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15
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 44 (15 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
The Equivalence of Constrained and Weighted Designs in Multiple Objective Design Problems
 Journal of the American Statistical Association
, 1996
"... Several competing objectives may be relevant in the design of an experiment. The competing objectives may not be easy to characterize in a single optimality criterion. One approach to these design problems has been to weight each criterion and find the design that optimizes the weighted average of t ..."
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Cited by 16 (3 self)
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Several competing objectives may be relevant in the design of an experiment. The competing objectives may not be easy to characterize in a single optimality criterion. One approach to these design problems has been to weight each criterion and find the design that optimizes the weighted average of the criteria. An alternative approach has been to optimize one criterion subject to constraints on the other criteria. An equivalence theorem is presented for the Bayesian constrained design problem. Equivalence theorems are essential in verifying optimality of proposed designs, especially when, as in most nonlinear design problems, numerical optimization is required. This theorem is used to show that the results of Cook and Wong on the equivalence of the weighted and constrained problems also apply much more generally. The results are applied to Bayesian nonlinear design problems with several objectives. KEY WORDS: Bayesian design, regression, nonlinear design 1. INTRODUCTION An experimen...
Bayesian adaptive exploration
 in Statistical Challenges in Astronomy, 2003
"... Abstract. I describe a framework for adaptive scientific exploration based on iterating an Observation–Inference–Design cycle that allows adjustment of hypotheses and observing protocols in response to the results of observation onthefly, as data are gathered. The framework uses a unified Bayesian ..."
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Cited by 8 (0 self)
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Abstract. I describe a framework for adaptive scientific exploration based on iterating an Observation–Inference–Design cycle that allows adjustment of hypotheses and observing protocols in response to the results of observation onthefly, as data are gathered. The framework uses a unified Bayesian methodology for the inference and design stages: Bayesian inference to quantify what we have learned from the available data and predict future data, and Bayesian decision theory to identify which new observations would teach us the most. When the goal of the experiment is simply to make inferences, the framework identifies a computationally efficient iterative “maximum entropy sampling ” strategy as the optimal strategy in settings where the noise statistics are independent of signal properties. Results of applying the method to two “toy ” problems with simulated data—measuring the orbit of an extrasolar planet, and locating a hidden onedimensional object—show the approach can significantly improve observational efficiency in settings that have welldefined nonlinear models. I conclude with a list of open issues that must be addressed to make Bayesian adaptive exploration a practical and reliable tool for optimizing scientific exploration.
Design issues for generalized linear models: A review
 Statistical Science
, 2006
"... Abstract. Generalized linear models (GLMs) have been used quite effectively in the modeling of a mean response under nonstandard conditions, where discrete as well as continuous data distributions can be accommodated. The choice of design for a GLM is a very important task in the development and bui ..."
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Cited by 7 (0 self)
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Abstract. Generalized linear models (GLMs) have been used quite effectively in the modeling of a mean response under nonstandard conditions, where discrete as well as continuous data distributions can be accommodated. The choice of design for a GLM is a very important task in the development and building of an adequate model. However, one major problem that handicaps the construction of a GLM design is its dependence on the unknown parameters of the fitted model. Several approaches have been proposed in the past 25 years to solve this problem. These approaches, however, have provided only partial solutions that apply in only some special cases, and the problem, in general, remains largely unresolved. The purpose of this article is to focus attention on the aforementioned dependence problem. We provide a survey of various existing techniques dealing with the dependence problem. This survey includes discussions concerning locally optimal designs, sequential designs, Bayesian designs and the quantile dispersion graph approach for comparing designs for GLMs. Key words and phrases: Bayesian design, dependence on unknown parameters, locally optimal design, logistic regression, response surface methodology, quantal dispersion graphs, sequential design. 1.
Optimal Design for Heart Defibrillators
 IN CASE STUDIES IN BAYESIAN STATISTICS, II
, 1993
"... ... this paper is on illustrating recent computational techniques, and on comparing nonsequential designs with certain sequential alternatives. In particular, the nonsequential design is analyzed in the fixed sample size case, and computed based on an approximation used by Chaloner and Larntz (1989) ..."
