Optimal Experimental Design Methods for Acquiring and Restricting Information to Improve Decision Making

The Bayesian optimal experimental design (OED) problem seeks to identify data, sensor configurations, or experiments which can optimally reduce uncertainty. The goal of OED is to find an experiment that maximizes the expected information gain (EIG) about quantities of interest given prior knowledge about expected data. Therefore, within the context of seismic monitoring, we can use Bayesian OED to configure sensor networks by choosing sensor locations, types, and fidelity in order to improve our ability to identify and locate seismic sources. In this work, we develop the framework necessary to use Bayesian OED to optimize the ability to locate seismic events from arrival time data of detected seismic phases. In order to do utilize Bayesian OED we must develop four elements:1. A likelihood function that describes the uncertainty of detection and travel times; 2. A Bayesian solver that takes a prior and likelihood to identify the posterior; 3. An algorithm to compute EIG; and, 4. An optimizer that finds a sensor network which maximizes EIG. Once we have developed this framework, we can explore many relevant questions to monitoring such as: how and what multiphenomenology data can be used to optimally reduce uncertainty, how to trade off sensor fidelity and earth model uncertainty, and how sensor types, number, and locations influence uncertainty

Optimal experimental design is an important methodology for most efficiently allocating resources in an experiment to best achieve some goal. Bayesian experimental design considers the potential impact that various choices of the controllable variables has on the posterior distribution of the unknowns. Optimal Bayesian design involves optimising an expected utility function, which is an analytically intractable integral over the prior predictive distribution. These integrals are typically estimated via standard Monte Carlo methods. In this paper, we demonstrate that the use of randomised quasi-Monte Carlo can bring significant reductions to the variance of the estimated expected utility. This variance reduction will then lead to a more efficient optimisation algorithm for maximising the expected utility.

Bayesian optimal experimental design (OED) plays an important role in minimizing model uncertainty with limited experimental data in a Bayesian framework. In many applications, rather than minimizing the uncertainty in the inference of model parameters, one seeks to minimize the uncertainty of a model-dependent quantity of interest (QoI). This is known as goal-oriented OED (GOOED). Here, we consider GOOED for linear Bayesian inverse problems governed by large-scale models represented by partial differential equations (PDE) that are computationally expensive to solve. In particular, we consider optimal sensor placement by maximizing an expected information gain (EIG) for the QoI. We develop an efficient method to solve such problems by deriving a new formulation of the goal-oriented EIG. Based on this formulation we propose an offline-online decomposition scheme that achieves significant computational reduction by computing all of the PDE-dependent quantities in an offline stage just once, and optimizing the sensor locations in an online stage without solving any PDEs. Moreover, in the offline stage we need only to compute low-rank approximations of two Hessian-related operators. The computational cost of these low-rank approximations, measured by the number of PDE solves, does not depend on the parameter or data dimensions for a large class of elliptic, parabolic, and sufficiently dissipative hyperbolic inverse problem that exhibit dimension-independent rapid spectra decay. We carry out detailed error analysis for the approximate goal-oriented EIG due to the low-rank approximations of the two operators. Furthermore, in the online stage we extend a swapping greedy method to optimize the sensor locations developed in our recent work that is demonstrated to be more efficient than a standard greedy method. We conduct a numerical experiment for a contaminant transport inverse problem with an infinite-dimensional parameter field to demonstrate the efficiency, accuracy, and both data- and parameter-dimension independence of the proposed algorithm.

Bayesian optimal experimental design for constitutive model calibration

Summary When Shannon entropy is used as a criterion in the optimal design of experiments, advantage can be taken of the classical identity representing the joint entropy of parameters and observations as the sum of the marginal entropy of the observations and the preposterior conditional entropy of the parameters. Following previous work in which this idea was used in spatial sampling, the method is applied to standard parameterized Bayesian optimal experimental design. Under suitable conditions, which include non-linear as well as linear regression models, it is shown in a few steps that maximizing the marginal entropy of the sample is equivalent to minimizing the preposterior entropy, the usual Bayesian criterion, thus avoiding the use of conditional distributions. It is shown using this marginal formulation that under normality assumptions every standard model which has a two-point prior distribution on the parameters gives an optimal design supported on a single point. Other results include a new asymptotic formula which applies as the error variance is large and bounds on support size.

