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20th World Congress of the International-Federation-of-Automatic-Control (IFAC)

作者： Kondo, Takaaki Yamaoka, Seiji Ohta, Yoshito Kyoto Univ Kyoto 6068501 Japan

This paper proposes a hyper-parameter estimation algorithm for the regularized least squares problem in the empirical Bayesian approach arising from FIR model identification using OBFs (orthonormal basis functions)-based kernels. The algorithm consists of two steps by dividing the decision variables into two groups and alternately minimizing with respect to each group. It is shown that DC (difference of convex functions) programming is effectively applicable in the algorithm because the search space is shown to be bounded. The paper includes a couple of numerical simulations to show the efficiency of the method. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

关键词： System identification Impulse response estimation Kernel-based regularization hyper-parameter estimation Laguerre basis Kautz basis

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Cost free hyper-parameter selection/averaging for Bayesian inverse problems with vanilla and Rao-Blackwellized SMC samplers

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STATISTICS AND COMPUTING 2023年第6期33卷 1-15页

作者： Viani, Alessandro Johansen, Adam M. Sorrentino, Alberto Univ Genoa Dipartimento Matemat I-16146 Genoa Italy Univ Warwick Dept Stat Coventry CV4 7AL England

In Bayesian inverse problems, one aims at characterizing the posterior distribution of a set of unknowns, given indirect measurements. For non-linear/non-Gaussian problems, analytic solutions are seldom available: Sequential Monte Carlo samplers offer a powerful tool for approximating complex posteriors, by constructing an auxiliary sequence of densities that smoothly reaches the posterior. Often the posterior depends on a scalar hyper-parameter, for which limited prior information is available. In this work, we show that properly designed Sequential Monte Carlo (SMC) samplers naturally provide an approximation of the marginal likelihood associated with this hyper-parameter for free, i.e. at a negligible additional computational cost. The proposed method proceeds by constructing the auxiliary sequence of distributions in such a way that each of them can be interpreted as a posterior distribution corresponding to a different value of the hyper-parameter. This can be exploited to perform selection of the hyper-parameter in Empirical Bayes (EB) approaches, as well as averaging across values of the hyper-parameter according to some hyper-prior distribution in Fully Bayesian (FB) approaches. For FB approaches, the proposed method has the further benefit of allowing prior sensitivity analysis at a negligible computational cost. In addition, the proposed method exploits particles at all the (relevant) iterations, thus alleviating one of the known limitations of SMC samplers, i.e. the fact that all samples at intermediate iterations are typically discarded. We show numerical results for two distinct cases where the hyper-parameter affects only the likelihood: a toy example, where an SMC sampler is used to approximate the full posterior distribution;and a brain imaging example, where a Rao-Blackwellized SMC sampler is used to approximate the posterior distribution of a subset of parameters in a conditionally linear Gaussian model.

关键词： Bayesian inverse problems hyper-parameter estimation Sequential Monte Carlo samplers Rao-Blackwellization Empirical Bayes Fully Bayesian

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Approximate Empirical Bayes estimation of the Regularization parameter in l1 Trend Filtering

Approximate Empirical Bayes Estimation of the Regularization...

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IEEE International Symposium on Information Theory (ISIT)

作者： Omae, Akiharu Watanabe, Kazuho Toyohashi Univ Technol Dept Comp Sci & Engn 1-1 HibarigaokaTempaku Cho Toyohashi Aichi 4418580 Japan

ISBN: (纸本)9781665421607;9781665421591

Trend filtering is often used in economics and other fields. l(1) trend filtering was proposed as a derivative of Hodrick-Prescott filtering based on the sparsity in the changes of trends. Although it has a regularization parameter, which needs to be set in advance, the non-conjugacy arising from the l(1) regularization term prohibits a tractable Bayesian treatment including the sequence-dependent estimation of the regularization parameter. In this study, we consider the empirical Bayes estimation of the regularization parameter by approximating the non-conjugate prior distribution in l(1) trend filtering by variational approximation.

关键词： l(1) trend filtering sparsity hyper-parameter estimation variational approximation

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Recursive Implementation of Gaussian Process Regression for Spatial-Temporal Data Modeling 11

Recursive Implementation of Gaussian Process Regression for ...

