@article{greenwade93,
    author  = "George D. Greenwade",
    title   = "The {C}omprehensive {T}ex {A}rchive {N}etwork ({CTAN})",
    year    = "1993",
    journal = "TUGBoat",
    volume  = "14",
    number  = "3",
    pages   = "342--351"
}


%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/


%% Created for Galyardt at 2014-08-21 22:49:39 -0400 


%% Saved with string encoding Unicode (UTF-8) 

@book{williams2006gaussian,
  title={Gaussian processes for machine learning},
  author={Williams, Christopher KI and Rasmussen, Carl Edward},
  volume={2},
  number={3},
  year={2006},
  publisher={MIT Press, Cambridge, MA}
}

@article{cho2009kernel,
  title={Kernel methods for deep learning},
  author={Cho, Youngmin and Saul, Lawrence},
  journal={Advances in Neural Information Processing Systems},
  volume={22},
  year={2009}
}

@article{chambers1976method,
  title={A method for simulating stable random variables},
  author={Chambers, John M and Mallows, Colin L and Stuck, BW4159820341},
  journal={Journal of the American Statistical Association},
  volume={71},
  number={354},
  pages={340--344},
  year={1976},
  publisher={Taylor \& Francis}
}


@article{PolsonScottBayesBridge,
  doi = {10.1111/rssb.12042},
  url = {https://doi.org/10.1111/rssb.12042},
  year = {2013},
  month = nov,
  publisher = {Wiley},
  volume = {76},
  number = {4},
  pages = {713--733},
  author = {Nicholas G. Polson and James G. Scott and Jesse Windle},
  title = {The {Bayesian} bridge}
}

@article{GomezSanchezBridge,
  doi = {10.1080/03610920701762754},
  url = {https://doi.org/10.1080/03610920701762754},
  year = {2008},
  month = feb,
  publisher = {Informa {UK} Limited},
  volume = {37},
  number = {6},
  pages = {972--985},
  author = {E. G{\'{o}}mez-S{\'{a}}nchez-Manzano and M. A. G{\'{o}}mez-Villegas and J. M. Mar{\'{\i}}n},
  title = {Multivariate Exponential Power Distributions as Mixtures of Normal Distributions with Bayesian Applications}
}


@article{MallickYiBridge,
  doi = {10.1080/02664763.2017.1324565},
  url = {https://doi.org/10.1080/02664763.2017.1324565},
  year = {2017},
  month = may,
  publisher = {Informa {UK} Limited},
  volume = {45},
  number = {6},
  pages = {988--1008},
  author = {Himel Mallick and Nengjun Yi},
  title = {Bayesian bridge regression}
}




@InProceedings{ArmaganBridge,
  title = 	 {Variational Bridge Regression},
  author = 	 {Armagan, Artin},
  booktitle = 	 {Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics},
  pages = 	 {17--24},
  year = 	 {2009},
  editor = 	 {van Dyk, David and Welling, Max},
  volume = 	 {5},
  series = 	 {Proceedings of Machine Learning Research},
  address = 	 {Hilton Clearwater Beach Resort, Clearwater Beach, Florida USA},
  month = 	 {16--18 Apr},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v5/armagan09a/armagan09a.pdf},
  url = 	 {https://proceedings.mlr.press/v5/armagan09a.html},
  abstract = 	 {Here we obtain approximate Bayes inferences through variational methods when an exponential power family type prior is specified for the regression coefficients to mimic the characteristics of the Bridge regression. We accomplish this through hierarchical modeling of such priors. Although the mixing distribution is not explicitly stated for scale normal mixtures, we obtain the required moments only to attain the variational distributions for the regression coefficients. By choosing specific values of hyper-parameters (tuning parameters) present in the model, we can mimic the model selection performance of best subset selection in sparse underlying settings. The fundamental difference between MAP, \emphmaximum a posteriori, estimation and the proposed method is that, here we can obtain approximate inferences besides a point estimator. We also empirically analyze the frequentist properties of the estimator obtained. Results suggest that the proposed method yields an estimator that performs significantly better in sparse underlying setups than the existing state-of-the-art procedures in both n&gt;p and p&gt;n scenarios.}
}
@book{barndorff2000probability,
  title={Probability densities and L{\'e}vy densities},
  author={Barndorff-Nielsen, Ole Eiler},
  year={2000},
  publisher={University of Aarhus. Centre for Mathematical Physics and Stochastics~…}
}

