Statistics for Spatial Data.

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Convex Analysisの二,三の進展について

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Modern Applied Statistics With S

Big Data applications are typically associated with systems involving large numbers of users, massive complex software systems, and large-scale heterogeneous computing and storage architectures. The construction of such systems involves many distributed design choices. The end products (e.g., recommendation systems, medical analysis tools, real-time game engines, speech recognizers) thus involve many tunable configuration parameters. These parameters are often specified and hard-coded into the software by various developers or teams. If optimized jointly, these parameters can result in significant improvements. Bayesian optimization is a powerful tool for the joint optimization of design choices that is gaining great popularity in recent years. It promises greater automation so as to increase both product quality and human productivity. This review paper introduces Bayesian optimization, highlights some of its methodological aspects, and showcases a wide range of applications.

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Taking the Human Out of the Loop: A Review of Bayesian Optimization

Foreword Stephen P. Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng Introduction to MCMC, Charles J. Geyer A short history of Markov chain Monte Carlo: Subjective recollections from in-complete data, Christian Robert and George Casella Reversible jump Markov chain Monte Carlo, Yanan Fan and Scott A. Sisson Optimal proposal distributions and adaptive MCMC, Jeffrey S. Rosenthal MCMC using Hamiltonian dynamics, Radford M. Neal Inference and Monitoring Convergence, Andrew Gelman and Kenneth Shirley Implementing MCMC: Estimating with confidence, James M. Flegal and Galin L. Jones Perfection within reach: Exact MCMC sampling, Radu V. Craiu and Xiao-Li Meng Spatial point processes, Mark Huber The data augmentation algorithm: Theory and methodology, James P. Hobert Importance sampling, simulated tempering and umbrella sampling, Charles J.Geyer Likelihood-free Markov chain Monte Carlo, Scott A. Sisson and Yanan Fan MCMC in the analysis of genetic data on related individuals, Elizabeth Thompson A Markov chain Monte Carlo based analysis of a multilevel model for functional MRI data, Brian Caffo, DuBois Bowman, Lynn Eberly, and Susan Spear Bassett Partially collapsed Gibbs sampling & path-adaptive Metropolis-Hastings in high-energy astrophysics, David van Dyk and Taeyoung Park Posterior exploration for computationally intensive forward models, Dave Higdon, C. Shane Reese, J. David Moulton, Jasper A. Vrugt and Colin Fox Statistical ecology, Ruth King Gaussian random field models for spatial data, Murali Haran Modeling preference changes via a hidden Markov item response theory model, Jong Hee Park Parallel Bayesian MCMC imputation for multiple distributed lag models: A case study in environmental epidemiology, Brian Caffo, Roger Peng, Francesca Dominici, Thomas A. Louis, and Scott Zeger MCMC for state space models, Paul Fearnhead MCMC in educational research, Roy Levy, Robert J. Mislevy, and John T. Behrens Applications of MCMC in fisheries science, Russell B. Millar Model comparison and simulation for hierarchical models: analyzing rural-urban migration in Thailand, Filiz Garip and Bruce Western

Handbook of Markov Chain Monte Carlo

Simulated tempering (ST) is an established Markov chain Monte Carlo (MCMC) method for sampling from a multimodal density $\pi(\theta)$ Typically, ST involves introducing an auxiliary variable $k$ taking values in a finite subset of $[0,1]$ and indexing a set of tempered distributions, say $\pi_k(\theta) \propto \pi(\theta)^k$ In this case, small values of $k$ encourage better mixing, but samples from $\pi$ are only obtained when the joint chain for $(\theta,k)$ reaches $k=1$ However, the entire chain can be used to estimate expectations under $\pi$ of functions of interest, provided that importance sampling (IS) weights are calculated Unfortunately this method, which we call importance tempering (IT), can disappoint This is partly because the most immediately obvious implementation is na\"ive and can lead to high variance estimators We derive a new optimal method for combining multiple IS estimators and prove that the resulting estimator has a highly desirable property related to the notion of effective sample size We briefly report on the success of the optimal combination in two modelling scenarios requiring reversible-jump MCMC, where the na\"ive approach fails

