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A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning
, 2010
"... We present a tutorial on Bayesian optimization, a method of finding the maximum of expensive cost functions. Bayesian optimization employs the Bayesian technique of setting a prior over the objective function and combining it with evidence to get a posterior function. This permits a utilitybased se ..."
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Cited by 91 (11 self)
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We present a tutorial on Bayesian optimization, a method of finding the maximum of expensive cost functions. Bayesian optimization employs the Bayesian technique of setting a prior over the objective function and combining it with evidence to get a posterior function. This permits a utilitybased selection of the next observation to make on the objective function, which must take into account both exploration (sampling from areas of high uncertainty) and exploitation (sampling areas likely to offer improvement over the current best observation). We also present two detailed extensions of Bayesian optimization, with experiments—active user modelling with preferences, and hierarchical reinforcement learning— and a discussion of the pros and cons of Bayesian optimization based on our experiences.
Bayesian Treed Gaussian Process Models with an Application to Computer Modeling
 Journal of the American Statistical Association
, 2007
"... This paper explores nonparametric and semiparametric nonstationary modeling methodologies that couple stationary Gaussian processes and (limiting) linear models with treed partitioning. Partitioning is a simple but effective method for dealing with nonstationarity. Mixing between full Gaussian proce ..."
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Cited by 88 (18 self)
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This paper explores nonparametric and semiparametric nonstationary modeling methodologies that couple stationary Gaussian processes and (limiting) linear models with treed partitioning. Partitioning is a simple but effective method for dealing with nonstationarity. Mixing between full Gaussian processes and simple linear models can yield a more parsimonious spatial model while significantly reducing computational effort. The methodological developments and statistical computing details which make this approach efficient are described in detail. Illustrations of our model are given for both synthetic and real datasets. Key words: recursive partitioning, nonstationary spatial model, nonparametric regression, Bayesian model averaging 1
A framework for validation of computer models
, 2002
"... In this paper, we present a framework that enables computer model evaluation oriented towards answering the question: Does the computer model adequately represent reality? The proposed validation framework is a sixstep procedure based upon Bayesian statistical methodology. The Bayesian methodology ..."
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Cited by 84 (16 self)
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In this paper, we present a framework that enables computer model evaluation oriented towards answering the question: Does the computer model adequately represent reality? The proposed validation framework is a sixstep procedure based upon Bayesian statistical methodology. The Bayesian methodology is particularly suited to treating the major issues associated with the validation process: quantifying multiple sources of error and uncertainty in computer models; combining multiple sources of information; and updating validation assessments as new information is acquired. Moreover, it allows inferential statements to be made about predictive error associated with model predictions in untested situations. The framework is implemented in two test bed models (a vehicle crash model and a resistance
Bayesian Analysis of Computer Code Outputs: A Tutorial
, 2004
"... The Bayesian approach to quantifying, analysing and reducing uncertainty in the application of complex process models is attracting increasing attention amongst users of such models. The range and power of the Bayesian methods is growing and there is already a sizeable literature on these methods. H ..."
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Cited by 81 (0 self)
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The Bayesian approach to quantifying, analysing and reducing uncertainty in the application of complex process models is attracting increasing attention amongst users of such models. The range and power of the Bayesian methods is growing and there is already a sizeable literature on these methods. However, most of it is in specialist statistical journals. The purpose of this tutorial is to introduce the more general reader to the Bayesian approach.
Lightweight emulators for multivariate deterministic functions
 FORTHCOMING IN THE JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS
, 2007
"... An emulator is a statistical model of a deterministic function, to be used where the function itself is too expensive to evaluate withintheloop of an inferential calculation. Typically, emulators are deployed when dealing with complex functions that have large and heterogeneous input and output sp ..."
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Cited by 42 (9 self)
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An emulator is a statistical model of a deterministic function, to be used where the function itself is too expensive to evaluate withintheloop of an inferential calculation. Typically, emulators are deployed when dealing with complex functions that have large and heterogeneous input and output spaces: environmental models, for example. In this challenging situation we should be sceptical about our statistical models, no matter how sophisticated, and adopt approaches that prioritise interpretative and diagnostic information, and the flexibility to respond. This paper presents one such approach, candidly rejecting the standard Smooth Gaussian Process approach in favour of a fullyBayesian treatment of multivariate regression which, by permitting sequential updating, allows for very detailed predictive diagnostics. It is argued directly and by illustration that the incoherence of such a treatment (which does not impose continuity on the model outputs) is more than compensated for by the wealth of available information, and the possibilities for generalisation.
