Bayesian parameter reconstruction using Bayesian optimization

Driver:

BayesianReconstruction

Download script:

bayesian_reconstruction.py

The target of the study is to showcase the solution of a Bayesian parameter reconstruction problembased on a set ot measurements \(\mathbf{t} \in \mathbb{R}^K\).

We assume that the measurement process can be accurately modeled by a function \(\mathbf{f}(\mathbf{b}) \in \mathbb{R}^K\). As an example we consider the MGH17 problem which consists of fitting a vectorial model \(\mathbf{f}(\mathbf{b}) \in \mathbb{R}^{33}\) with

\[f_i(\mathbf{b}) = b_1 + b_2 \exp(-i \cdot b_4) + b_3 \exp(-i \cdot b_5),\, i = 0, \dots, 32\]

to a measurement vector with \(K = 33\) entries.

Due to random measurement noise \(\mathbf{w} \sim \mathcal{N}(0, {\rm diag}{[\eta_1^2,...,\eta_K^2]})\), the vector of measurements \(\mathbf{t} = \mathbf{f}(\mathbf{b}) + \mathbf{w}\) is a random vector with probability density

\[P(\mathbf{t}) = \prod_{i=1}^K \frac{1}{\sqrt{2\pi}\eta_i}\exp\left[-\frac{1}{2}\left(\frac{t_i - f_i(\mathbf{b})}{\eta_i}\right)^2\right].\]

The noise variances \(\eta_i^2\) shall be unknown and estimated from the measurement data. To this end we assume that the variance is composed of a background term \(c_1^2\) and a noise contribution which scales linearly with \(f_i(\mathbf{b})\), i.e.

\[\eta_i^2(\mathbf{c}) = c_1^2 + \left[c_2 f_i(\mathbf{b})\right]^2.\]

Taking non-uniform prior distributions for the design parameter vector \(P_\text{prior}(\mathbf{b})\) and the error model parameters \(P_\text{prior}(\mathbf{c})\) into account the posterior distribution is then proportional to \(P(\mathbf{b}, \mathbf{c} | \mathbf{t}) \propto P(\mathbf{t} | \mathbf{b}, \mathbf{c}) P_\text{prior}(\mathbf{b}) P_\text{prior}(\mathbf{c})\).

Alltogether, the target of finding the parameters with maximum posterior probability density is equivalent of minimizing the value of the negative log-probability

\[\begin{split}\begin{split} -\log\left(P(\mathbf{b}, \mathbf{c}| \mathbf{t})\right) = & \frac{1}{2} K\log(2\pi) +\sum_{i=1}^K\log\left(\eta_i(\mathbf{c})\right) +\frac{1}{2}\sum_{i=1}^K \left( \frac{t_i - f_i(\mathbf{b})}{\eta_i(\mathbf{c})} \right)^2 \\ &-\log\left(P_\text{prior}(\mathbf{d})\right) -\log\left(P_\text{prior}(\mathbf{c})\right). \end{split}\end{split}\]

In the following, we show how this negative log-probability can be minimized using the BayesianReconstruction driver. We compare the results of the dirver’s MCMC sampling of the posterior probability to an MCMC sampling of the analytic value.

import sys,os
import numpy as np
import time
from scipy.stats import multivariate_normal, uniform
import pandas as pd
import torch
torch.set_default_dtype(torch.float64)
import matplotlib.pyplot as plt
import corner #run "pip install corner" if not installed
import emcee #run "pip install emcee" if not installed


from jcmoptimizer import Client, Study, Obseravtion
client = Client()


# Definition of the search domain
design_space = [
    {'name': 'b1', 'type': 'continuous', 'domain': (0,10)}, 
    {'name': 'b2', 'type': 'continuous', 'domain': (0.1,4)},
    {'name': 'b3', 'type': 'continuous', 'domain': (-4,-0.1)},
    {'name': 'b4', 'type': 'continuous', 'domain': (0.05,1)},
    {'name': 'b5', 'type': 'continuous', 'domain': (0.05,1)}
]

# Creation of the study object with study_id 'bayesian_reconstruction'
study = client.create_study(
    design_space=design_space,
    driver="BayesianReconstruction",
    study_name="Bayesian parameter reconstruction using Bayesian optimization",
    study_id="bayesian_reconstruction"
)
#The vectorial model function of the MGH17 problem
def model(x: torch.Tensor) -> torch.Tensor:
    s = torch.arange(33)
    return x[0] + x[1]*torch.exp(-s*x[3]) + x[2]*torch.exp(-s*x[4])

#The forward model parameters b to be reconstructed
b_true = torch.tensor([3.7541005211E-01, 1.9358469127E+00, -1.4646871366E+00,
              1.2867534640E-01,2.2122699662E-01])

#The error model parameters in log-space to be reconstructed
log_c1, log_c2 = np.log(0.005), np.log(0.01)

#The error model, i.e. the noise stddev depending on the model value y=f(b)
def error_model(log_c1: float, log_c2: float, y: torch.Tensor) -> torch.Tensor:
    return torch.sqrt( np.exp(log_c1)**2 + (np.exp(log_c2)*y)**2)

