)
]

Each sampler must be made aware of the likelihood. Otherwise we would sample from the prior!

for mh in mh_coll:
    mh.setLikelihood(conditional, y_obs, linkFunction, x_obs)

Finally, the Gibbs algorithm is constructed from all Metropolis-Hastings samplers.

sampler = ot.Gibbs(mh_coll)

Generate a sample from the posterior distribution of the parameters $\vect \theta$ .

sampleSize = 10000
sample = sampler.getSample(sampleSize)

Look at the acceptance rate (basic check of the sampling efficiency: values close to $0.2$ are usually recommended for Normal posterior distributions).

[mh.getAcceptanceRate() for mh in sampler.getMetropolisHastingsCollection()]

[0.282, 0.2944, 0.3035]

Build the distribution of the posterior by kernel smoothing.

kernel = ot.KernelSmoothing()
posterior = kernel.build(sample)

Display prior vs posterior for each parameter.

fig = pl.figure(figsize=(12, 4))

for parameter_index in range(paramDim):
    graph = posterior.getMarginal(parameter_index).drawPDF()
    priorGraph = prior.getMarginal(parameter_index).drawPDF()
    priorGraph.setColors(["blue"])
    graph.add(priorGraph)
    graph.setLegends(["Posterior", "Prior"])
    ax = fig.add_subplot(1, paramDim, parameter_index + 1)
    _ = ot.viewer.View(graph, figure=fig, axes=[ax])

_ = fig.suptitle("Bayesian calibration")

plt.show()

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Bayesian calibration of a computer code¶

Test the Metropolis-Hastings sampler¶