Using Approximate Bayesian Computation by Subset Simulation for Efficient Posterior Assessment of Dynamic State-Space Model Classes

Approximate Bayesian Computation (ABC) methods have gained in popularity over the last decade because they expand the horizon of Bayesian parameter inference methods to the range of models for which an analytical formula for the likelihood function might be difficult, or even impossible, to establish. The majority of the ABC methods rely on the choice of a set of summary statistics to reduce the dimension of the data. However, as has been noted in the ABC literature, the lack of convergence guarantees induced by the absence of a vector of sufficient summary statistics that assures intermodel sufficiency over the set of competing models hinders the use of the usual ABC methods when applied to Bayesian model selection or assessment. In this paper, we present a novel ABC model selection procedure for dynamical systems based on a recently introduced multilevel Markov chain Monte Carlo method, self-regulating ABC-SubSim, and a hierarchical state-space formulation of dynamic models. We show that this formulation makes it possible to independently approximate the model evidence required for assessing the posterior probability of each of the competing models. We also show that ABC-SubSim not only provides an estimate of the model evidence as a simple by-product but also gives the posterior probability of each model as a function of the tolerance level, which allows the ABC model choices made in previous studies to be understood. We illustrate the performance of the proposed framework for ABC model updating and model class selection by applying it to two problems in Bayesian system identification: a single-degree-of-freedom bilinear hysteretic oscillator and a three-story shear building with Masing hysteresis, both of which are subject to a seismic excitation

Approximate Bayesian Computation by Subset Simulation

A new Approximate Bayesian Computation (ABC) algorithm for Bayesian updating of model parameters is proposed in this paper, which combines the ABC principles with the technique of Subset Simulation for efficient rare-event simulation, first developed in S.K. Au and J.L. Beck [1]. It has been named ABC- SubSim. The idea is to choose the nested decreasing sequence of regions in Subset Simulation as the regions that correspond to increasingly closer approximations of the actual data vector in observation space. The efficiency of the algorithm is demonstrated in two examples that illustrate some of the challenges faced in real-world applications of ABC. We show that the proposed algorithm outperforms other recent sequential ABC algorithms in terms of computational efficiency while achieving the same, or better, measure of ac- curacy in the posterior distribution. We also show that ABC-SubSim readily provides an estimate of the evidence (marginal likelihood) for posterior model class assessment, as a by-product

Bayesian Updating, Model Class Selection and Robust Stochastic Predictions of Structural Response

A fundamental issue when predicting structural response by using mathematical models is how to treat both modeling and excitation uncertainty. A general framework for this is presented which uses probability as a multi-valued conditional logic for quantitative plausible reasoning in the presence of uncertainty due to incomplete information. The fundamental probability models that represent the structure’s uncertain behavior are specified by the choice of a stochastic system model class: a set of input-output probability models for the structure and a prior probability distribution over this set that quantifies the relative plausibility of each model. A model class can be constructed from a parameterized deterministic structural model by stochastic embedding utilizing Jaynes’ Principle of Maximum Information Entropy. Robust predictive analyses use the entire model class with the probabilistic predictions of each model being weighted by its prior probability, or if structural response data is available, by its posterior probability from Bayes’ Theorem for the model class. Additional robustness to modeling uncertainty comes from combining the robust predictions of each model class in a set of competing candidates weighted by the prior or posterior probability of the model class, the latter being computed from Bayes’ Theorem. This higherlevel application of Bayes’ Theorem automatically applies a quantitative Ockham razor that penalizes the data-fit of more complex model classes that extract more information from the data. Robust predictive analyses involve integrals over highdimensional spaces that usually must be evaluated numerically. Published applications have used Laplace's method of asymptotic approximation or Markov Chain Monte Carlo algorithms

Computational statistics using the Bayesian Inference Engine

This paper introduces the Bayesian Inference Engine (BIE), a general parallel, optimised software package for parameter inference and model selection. This package is motivated by the analysis needs of modern astronomical surveys and the need to organise and reuse expensive derived data. The BIE is the first platform for computational statistics designed explicitly to enable Bayesian update and model comparison for astronomical problems. Bayesian update is based on the representation of high-dimensional posterior distributions using metric-ball-tree based kernel density estimation. Among its algorithmic offerings, the BIE emphasises hybrid tempered MCMC schemes that robustly sample multimodal posterior distributions in high-dimensional parameter spaces. Moreover, the BIE is implements a full persistence or serialisation system that stores the full byte-level image of the running inference and previously characterised posterior distributions for later use. Two new algorithms to compute the marginal likelihood from the posterior distribution, developed for and implemented in the BIE, enable model comparison for complex models and data sets. Finally, the BIE was designed to be a collaborative platform for applying Bayesian methodology to astronomy. It includes an extensible object-oriented and easily extended framework that implements every aspect of the Bayesian inference. By providing a variety of statistical algorithms for all phases of the inference problem, a scientist may explore a variety of approaches with a single model and data implementation. Additional technical details and download details are available from http://www.astro.umass.edu/bie. The BIE is distributed under the GNU GPL.Comment: Resubmitted version. Additional technical details and download details are available from http://www.astro.umass.edu/bie. The BIE is distributed under the GNU GP

Simulation methods for reliability-based design optimization and model updating of civil engineering structures and systems