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11th IEEE International Conference on Wireless Communications and Signal Processing (WCSP)

作者： Kuang, Ye Chen, Tianshi Yin, Feng Zhong, Renxin Chinese Univ Hong Kong Sch Sci & Engn Shenzhen Guangdong Peoples R China Chinese Univ Hong Kong Shenzhen Res Inst Big Data Shenzhen Guangdong Peoples R China Sun Yat Sen Univ Sch Intelligent Syst Engn Guangzhou Guangdong Peoples R China

ISBN: (纸本)9781728135557

In this paper, we consider the spatial-temporal data modeling problem with large number of time instants and moderate number of locations. The problem is formulated as a function estimation problem and then handled by the Gaussian process regression method. To lower the computational complexity, we first sample the continuous-time kernel to get a discrete-time kernel, and then we derive its discrete-time state-space model realization, which is free of numerical problems. Then we convert the Gaussian process regression problem to a Kalman filtering and smoothing problem including both the hyper-parameter estimation and prediction. We consider three hyper-parameter estimation methods: the marginal likelihood maximization method, the generalized cross validation method, and the Stein's unbiased risk estimation method. The proposed implementation is tested over a simulated data set and the Colorado weather data set.

关键词： Gaussian Process Kalman filter and smoother hyper-parameter estimation marginal likelihood maximization SURE generalized cross validation

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Sequential optimization using multi-level cokriging and extended expected improvement criterion

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2018年第3期58卷 1155-1173页

作者： Liu, Yixin Chen, Shishi Wang, Fenggang Xiong, Fenfen Beijing Inst Technol Sch Aerosp Engn Beijing 100081 Peoples R China Minist Educ Key Lab Dynam & Control Flight Vehicle Beijing 100081 Peoples R China Beijing Electromech Engn Inst Beijing 100074 Peoples R China

To reduce the computational cost of metamodel based design optimization that directly relies on the computationally expensive simulation, the multi-fidelity cokriging method has gained increasing attention by fusing data from two or more models with different levels of fidelity. In this paper, an enhanced cokriging based sequential optimization method is proposed. Firstly, the impact of considering full correlation of data among all models on the hyper-parameter estimation during cokriging modeling is investigated by setting up a unified maximum likelihood function. Then, to reduce the computational cost, an extended expected improvement function is established to more reasonably identify the location and fidelity level of the next response evaluation based on the original expected improvement criterion. The results from comparative studies and one airfoil aerodynamic optimization application show that the proposed cokriging based sequential optimization method is more accurate in modeling and efficient in model evaluation than some existing popular approaches, demonstrating its effectiveness and relative merits.

关键词： Cokriging Gaussian process hyper-parameter estimation Sequential sampling

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Multiple Kernel Based Regularized System Identification with SURE hyper-parameter Estimator

Multiple Kernel Based Regularized System Identification with...

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18th IFAC Symposium on System Identification (SYSID)

作者： Hong, Shiying Mu, Biqiang Yin, Feng Andersen, Martin S. Chen, Tianshi Chinese Univ Hong Kong Sch Sci & Engn Shenzhen Peoples R China Chinese Univ Hong Kong Shenzhen Res Inst Big Data Shenzhen Peoples R China Linkoping Univ Dept Elect Engn Linkoping Sweden Tech Univ Denmark Dept Appl Math & Comp Sci Copenhagen Denmark

In this work, we study the multiple kernel based regularized system identification with the hyper-parameter estimated by using the Stein's unbiased risk estimators (SURE). To approach the problem, a QR factorization is first employed to compute SURE's objective function and its gradient in an efficient and accurate way. Then we propose an algorithm to solve the SURE problem, which contains two parts: the outer optimization part and the inner optimization part. For the outer optimization part, the coordinate descent algorithm is used and for the inner optimization part, the projection gradient algorithm is used. Finally, the efficacy of the proposed algorithm is demonstrated by numerical simulations. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

关键词： Linear system identification regularization methods hyper-parameter estimation SURE multiple kernel

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ON SATURATION OF THE CRAMER RAO BOUND FOR SPARSE BAYESIAN LEARNING

ON SATURATION OF THE CRAMER RAO BOUND FOR SPARSE BAYESIAN LE...