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Cited by 4 (1 self)
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... this paper is on illustrating recent computational techniques, and on comparing nonsequential designs with certain sequential alternatives. In particular, the nonsequential design is analyzed in the fixed sample size case, and computed based on an approximation used by Chaloner and Larntz (1989). The sequential design is an heuristic updown design implemented via a simulation based numerical optimization scheme.
Bayesian Experimental Design and Shannon Information
 In 1997 Proceedings of the Section on Bayesian Statistical Science
, 1997
"... The information theoretic approach to optimal design of experiments yields a simple design criterion: the optimal design minimizes the expected posterior entropy of the parameters. Unfortunately, this strategy is often computational infeasible for nonlinear problems and numerical approximations are ..."
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Cited by 4 (0 self)
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The information theoretic approach to optimal design of experiments yields a simple design criterion: the optimal design minimizes the expected posterior entropy of the parameters. Unfortunately, this strategy is often computational infeasible for nonlinear problems and numerical approximations are required. This paper reviews the information theoretic approach to design of experiments, and examines computational issues related to the minimization of the expected posterior entropy and its asymptotic approximations. It is shown that Maximum Entropy Sampling simplifies both the formulation of the design criterion and the optimization problem. Numerical advantages are shown in an example, where the exact solution is compared to asymptotic optimal solution.
Optimal Design for Quantal Bioassay via Monte Carlo Methods
 In
, 1999
"... this paper we present a general decision theoretic setup for the design problem and develop a solution using Monte Carlobased methods. Following Kuo (1983), to relax the restrictive assumptions on the potency curve, we adopt a nonparametric Bayesian approach and assume a Dirichlet process prior (Fe ..."
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this paper we present a general decision theoretic setup for the design problem and develop a solution using Monte Carlobased methods. Following Kuo (1983), to relax the restrictive assumptions on the potency curve, we adopt a nonparametric Bayesian approach and assume a Dirichlet process prior (Ferguson, 1973) with parameter ffF 0 on the potency curve, where ff represents our strength of prior belief and F 0 is the prior mean of the random tolerance distribution. In particular, we model the potency curve as a random discrete distribution function with random locations and random size of the jumps with the specific distribution given in Sethuraman and Tiwari (1982). The prior mean of this random distribution evaluated at t is assumed to be F 0 (t) and the variance of this random distribution evaluated at t is assumed to be F 0 (t)(1 \Gamma F 0 (t))=(ff + 1). So ff can be interpreted as the strength of prior belief or the degree of concentration of the random distribution around F 0 . The larger the ff is, the more concentrated F is around F 0 . In Section 2, we describe the design problem and introduce its ingredients. We present a decision theoretic setup for the design problem in Section 3 and illustrate the difficulties involved in obtaining an analytical solution to the preposterior analysis. In Section 4 we discuss how MCMC methods can be used for evaluating the posterior expected loss. We present a Monte Carlo integration technique to evaluate the preposterior expected loss and discuss the potential computational burden involved in finding the optimal design. To alleviate such computational burden, we adopt the simulationbased approach of Muller and Parmigiani (1995) where the preposterior analysis is replaced by a curve (surface)fitting technique. This simulat...
Simulation Approach to Onestage and Sequential Optimal Design Problems
, 1994
"... this paper. We refer to West, Muller and Escobar (1994) and Muller, Erkanli and West (1994). 3 EXAMPLES FOR ONESTAGE OPTIMAL DESIGNS 3.1 Example 1: An information theoretic stopping rule ..."
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this paper. We refer to West, Muller and Escobar (1994) and Muller, Erkanli and West (1994). 3 EXAMPLES FOR ONESTAGE OPTIMAL DESIGNS 3.1 Example 1: An information theoretic stopping rule
Numerical Evaluation of Information Theoretic Measures
"... ... In this paper we discuss implementation strategies for fast numerical computations of Entropies and KullbackLeibler divergences that are relevant to Bayesian inference and design problems. We illustrate the methods proposed with examples in model diagnostics and information theoretic design. ..."
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Cited by 1 (1 self)
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... In this paper we discuss implementation strategies for fast numerical computations of Entropies and KullbackLeibler divergences that are relevant to Bayesian inference and design problems. We illustrate the methods proposed with examples in model diagnostics and information theoretic design.