This thesis presents a collection of original contributions pertaining to the subjects of reliability-based design optimization (RBDO) and model updating of civil engineering structures and systems. In this regard, probability theory concepts and tools are instrumental in the formulation of the herein reported developments. Firstly, two approaches are devised for the RBDO of structural dynamical systems under stochastic excitation. Namely, a stochastic search technique is proposed for constrained and unconstrained RBDO problems involving continuous, discrete and mixed discrete-continuous design spaces, whereas an efficient sensitivity assessment framework for linear stochastic structures is implemented to identify optimal designs and evaluate their sensitivities. Moreover, two classes of model updating problems are considered. In this context, the Bayesian interpretation of probability theory plays a key role in the proposed solution schemes. Specifically, contaminant source detection in water distribution networks is addressed by resorting to a sampling-based Bayesian model class selection framework. Furthermore, an effective strategy for Bayesian model updating with structural reliability methods is presented to treat identification problems involving structural dynamical systems, measured response data, and high-dimensional parameter spaces. The approaches proposed in this thesis integrate stochastic simulation techniques as an essential part of their formulation, which allows obtaining non-trivial information about the systems of interest as a byproduct of the solution processes. Overall, the findings presented in this thesis suggest that the reported methods can be potentially adopted as supportive tools for a number of practical decision-making processes in civil engineering.Diese Arbeit stellt eine Sammlung von Beiträgen vor, die sich mit der Reliability-based-Design-Optimization (RBDO) und dem Model updating von Strukturen und Systemen im Bauwesen befassen. In diesem Zusammenhang sind wahrscheinlichkeitstheoretische Konzepte für die Formulierung der hier vorgestellten Entwicklungen von entscheidender Bedeutung. Zunächst werden zwei Ansätze für eine RBDO von strukturdynamischen Systemen unter stochastischer Anregung entwickelt. Es wird eine stochastische Suchtechnik für beschränkte und unbeschränkte RBDO-Probleme vorgeschlagen. Diese beziehen kontinuierliche, diskrete und gemischt diskret-kontinuierliche Designräume ein. Gleichzeitig wird ein effizientes Framework zur Bewertung der Sensitivität lineare stochastische Strukturen implementiert, um optimale Designs zu identifizieren und ihre Sensitivitäten zu bewerten. Darüber hinaus werden zwei Klassen von Problem aus dem Model updating betrachtet. Der Fokus wird hierbei auf die Erkennung von Kontaminationsquellen in Wasserverteilungsnetzen mithilfe eines auf Stichproben basierenden Bayesian-Model-Class-selection-Framework gelegt. Ferner wird eine effektive Strategie zur Bearbeitung von Problemen des Bayesian-Model-updating, die strukturdynamischen Systeme, gemessene Systemantwortdaten und hochdimensionale Parameterräume umfassen, vorgestellt. Die beschriebenen Ansätze verwenden stochastische Simulationstechniken als wesentlicher Bestandteil ihrer Formulierung, wodurch nicht-triviale Informationen über betrachtete Systeme als Nebenprodukt der Lösungsprozesse gewonnen werden können. Insgesamt deuten die vorgestellten Ergebnisse dieser Arbeit darauf hin, dass die beschriebenen Methoden potenziell als unterstützende Elemente in praktischen Entscheidungsproblemen im Zusammenhang mit Strukturen und Systemen im Bauwesen eingesetzt werden können

Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

Reduction of Petri net maintenance modeling complexity via Approximate Bayesian Computation

This paper is part of the ENHAnCE ITN project (https://www.h2020-enhanceitn.eu/) funded by the European Union's Horizon 2020 research and innovation programme under the Marie SklodowskaCurie grant agreement No. 859957. The authors would like to thank the Lloyd's Register Foundation (LRF), a charitable foundation in the U.K. helping to protect life and property by supporting engineeringrelated education, public engagement, and the application of research. The authors gratefully acknowledge the support of these organizations which have enabled the research reported in this paper.The accurate modeling of engineering systems and processes using Petri nets often results in complex graph representations that are computationally intensive, limiting the potential of this modeling tool in real life applications. This paper presents a methodology to properly define the optimal structure and properties of a reduced Petri net that mimic the output of a reference Petri net model. The methodology is based on Approximate Bayesian Computation to infer the plausible values of the model parameters of the reduced model in a rigorous probabilistic way. Also, the method provides a numerical measure of the level of approximation of the reduced model structure, thus allowing the selection of the optimal reduced structure among a set of potential candidates. The suitability of the proposed methodology is illustrated using a simple illustrative example and a system reliability engineering case study, showing satisfactory results. The results also show that the method allows flexible reduction of the structure of the complex Petri net model taken as reference, and provides numerical justification for the choice of the reduced model structure.European Commission 859957Lloyd's Register Foundation (LRF), a charitable foundation in the U.K

Statistical Inference for Partially Observed Markov Processes via the R Package pomp

Partially observed Markov process (POMP) models, also known as hidden Markov models or state space models, are ubiquitous tools for time series analysis. The R package pomp provides a very flexible framework for Monte Carlo statistical investigations using nonlinear, non-Gaussian POMP models. A range of modern statistical methods for POMP models have been implemented in this framework including sequential Monte Carlo, iterated filtering, particle Markov chain Monte Carlo, approximate Bayesian computation, maximum synthetic likelihood estimation, nonlinear forecasting, and trajectory matching. In this paper, we demonstrate the application of these methodologies using some simple toy problems. We also illustrate the specification of more complex POMP models, using a nonlinear epidemiological model with a discrete population, seasonality, and extra-demographic stochasticity. We discuss the specification of user-defined models and the development of additional methods within the programming environment provided by pomp.Comment: In press at the Journal of Statistical Software. A version of this paper is provided at the pomp package website: http://kingaa.github.io/pom