@article{gelman2006prior,
  title={Prior distributions for variance parameters in hierarchical models (comment on article by {B}rowne and {D}raper)},
  author={Gelman, Andrew},
  journal={Bayesian Analysis},
  volume={1},
  pages={515--534},
  year={2006}
}

@book{ContTankov,
author = {Cont, Rama. and Tankov, Peter},
address = {Boca Raton, Fla},
booktitle = {Financial modelling with jump processes},
isbn = {1584884134},
keywords = {Finance -- Mathematical models},
language = {eng},
lccn = {2003063470},
publisher = {Chapman \& Hall/CRC},
series = {Chapman \& Hall/CRC financial mathematics series},
title = {Financial modelling with jump processes },
year = {2004},
}



@article{Efron2004,
  doi = {10.1198/016214504000000692},
  url = {https://doi.org/10.1198/016214504000000692},
  year = {2004},
  month = sep,
  publisher = {Informa {UK} Limited},
  volume = {99},
  number = {467},
  pages = {619--632},
  author = {Bradley Efron},
  title = {The Estimation of Prediction Error}
}


@article{West1987,
  doi = {10.1093/biomet/74.3.646},
  url = {https://doi.org/10.1093/biomet/74.3.646},
  year = {1987},
  publisher = {Oxford University Press ({OUP})},
  volume = {74},
  number = {3},
  pages = {646--648},
  author = {Mike West},
  title = {On scale mixtures of normal distributions}
}



@article{ChambersMallows1976,
  doi = {10.1080/01621459.1976.10480344},
  url = {https://doi.org/10.1080/01621459.1976.10480344},
  year = {1976},
  month = jun,
  publisher = {Informa {UK} Limited},
  volume = {71},
  number = {354},
  pages = {340--344},
  author = {J. M. Chambers and C. L. Mallows and B. W. Stuck},
  title = {A Method for Simulating Stable Random Variables}
}

@article{PericchiSmith,
 ISSN = {00359246},
 URL = {http://www.jstor.org/stable/2345859},
 abstract = {The forms of first and second posterior moments for a normal location parameter are identified for a rather general class of prior distributions. Exact and approximate illustrations are given where the prior distribution is double exponential or Student t respectively.},
 author = {L. R. Pericchi and A. F. M. Smith},
 journal = {Journal of the Royal Statistical Society. Series B (Methodological)},
 number = {3},
 pages = {793--804},
 publisher = {[Royal Statistical Society, Wiley]},
 title = {Exact and Approximate Posterior Moments for a Normal Location Parameter},
 volume = {54},
 year = {1992}
}
@article{devroye2009polynomiallytiltedStable,
  title={Random variate generation for exponentially and polynomially tilted stable distributions},
  author={Devroye, Luc},
  journal={ACM Transactions on Modeling and Computer Simulation (TOMACS)},
  volume={19},
  number={4},
  pages={1--20},
  year={2009},
  publisher={ACM New York, NY, USA}
}

@misc{matrixcookbook,
    author = {K. B. Petersen and M. S. Pedersen},
    title = {The Matrix Cookbook},
    year = {2012},
    month = nov,
    keywords = "Matrix identity, matrix relations, inverse, matrix derivative",
    publisher = "Technical University of Denmark",
    address = "",
    note = "Version 20121115",
    url = {http://www2.compute.dtu.dk/pubdb/pubs/3274-full.html},
    abstract = "Matrix identities, relations and approximations. A desktop reference for quick overview of mathematics of matrices."
}
 
@article{GriffinBrown,
author = {Philip J. Brown and Jim E. Griffin},
title = {{Inference with normal-gamma prior distributions in regression problems}},
volume = {5},
journal = {Bayesian Analysis},
number = {1},
publisher = {International Society for Bayesian Analysis},
pages = {171 -- 188},
keywords = {"Spike-and-slab" prior, $p>n$, Bayesian lasso, Markov chain Monte Carlo, multiple regression, normal-gamma prior, Posterior moments, Scale mixture of normals, shrinkage},
year = {2010},
doi = {10.1214/10-BA507},
URL = {https://doi.org/10.1214/10-BA507}
}