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IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： Koochakzadeh, Ali Pal, Piya Univ Calif San Diego Dept Elect & Comp Engn San Diego CA 92103 USA

ISBN: (纸本)9781509041176

This paper analyzes the Cramer-Rao Bound associate? with the estimation of certain sparse hyper-parameters 10 the Sparse Bayesian Learning (SBL) framework, that crucially control the sparsity of the desired signal. The CRB IS shown to exhibit saturation with respect to the number of measurements, i.e., it can be lower bounded by a non-negative quantity that does not go to zero even when the number of measurements tends to infinity. Moreover, the CRB corresponding to the nonzero and zero elements of the sparse hyper-parameter can exhibit different behaviors. While the CRE for the non-zero elements always saturate regardless of the type of dictionary, saturation of the CRB for zero. elements provably happens when the dictionary has normahzed columns. For an un normalized dictionary, singular values of certain sub-dictionaries determine if saturation can happen, prompting future research into this interesting phenomenon.(1)

关键词： Sparse Bayesian Learning Cramer-Rao Bounds Compressed Sensing hyper-parameter estimation mean squared error

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On saturation of the Cramer Rao Bound for Sparse Bayesian Learning

On saturation of the Cramer Rao Bound for Sparse Bayesian Le...

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IEEE International Conference on Acoustics, Speech and Signal Processing

作者： Ali Koochakzadeh Piya Pal Dept. of Electrical and Computer Engineering University of California San Diego USA

ISBN: (纸本)9781509041183

This paper analyzes the Cramer-Rao Bound associated with the estimation of certain sparse hyper-parameters in the Sparse Bayesian Learning (SBL) framework, that crucially control the sparsity of the desired signal. The CRB is shown to exhibit saturation with respect to the number of measurements, i.e., it can be lower bounded by a non-negative quantity that does not go to zero even when the number of measurements tends to infinity. Moreover, the CRB corresponding to the nonzero and zero elements of the sparse hyper-parameter can exhibit different behaviors. While the CRB for the non-zero elements always saturate regardless of the type of dictionary, saturation of the CRB for zero elements provably happens when the dictionary has normalized columns. For an unnormalized dictionary, singular values of certain sub-dictionaries determine if saturation can happen, prompting future research into this interesting phenomenon.

关键词： Sparse Bayesian Learning Cramer-Rao Bounds Compressed Sensing hyper-parameter estimation mean squared error

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Cross Validation and Maximum Likelihood estimations of hyper-parameters of Gaussian processes with model misspecification

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2013年 66卷 55-69页

作者： Bachoc, Francois CEA Saclay DEN DM2S STMFLGLS F-91191 Gif Sur Yvette France Univ Paris 07 Lab Probabilites & Modeles Aleatoires F-75205 Paris 13 France

The Maximum Likelihood (ML) and Cross Validation (CV) methods for estimating covariance hyper-parameters are compared, in the context of Kriging with a misspecified covariance structure. A two-step approach is used. First, the case of the estimation of a single variance hyper-parameter is addressed, for which the fixed correlation function is misspecified. A predictive variance based quality criterion is introduced and a closed-form expression of this criterion is derived. It is shown that when the correlation function is misspecified, the CV does better compared to ML, while ML is optimal when the model is well-specified. In the second step, the results of the first step are extended to the case when the hyper-parameters of the correlation function are also estimated from data. (c) 2013 Elsevier B.V. All rights reserved.

关键词： Uncertainty quantification Metamodel Kriging hyper-parameter estimation Maximum Likelihood Leave-one-out

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A HIERARCHICAL BAYESIAN MODEL FOR FRAME REPRESENTATION

A HIERARCHICAL BAYESIAN MODEL FOR FRAME REPRESENTATION

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2010 IEEE International Conference on Acoustics, Speech, and Signal Processing

作者： Chaari, Lotfi Pesquet, Jean-Christophe Tourneret, Jean-Yves Ciuciu, Philippe Benazza-Benyahia, Amel Univ Paris Est IGM F-77454 Marne La Vallee France Univ Toulouse IRIT ENSEEIHT TSA F-31071 Toulouse France CEA Saclay I2BM DSV F-91191 Gif Sur Yvette France Ecole Super Commun Tunis Cite Technol Commun URISA Tunis 2083 Tunisia

ISBN: (纸本)9781424442966

In many signal processing problems, it may be fruitful to represent the signal under study in a redundant linear decomposition called a frame. If a probabilistic approach is adopted, it becomes then necessary to estimate the hyper-parameters characterizing the probability distribution of the frame coefficients. This problem is difficult since in general, the frame synthesis operator is not bijective and consequently, the frame coefficients are not directly observable. In this work, a hierarchical Bayesian model is introduced to solve this problem. A hybrid MCMC algorithm is subsequently proposed to sample from the derived posterior distribution. We show that through classical Bayesian estimators, this algorithm allows us to determine these hyper-parameters, as well as the frame coefficients in applications to image denoising with uniform noise.

关键词： frame representations Bayesian estimation MCMC hyper-parameter estimation sparsity wavelets

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