Stochastic Kriging for Simulation Metamodeling
 Operations Research
, 2010
"... We extend the basic theory of kriging, as applied to the design and analysis of deterministic computer experiments, to the stochastic simulation setting. Our goal is to provide flexible, interpolationbased metamodels of simulation output performance measures as functions of the controllable design ..."
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Cited by 41 (9 self)
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We extend the basic theory of kriging, as applied to the design and analysis of deterministic computer experiments, to the stochastic simulation setting. Our goal is to provide flexible, interpolationbased metamodels of simulation output performance measures as functions of the controllable design or decision variables. To accomplish this we characterize both the intrinsic uncertainty inherent in a stochastic simulation and the extrinsic uncertainty about the unknown response surface. We use tractable examples to demonstrate why it is critical to characterize both types of uncertainty, derive general results for experiment design and analysis, and present a numerical example that illustrates the stochastic kriging method. 1
Active policy learning for robot planning and exploration under uncertainty
 IN PROCEEDINGS OF ROBOTICS: SCIENCE AND SYSTEMS
, 2007
"... This paper proposes a simulationbased active policy learning algorithm for finitehorizon, partiallyobserved sequential decision processes. The algorithm is tested in the domain of robot navigation and exploration under uncertainty. In such a setting, the expected cost, that must be minimized, is ..."
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Cited by 39 (5 self)
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This paper proposes a simulationbased active policy learning algorithm for finitehorizon, partiallyobserved sequential decision processes. The algorithm is tested in the domain of robot navigation and exploration under uncertainty. In such a setting, the expected cost, that must be minimized, is a function of the belief state (filtering distribution). This filtering distribution is in turn nonlinear and subject to discontinuities, which arise because constraints in the robot motion and control models. As a result, the expected cost is nondifferentiable and very expensive to simulate. The new algorithm overcomes the first difficulty and reduces the number of required simulations as follows. First, it assumes that we have carried out previous simulations which returned values of the expected cost for different corresponding policy parameters. Second, it fits a Gaussian process (GP) regression model to these values, so as to approximate the expected cost as a function of the policy parameters. Third, it uses the GP predicted mean and variance to construct a statistical measure that determines which policy parameters should be used in the next simulation. The process is then repeated using the new parameters and the newly gathered expected cost observation. Since the objective is to find the policy parameters that minimize the expected cost, this iterative active learning approach effectively tradesoff between exploration (in regions where the GP variance is large) and exploitation (where the GP mean is low). In our experiments, a robot uses the proposed algorithm to plan an optimal path for accomplishing a series of tasks, while maximizing the information about its pose and map estimates. These estimates are obtained with a standard filter for simultaneous localization and mapping. Upon gathering new observations, the robot updates the state estimates and is able to replan a new path in the spirit of openloop feedback control.
Reified bayesian modelling and inference for physical systems (with discussion
 Journal of Statistical Planning and Inference
, 2009
"... We describe an approach, termed reified analysis, for linking the behaviour of mathematical models with inferences about the physical systems which the models represent. We describe the logical basis for the approach, based on coherent assessment of the implications of deficiencies in the mathemat ..."
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Cited by 38 (18 self)
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We describe an approach, termed reified analysis, for linking the behaviour of mathematical models with inferences about the physical systems which the models represent. We describe the logical basis for the approach, based on coherent assessment of the implications of deficiencies in the mathematical model. We show how the statistical analysis may be carried out by specifying stochastic relationships between the model that we have, improved versions of the model that we might construct, and the system itself. We illustrate our approach with an example concerning the potential shutdown of the Thermohaline Circulation in the Atlantic Ocean.
Subjective Bayesian Analysis: Principle and practice
 BAYESIAN ANALYSIS
, 2006
"... We address the position of subjectivism within Bayesian statistics. We argue, first, that the subjectivist Bayes approach is the only feasible method for tackling many important practical problems. Second, we describe the essential role of the subjectivist approach in scientific analysis. Third, we ..."
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Cited by 35 (0 self)
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We address the position of subjectivism within Bayesian statistics. We argue, first, that the subjectivist Bayes approach is the only feasible method for tackling many important practical problems. Second, we describe the essential role of the subjectivist approach in scientific analysis. Third, we consider possible modifications to the Bayesian approach from a subjectivist viewpoint. Finally, we address the issue of pragmatism in implementing the subjectivist approach.