 
@article{PSLocalShrinkage,
author = {Polson, Nicholas G. and Scott, James G.},
title = {Local shrinkage rules, Lévy processes and regularized regression},
journal = {Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
volume = {74},
number = {2},
pages = {287-311},
keywords = {Lévy processes, Normal scale mixtures, Partial least squares, Principal components regression, Shrinkage, Sparsity},
doi = {https://doi.org/10.1111/j.1467-9868.2011.01015.x},
url = {https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2011.01015.x},
eprint = {https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9868.2011.01015.x},
abstract = {Summary.  We use Lévy processes to generate joint prior distributions, and therefore penalty functions, for a location parameter as p grows large. This generalizes the class of local–global shrinkage rules based on scale mixtures of normals, illuminates new connections between disparate methods and leads to new results for computing posterior means and modes under a wide class of priors. We extend this framework to large-scale regularized regression problems where p>n, and we provide comparisons with other methodologies.},
year = {2012}
}


@book{ISLR,
booktitle = {An Introduction to statistical learning : with applications in R},
abstract = {Presents an essential statistical learning toolkit for practitioners in science, industry, and other fields. Demonstrates application of the statistical learning methods in R. Includes new chapters on deep learning, survival analysis, and multiple testing. Covers a range of topics, such as linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, and deep learning. Features extensive color graphics for a dynamic learning experience.},
address = {New York},
author = "James, Gareth and Witten, Daniela and Hastie Trevor and Tibshirani Robert",
edition = {2nd ed.},
isbn = {1-0716-1418-5},
keywords = {Statistical -- Theory and Methods},
language = {eng},
publisher = {Springer},
series = {Springer Texts in Statistics},
title = {An Introduction to statistical learning : with applications in R },
year = {2021},
}


@book{Goodfellow,
    title={Deep Learning},
    author={Ian Goodfellow and Yoshua Bengio and Aaron Courville},
    publisher={MIT Press},
    note={\url{http://www.deeplearningbook.org}},
    year={2016}
}




@InProceedings{Peluchetti,
  title = 	 {Stable behaviour of infinitely wide deep neural networks},
  author =       {Peluchetti, Stefano and Favaro, Stefano and Fortini, Sandra},
  booktitle = 	 {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
  pages = 	 {1137--1146},
  year = 	 {2020},
  editor = 	 {Chiappa, Silvia and Calandra, Roberto},
  volume = 	 {108},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {26--28 Aug},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v108/peluchetti20b/peluchetti20b.pdf},
  url = 	 {https://proceedings.mlr.press/v108/peluchetti20b.html},
  abstract = 	 {We consider fully connected feed-forward deep neural networks (NNs) where weights and biases are independent and identically distributed as symmetric centered stable distributions. Then, we show that the infinite wide limit of the NN, under suitable scaling on the weights, is a stochastic process whose finite-dimensional distributions are multivariate stable distributions. The limiting process is referred to as the stable process, and it generalizes the class of Gaussian processes recently obtained as infinite wide limits of NNs (Matthews at al., 2018b). Parameters of the stable process can be computed via an explicit recursion over the layers of the network. Our result contributes to the theory of fully connected feed-forward deep NNs, and it paves the way to expand recent lines of research that rely on Gaussian infinite wide limits.}
}


@book{SamorodnitskyTaqqu,
booktitle = {Stable non-Gaussian random processes : stochastic models with infinite variance},
abstract = {"The familiar Gaussian models do not allow for large deviations and are thus often inadequate for modeling high variability. Non-Gaussian stable models do not possess such limitations. They all share a familiar feature which differentiates them from the Gaussian ones. Their marginal distributions possess heavy "probability tails," always with infinite variance and in some cases with infinite first moment."--BOOK JACKET. "The aim of this book is to make this exciting material easily accessible to graduate students and practitioners. Assuming only a first-year graduate course in probability, it includes material which has appeared only recently in journals and unpublished materials."--BOOK JACKET. "Each chapter begins with a brief overview and concludes with a range of exercises at varying levels of difficulty. Proofs are spelled out in detail. The book includes a discussion of self-similar processes, ARMA, and fractional ARIMA time series with stable innovations."--Jacket.},
author = {Samorodnitsky, G. and Taqqu, Murad.},
address = {New York},
isbn = {0412051710},
keywords = {Gaussian processes},
language = {eng},
lccn = {94013685},
publisher = {Chapman \& Hall},
series = {Stochastic modeling},
title = {Stable non-Gaussian random processes : stochastic models with infinite variance },
year = {1994},
}


@article{FanLiSCADJASA,
author = {Jianqing Fan and Runze Li},
title = {Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties},
journal = {Journal of the American Statistical Association},
volume = {96},
number = {456},
pages = {1348-1360},
year  = {2001},
publisher = {Taylor & Francis},
doi = {10.1198/016214501753382273},