Importance Tempering

We consider adapting a canonical computer model calibration apparatus, involving coupled Gaussian process (GP) emulators, to a computer experiment simulating radiative shock hydrodynamics that is orders of magnitude larger than what can typically be accommodated. The conventional approach calls for thousands of large matrix inverses to evaluate the likelihood in an MCMC scheme. Our approach replaces that costly ideal with a thrifty take on essential ingredients, synergizing three modern ideas in emulation, calibration and optimization: local approximate GP regression, modularization, and mesh adaptive direct search. The new methodology is motivated both by necessity - considering our particular application - and by recent trends in the supercomputer simulation literature. A synthetic data application allows us to explore the merits of several variations in a controlled environment and, together with results on our motivating real-data experiment, lead to noteworthy insights into the dynamics of radiative shocks as well as the limitations of the calibration enterprise generally.

Calibrating a large computer experiment simulating radiative shock hydrodynamics

Recognizing the successes of treed Gaussian process (TGP) models as an interpretable and thrifty model for nonparametric regression, we seek to extend the model to classification. Both treed models and Gaussian processes (GPs) have, separately, enjoyed great success in application to classification problems. An example of the former is Bayesian CART. In the latter, real-valued GP output may be utilized for classification via latent variables, which provide classification rules by means of a softmax function. We formulate a Bayesian model averaging scheme to combine these two models and describe a Monte Carlo method for sampling from the full posterior distribution with joint proposals for the tree topology and the GP parameters corresponding to latent variables at the leaves. We concentrate on efficient sampling of the latent variables, which is important to obtain good mixing in the expanded parameter space. The tree structure is particularly helpful for this task and also for developing an efficient scheme for handling categorical predictors, which commonly arise in classification problems. Our proposed classification TGP (CTGP) methodology is illustrated on a collection of synthetic and real data sets. We assess performance relative to existing methods and thereby show how CTGP is highly flexible, offers tractable inference, produces rules that are easy to interpret, and performs well out of sample.

Classification and Categorical Inputs with Treed Gaussian Process Models

Applications of copula models have been increasing in number in recent years. This class of models provides a modular parameterization of joint distributions: the specification of the marginal distributions is parameterized separately from the dependence structure of the joint, a convenient way of encoding a model for domains such as finance. Some recent advances on how to specify copulas for arbitrary dimensions have been proposed, by means of mixtures of decomposable graphical models. This paper introduces a Bayesian approach for dealing with mixtures of copulas which, due to the lack of prior conjugacy, raise computational challenges. We motivate and present families of Markov chain Monte Carlo (MCMC) proposals that exploit the particular structure of mixtures of copulas. Different algorithms are evaluated according to their mixing properties, and an application in financial forecasting with missing data illustrates the usefulness of the methodology.

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MCMC Methods for Bayesian Mixtures of Copulas

Constrained blackbox optimization is a difficult problem, with most approaches coming from the mathematical programming literature The statistical literature is sparse, especially in addressing problems with nontrivial constraints This situation is unfortunate because statistical methods have many attractive properties: global scope, handling noisy objectives, sensitivity analysis, and so forth To narrow that gap, we propose a combination of response surface modeling, expected improvement, and the augmented Lagrangian numerical optimization framework This hybrid approach allows the statistical model to think globally and the augmented Lagrangian to act locally We focus on problems where the constraints are the primary bottleneck, requiring expensive simulation to evaluate and substantial modeling effort to map out In that context, our hybridization presents a simple yet effective solution that allows existing objective-oriented statistical approaches, like those based on Gaussian process surrogates and expected improvement heuristics, to be applied to the constrained setting with minor modification This work is motivated by a challenging, real-data benchmark problem from hydrology where, even with a simple linear objective function, learning a nontrivial valid region complicates the search for a global minimum

/pdf/modeling-an-augmented-lagrangian-for-improved-blackbox-3ogcetrvqi.pdf

Robert B. Gramacy

Papers

Importance Tempering

Calibrating a large computer experiment simulating radiative shock hydrodynamics

Classification and Categorical Inputs with Treed Gaussian Process Models

MCMC Methods for Bayesian Mixtures of Copulas

Modeling an augmented Lagrangian for improved blackbox constrained optimization