#Generate a rantom target vector of measurements
torch.manual_seed(0)
model_vector = model(b_true)
err = error_model(log_c1,log_c2,model_vector)
measurement_vector = model_vector + err*torch.randn(model_vector.shape)

study.configure(
    max_iter=80,
    target_vector=measurement_vector.tolist(),
    error_model=dict(
        #error model expression
        expression='sqrt(exp(log_c1)^2 + (exp(log_c2)*y_model)^2)',
        #prior distribution of error model parameters
        distributions=[
            {'type': 'normal', 'parameter': 'log_c1', 'mean': -5.0, 'stddev': 1},
            {'type': 'normal', 'parameter': 'log_c2', 'mean': -4.0, 'stddev': 1},
        ],
        #initial values and parameter bounds for fitting the error model parameters
        initial_parameters=[-5.0, -4.0],
        parameter_bounds=[(-7,-2), (-6.0,-1)]
    ),
    #Multivariate normal prior distribution of forward model parameters.
    #Unspecified parameters (b1,b4,b5) are uniformly distributed in the design space
    parameter_distribution=dict(
         distributions=[
             dict(type='mvn', parameters=['b2','b3'], mean=[2.25,-2.0], 
                  covariance=[[0.5,-0.01], [-0.01,0.5]]),                  
         ]
    )    
)
# Evaluation of the black-box function for specified design parameters
def evaluate(study: Study, b1: float, b2: float, b3: float, b4: float, b5: float) -> Observation:

    observation = study.new_observation()
    #tensor of design values to reconstruct
    x = torch.tensor([b1, b2, b3, b4, b5])    
    observation.add(model(x).tolist())

    return observation

# Run the minimization
study.set_evaluator(evaluate)
study.run()
#best sample of forward model parameters b
best_b_sample = study.driver.best_sample
#minimum of negative log-probability
min_neg_log_prob = study.driver.min_objective

#determine sample [b1, b2, b3, b4, b5, log_c1, log_c2] that minimizes the negative
#log-probability
minimum = list(best_b_sample.values())

#path to negative log-probability variable
path = "driver.acquisition_function.main_objective.variable"
neg_log_probs = study.historic_parameter_values(f"{path}.observed_value")
idx = np.argmin(neg_log_probs)

logs_c1 = study.historic_parameter_values(f"{path}.error_model_parameters.log_c1")
logs_c2 = study.historic_parameter_values(f"{path}.error_model_parameters.log_c2")

minimum += [logs_c1[idx], logs_c2[idx]]
minimum = np.array(minimum)


# Before running a Markov-chain Monte-Carlo (MCMC) sampling we converge the surrogate
# models by sampling around the minimum. To make the study more explorative, the
# scaling parameter is increased and the effective degrees of freedom is set to one.
study.configure(
    scaling=10.0,
    effective_DOF=1.0, 
    min_uncertainty=1e-8*np.abs(min_neg_log_prob),
    max_iter=120,
)
study.run()

# Run the MCMC sampling with 32 walkers
num_walkers, max_iter = 32, 20000
mcmc_result = study.driver.run_mcmc(
    rel_error=0.01,
    num_walkers=num_walkers,
    max_iter=max_iter
)
fig = corner.corner(
    np.array(mcmc_result['samples']),
    quantiles=(0.16, 0.5, 0.84),
    levels=(1-np.exp(-1.0), 1-np.exp(-0.5)),
    show_titles=True, scale_hist=False,
    title_fmt=".3f",
    labels=[d['name'] for d in design_space] + ["log_c1", "log_c2"],
    truths=minimum
)
plt.savefig("corner_surrogate.svg", transparent=True) 


# As a comparison, we run the MCMC sampling directly on the analytic model.
p0 = 0.05*np.random.randn(num_walkers, len(design_space)+2)
p0 += minimum

# Uniform prior domain for the model parameters b1, b4, b5
uniform_domain = np.array([design_space[idx]["domain"] for idx in [0,3,4]])

#log probability function
def log_prob(x: np.ndarray) -> np.ndarray:

    y = model(x[:5])
    res = y - measurement_vector
    err = error_model(x[5], x[6], y)

    #log-likelihood
    ll = -0.5*(
        torch.log(2*torch.tensor(torch.pi)) +
        torch.log(err**2) +
        (res/err)**2
    ).sum()

    #log prior
    lp = (
        multivariate_normal.logpdf(
            x[1:3],
            mean=[2.25,-2.0], 
            cov=[[0.5,-0.01], [-0.01,0.5]]
        ) + 
        uniform.logpdf(
            [x[0],x[3],x[4]],
            loc=uniform_domain[:,0],
            scale=uniform_domain[:,1] - uniform_domain[:,0]
        ).sum() +
        multivariate_normal.logpdf(
            x[5:],
            mean=[-5.0,-4.0], 
            cov=[[1,0.0], [0.0,1]]
        )
    )

    #log probability
    return ll + lp
sampler = emcee.EnsembleSampler(
    nwalkers=num_walkers, ndim=len(design_space)+2, log_prob_fn=log_prob
)

#burn-in phase
state = sampler.run_mcmc(p0, 100)
sampler.reset()
#actual MCMC sampling 
sampler.run_mcmc(state, max_iter, progress=True)
samples = sampler.get_chain(flat=True)
fig = corner.corner(
    samples, quantiles=(0.16, 0.5, 0.84),
    levels=(1-np.exp(-1.0), 1-np.exp(-0.5)),
    show_titles=True, scale_hist=False,
    title_fmt=".3f",
    labels=[d['name'] for d in design_space] + ["log_c1", "log_c2"],
    truths=minimum
)
plt.savefig("corner_analytic.svg", transparent=True)


client.shutdown_server()

Markov-Chain Monte-Carlo (MCMC) sampling of the probability density of the model parameters \(b_1,\dots,b_5\) and the log value of the error model parameters \(c_1,\dots,c_2\) based on the analytic model function.

Markov-Chain Monte-Carlo (MCMC) sampling of the probability density of the model parameters \(b_1,\dots,b_5\) and the log value of the error model parameters \(c_1,\dots,c_2\) based on the trained surrogate of the study. A comparison between the analytic and the surrogate model function shows a good quantitative agreement.