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Simulation-based optimal Bayesian experimental design for nonlinear systems (1108.4146v3)

Published 20 Aug 2011 in stat.ML, stat.CO, and stat.ME

Abstract: The optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction, particularly in situations where experiments are time-consuming and expensive to conduct. We propose a general mathematical framework and an algorithmic approach for optimal experimental design with nonlinear simulation-based models; in particular, we focus on finding sets of experiments that provide the most information about targeted sets of parameters. Our framework employs a Bayesian statistical setting, which provides a foundation for inference from noisy, indirect, and incomplete data, and a natural mechanism for incorporating heterogeneous sources of information. An objective function is constructed from information theoretic measures, reflecting expected information gain from proposed combinations of experiments. Polynomial chaos approximations and a two-stage Monte Carlo sampling method are used to evaluate the expected information gain. Stochastic approximation algorithms are then used to make optimization feasible in computationally intensive and high-dimensional settings. These algorithms are demonstrated on model problems and on nonlinear parameter estimation problems arising in detailed combustion kinetics.

Citations (406)

Summary

  • The paper introduces a Bayesian framework that quantifies expected information gain via Shannon measures to guide experimental design in nonlinear systems.
  • It implements a two-stage Monte Carlo sampling approach with Polynomial Chaos surrogates to efficiently optimize experiments in high-dimensional settings.
  • Stochastic approximation techniques are employed to navigate noisy design landscapes, significantly reducing posterior uncertainty in parameter inference.

Simulation-Based Optimal Bayesian Experimental Design for Nonlinear Systems

The paper by Huan and Marzouk presents an innovative approach to Bayesian optimal experimental design in nonlinear systems. It provides a comprehensive framework for the optimal selection of experimental conditions, specifically emphasizing applications where experiments are costly and time-consuming. The primary focus is on systems modeled by nonlinear simulation-based models, aiming to maximize information gain about targeted parameter sets.

Key Contributions

  1. Bayesian Framework and Information Gain: The paper adopts a Bayesian framework to handle the inherent uncertainties in experimental data. By employing Shannon information measures, the method quantifies the expected information gain from experiments. This choice of design criterion leverages relative entropy to reflect the qualitative and quantitative value of data, providing a robust decision-theoretic foundation to the experimental design process.
  2. Numerical Strategy: A two-stage Monte Carlo sampling approach is implemented to evaluate expected information gain, facilitating optimization in high-dimensional and computationally intensive scenarios. The integration of Polynomial Chaos (PC) expansions as surrogate models is crucial here, as it allows capturing the dependency between observables, parameter sets, and design conditions effectively.
  3. Stochastic Optimization: The paper employs stochastic approximation techniques to navigate the complex landscape of the design space. These methods efficiently exploit the noisy estimates of information gain, making optimization feasible for computationally demanding models.
  4. Application and Examples: Demonstrative applications explore various parameter inferences in nonlinear models, including intricacies found in combustion kinetics. The chosen examples highlight the capacity of the framework to deliver precise experimental designs, facilitating significant reduction in posterior uncertainty.

Implications and Future Directions

Practically, this framework serves as a versatile tool for scientists and engineers seeking optimal strategies for data collection, particularly in domains such as chemical kinetics and material science. Theoretically, it enriches the dialogue on Bayesian experimental design by coupling rigorous information-theoretic criteria with efficient computational methods.

Looking forward, several promising avenues for extension are suggested. Enhancements in surrogate modeling, possibly integrating adaptive refinement in PC expansions, could further streamline computational costs. Addressing uncertainties not only in model parameters but also in design conditions (e.g., equipment tolerances) offers another layer for exploration. Furthermore, connecting the developed framework with sequential experimental design protocols could open pathways to real-time decision-making applications.

In summary, Huan and Marzouk's work stands as a valuable contribution to Bayesian experimental design, offering a robust, computationally feasible method for optimizing nonlinear system investigations. By effectively leveraging statistical methodologies and stochastic computation, the paper advances the state of the art, facilitating deeper insights into parameter inference and model calibration across complex systems.