We consider the problem of learning structures and parameters of continuous-time Bayesian networks (CTBNs) from time-course data under minimal experimental resources. In practice, the cost of generating experimental data poses a bottleneck, especially in the natural and social sciences. A popular approach to overcome this is Bayesian optimal experimental design (BOED). However, BOED becomes infeasible in high-dimensional settings, as it involves integration over all possible experimental outcomes. We propose a novel criterion for experimental design based on a variational approximation of the expected information gain. We show that for CTBNs, a semi-analytical expression for this criterion can be calculated for structure and parameter learning. By doing so, we can replace sampling over experimental outcomes by solving the CTBNs master-equation, for which scalable approximations exist. This alleviates the computational burden of integrating over possible experimental outcomes in high-dimensions. We employ this framework in order to recommend interventional sequences. In this context, we extend the CTBN model to conditional CTBNs in order to incorporate interventions. We demonstrate the performance of our criterion on synthetic and real-world data.

We develop a framework for goal-oriented optimal design of experiments (GOODE) for large-scale Bayesian linear inverse problems governed by PDEs. This framework differs from classical Bayesian optimal design of experiments (ODE) in the following sense: we seek experimental designs that minimize the posterior uncertainty in the experiment end-goal, e.g. a quantity of interest (QoI), rather than the estimated parameter itself. This is suitable for scenarios in which the solution of an inverse problem is an intermediate step and the estimated parameter is then used to compute a QoI. In such problems, a GOODE approach has two benefits: the designs can avoid wastage of experimental resources by a targeted collection of data, and the resulting design criteria are computationally easier to evaluate due to the often low-dimensionality of the QoIs. We present two modified design criteria, A-GOODE and D-GOODE, which are natural analogues of classical Bayesian A- and D-optimal criteria. We analyze the connections to other ODE criteria, and provide interpretations for the GOODE criteria by using tools from information theory. Then, we develop an efficient gradient-based optimization framework for solving the GOODE optimization problems. Additionally, we present comprehensive numerical experiments testing the various aspects of the presented approach. The driving application is the optimal placement of sensors to identify the source of contaminants in a diffusion and transport problem. We enforce sparsity of the sensor placements using an -norm penalty approach, and propose a practical strategy for specifying the associated penalty parameter.

In this chapter, we provide a general overview on the Bayesian experimental design of various statistical models in the recent years. The Bayesian optimal designs incorporate the prior information and uncertainties of the models by using various utility functions, which describe various aims of the experiments. Although there are difficulties in obtaining the analytical expression of the expected utilities and their corresponding numerical approximations under Bayesian framework, the development of fast computational algorithms will bring wider applications of Bayesian experimental design to more complicated models.

The sequential optimal experimental design formulated as an information-theoretic sensitivity analysis is applied to the ignition delay problem using real experimental. The optimal design is obtained by maximizing the statistical dependence between the model parameters and observables, which is quantified in this study using mutual information. This is naturally posed in the Bayesian framework. The study shows that by monitoring the information gain after each measurement update, one can design a stopping criteria for the experimental process which gives a minimal set of experiments to efficiently learn the Arrhenius parameters.

Derivative-informed projected neural networks for high-dimensional parametric maps governed by PDEs

We consider the problem of A-optimal design of experiment under a Bayesian probabilistic model with both categorical and continuous response variables. The utility function of the local design problem is derived by applying Bayesian experimental design framework. We also develop an efficient optimization algorithm to obtain the local optimal design by combining the particle swarm optimization and the blocked coordinate descent methods. In addition, we discuss two different ways of constructing the global optimal design based on the algorithm for local optimal design. Simulation studies are presented to illustrate the efficiency of our approach.