URL = { 
        https://doi.org/10.1198/016214501753382273
    
},
eprint = { 
        https://doi.org/10.1198/016214501753382273
    
}

}

@article{Lin2022,
  doi = {10.3390/plants11050676},
  url = {https://doi.org/10.3390/plants11050676},
  year = {2022},
  month = mar,
  publisher = {{MDPI} {AG}},
  volume = {11},
  number = {5},
  pages = {676},
  author = {Meng-Yang Lin and Valerie Lynch and Dongdong Ma and Hideki Maki and Jian Jin and Mitchell Tuinstra},
  title = {Multi-Species Prediction of Physiological Traits with Hyperspectral Modeling},
  journal = {Plants}
}


@article{ArmaganArtin2013GDPS,
abstract = {We propose a generalized double Pareto prior for Bayesian shrinkage estimation and inferences in linear models. The prior can be obtained via a scale mixture of Laplace or normal distributions, forming a bridge between the Laplace and Normal-Jeffreys' priors. While it has a spike at zero like the Laplace density, it also has a Student's t-like tail behavior. Bayesian computation is straightforward via a simple Gibbs sampling algorithm. We investigate the properties of the maximum a posteriori estimator, as sparse estimation plays an important role in many problems, reveal connections with some well-established regularization procedures, and show some asymptotic results. The performance of the prior is tested through simulations and an application.},
author = {Armagan, Artin and Dunson, David B and Lee, Jaeyong},
address = {China (Republic : 1949- )},
copyright = {2013 STATISTICA SINICA},
issn = {1017-0405},
journal = {Statistica Sinica},
keywords = {A posteriori knowledge ; Analytical estimating ; Density estimation ; Estimators ; Frequentism ; General ; Heavy tails ; high-dimensional data ; Horseshoes ; LASSO ; Machine learning ; maximum a posteriori estimation ; Modeling ; relevance vector machine ; robust prior ; shrinkage estimation ; Standard error ; Threshing},
language = {eng},
number = {1},
pages = {119-143},
publisher = {Institute of Statistical Science, Academia Sinica and International Chinese Statistical Association},
title = {GENERALIZED DOUBLE PARETO SHRINKAGE},
volume = {23},
year = {2013},
}

@incollection{RadfordNeal1996,
abstract = {In this chapter, I show that priors over network parameters can be defined in such a way that the corresponding priors over functions computed by the network reach reasonable limits as the number of hidden units goes to infinity. When using such priors,there is thus no need to limit the size of the network in order to avoid “overfitting”. The infinite network limit also provides insight into the properties of different priors. A Gaussian prior for hidden-to-output weights results in a Gaussian process prior for functions,which may be smooth, Brownian, or fractional Brownian. Quite different effects can be obtained using priors based on non-Gaussian stable distributions. In networks with more than one hidden layer, a combination of Gaussian and non-Gaussian priors appears most interesting.},
author = {Neal, Radford M},
year = {1996},
address = {New York, NY},
booktitle = {Bayesian Learning for Neural Networks},
copyright = {Springer Science+Business Media New York 1996},
isbn = {0387947248},
issn = {0930-0325},
keywords = {Gaussian Process ; Hide Layer ; Hide Unit ; Output Unit ; Prior Distribution},
language = {eng},
pages = {29-53},
publisher = {Springer New York},
series = {Lecture Notes in Statistics},
title = {Priors for Infinite Networks},
}

@inproceedings{DerLee2005,
author = {Der, Ricky and Lee, Daniel},
title = {Beyond {Gaussian} Processes: On the Distributions of Infinite Networks},
year = {2005},
publisher = {MIT Press},
address = {Cambridge, MA, USA},
abstract = {A general analysis of the limiting distribution of neural network functions is performed, with emphasis on non-Gaussian limits. We show that with i.i.d. symmetric stable output weights, and more generally with weights distributed from the normal domain of attraction of a stable variable, that the neural functions converge in distribution to stable processes. Conditions are also investigated under which Gaussian limits do occur when the weights are independent but not identically distributed. Some particularly tractable classes of stable distributions are examined, and the possibility of learning with such processes.},
booktitle = {Proceedings of the 18th International Conference on Neural Information Processing Systems},
pages = {275–282},
numpages = {8},
location = {Vancouver, British Columbia, Canada},
series = {NIPS'05}
}