Earthquakes arise from tectonic loading of complex fault systems consisting of heterogeneous material parameters, geometry, rheology, and prestress. All of those are subject to uncertainties, and their interactions and sensitivities for the dynamic rupture problem are incompletely understood. Here, we apply Graph Neural Networks (GNNs) to approximate the behavior learned from more computationally intensive, physics‐based (“high‐fidelity”) computations to build a GNN‐based simulator (GNS) for earthquake rupture dynamics. Given only a minimum input –the hypocenter location– our GNS can reproduce rate‐weakening friction governed dynamic rupture behavior, from nucleation to propagation and termination. Outside the training set, the GNS can generalize well to different hypocenter locations, fault sizes, and pre‐stress state levels while achieving a factor ∼20–40 per‐time‐step computational speedup. This may allow for more efficient estimates of the mapping from pre‐earthquake state, as might be inferred from geodesy, to expected rupture dynamics, for example. By extracting a coarse‐grained version of the underlying dynamics, the GNS provides new perspectives to explore the physics of rupture. Further development of GNS may enable new kinds of parameter space exploration and provide surrogates for Bayesian model inference, uncertainty quantification, and optimal experimental design.

Bayesian OED for Sensor Placement: Analysis and Optimization of Seismo-Acoustic Monitoring Networks with Bayesian Optimal Experimental Design

Efficient Sampling of Climate Models using Bayesian Optimal Experimental Design

We are honored to be guest editors for this special issue of Applied Stochastic Models in Business and Industry, dedicated to Kathryn Chaloner's life and achievements. Contributions from her colleagues and friends in this volume are devoted to Kathryn's early research interest in Bayesian optimal design. She was a pioneer in this field and had a broad impact on the subsequent literature on experimental design. Kathryn was internationally known for her research in Bayesian Statistics, which included design of experiments, outlier detection, prior elicitation, and clinical trials, and for her advancements in the study of HIV/AIDS. She was an accomplished researcher, teacher, and mentor. She actively encouraged women and under-represented minority students to join the field of biostatistics and provided mentorship, support, and encouragement. The special issue contains seven papers. The first three papers present some of the recent work in Bayesian optimal designs for nonlinear models, binary responses, and experiments with adversarial components. This is followed by two articles on Bayesian designs in clinical trials. The last two articles of the special issue involve engineering design problems. The paper by Giovagnoli considers the binary response models of Chaloner and Larntz (1989, Journal of Statistical Planning and Inference, 21:191–208) and develops an adaptive version of A-optimal Bayesian designs that are revised as additional data become available during the experiment. Adaptive Bayesian compound designs is also discussed as another extension. Konstantinou and Dette develop approximate Bayesian D-optimal designs for nonlinear regression models that involve covariates that cannot be observed directly. Applications to specific nonlinear models including the exponential regression model are considered and characterizations of the D-optimal saturated designs are presented. Optimal sample size selection for experiments is considered by De Santis and Gubbiotti in an adversarial setting where the decision makers have different prior opinions while having common utility functions and data. Using a Bayesian decision approach with a quadratic loss function, the authors present results for the one-parameter exponential family and investigate effect of the priors on the optimal sample size. Müller, Xu, and Thall discuss advantages and limitations of Bayesian decision theoretic approach in design of clinical trials. In doing so, the authors present a specific case study and point out the limitations such as the computational difficulties involved in solving sequential problems as well as selection of utility functions capturing different stakeholder interests. The article by Ventz, Parmigiani, and Trippa also consider limitations of the Bayesian framework in design of clinical trials and propose a strategy to alleviate these. The proposed strategy involves combining Bayesian designs with frequentist metrics such as error rates, confidence intervals, etc. that are commonly used by medical community and regulatory agencies. The authors show that this can be achieved by an inclusion of frequentist constraints into the Bayesian formulation. The constrained decision theoretic framework is illustrated via several applications and computational algorithms are presented for implementing the proposed approach. Nakamura, Seepaul, Kadane, and Reeja-Jayan discuss design of experiments in materials science. The authors consider use of a batch-sequential experiment and a regression model to determine optimal levels of input variables for synthesis of ceramic materials. It is illustrated that the optimal settings produce more reliable results than other experiments. The final article by Polson and Soyer considers Bayesian design of accelerated life tests for reliability assessment. An augmented probability simulation (APS) approach using Lindley's (1976, Annals of Statistics, 4:1–10) conjugate utility functions is proposed to compute optimal designs in an efficient manner. The proposed approach is illustrated using single and multiple point designs. Use of particle based methods for APS is also discussed as an alternative to Markov chain Monte Carlo methods.