@article{FranklldikoE1993ASVo,
abstract = {Chemometrics is a field of chemistry that studies the application of statistical methods to chemical data analysis. In addition to borrowing many techniques from the statistics and engineering literatures, chemometrics itself has given rise to several new data-analytical methods. This article examines two methods commonly used in chemometrics for predictive modeling-partial least squares and principal components regression-from a statistical perspective. The goal is to try to understand their apparent successes and in what situations they can be expected to work well and to compare them with other statistical methods intended for those situations. These methods include ordinary least squares, variable subset selection, and ridge regression.},
author = {Frank, lldiko E and Friedman, Jerome H},
copyright = {Copyright Taylor & Francis Group, LLC 1993},
issn = {0040-1706},
journal = {Technometrics},
keywords = {Collinearity ; Covariance matrices ; Eigenvalues ; Error rates ; Least squares ; Linear regression ; Mathematical vectors ; Modeling ; Multiple response regression ; Partial least squares ; Principal components regression ; Regression coefficients ; Ridge regression ; Variable coefficients ; Variable subset selection},
language = {eng},
number = {2},
pages = {109-135},
publisher = {Taylor & Francis Group},
title = {A Statistical View of Some Chemometrics Regression Tools},
volume = {35},
year = {1993},
}

@article{EfronPredictionsEstimationAttribution,
author = {Bradley Efron},
title = {Prediction, Estimation, and Attribution},
journal = {Journal of the American Statistical Association},
volume = {115},
number = {530},
pages = {636-655},
year  = {2020},
publisher = {Taylor & Francis},
doi = {10.1080/01621459.2020.1762613},

URL = { 
        https://doi.org/10.1080/01621459.2020.1762613
    
},
eprint = { 
        https://doi.org/10.1080/01621459.2020.1762613
    
}

}


@book{UchaikinZolotarev,
  url = {https://doi.org/10.1515/9783110935974},
  year = {1999},
  month = dec,
  publisher = {De Gruyter},
  author = {Vladimir V. Uchaikin and Vladimir M. Zolotarev},
  title = {Chance and Stability}
}


@article{PolsonScottHalfCauchyPrior,
author = {Nicholas G. Polson and James G. Scott},
title = {{On the Half-Cauchy Prior for a Global Scale Parameter}},
volume = {7},
journal = {Bayesian Analysis},
number = {4},
publisher = {International Society for Bayesian Analysis},
pages = {887 -- 902},
keywords = {hierarchical models, normal scale mixtures, shrinkage},
year = {2012},
doi = {10.1214/12-BA730},
URL = {https://doi.org/10.1214/12-BA730}
}



@book{LevyPaul1925Cdpp,
author = {Lévy, Paul},
address = {France},
language = {fre},
publisher = {Gauthier-Villars et cie},
title = {Calcul des probabilités},
year = {1925},
}

@article{CybenkoG,
author = {Cybenko, G.},
issn = {0932-4194},
journal = {Mathematics of control, signals, and systems},
language = {eng},
number = {4},
pages = {303-314},
title = {Approximation by superpositions of a sigmoidal function},
volume = {2},
year = {1989},
}

@article{srivastava2014dropout,
  title={Dropout: a simple way to prevent neural networks from overfitting},
  author={Srivastava, Nitish and Hinton, Geoffrey and Krizhevsky, Alex and Sutskever, Ilya and Salakhutdinov, Ruslan},
  journal={The journal of machine learning research},
  volume={15},
  number={1},
  pages={1929--1958},
  year={2014},
  publisher={JMLR. org}
}

@book{SeberLee,
abstract = {Concise, mathematically clear, and comprehensive treatment of the subject. * Expanded coverage of diagnostics and methods of model fitting. * Requires no specialized knowledge beyond a good grasp of matrix algebra and some acquaintance with straight-line regression and simple analysis of variance models. * More than 200 problems throughout the book plus outline solutions for the exercises. * This revision has been extensively class-tested.},
author = {Seber, George A. F and Lee, Alan J},
address = {Hoboken},
isbn = {0471415405},
keywords = {Multivariate analysis ; Regression analysis},
language = {eng},
publisher = {John Wiley & Sons, Incorporated},
title = {Linear Regression Analysis},
year = {2003},
}


@article{TibshiraniLasso,
 ISSN = {00359246},
 URL = {http://www.jstor.org/stable/2346178},
 abstract = {We propose a new method for estimation in linear models. The `lasso' minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models. Our simulation studies suggest that the lasso enjoys some of the favourable properties of both subset selection and ridge regression. It produces interpretable models like subset selection and exhibits the stability of ridge regression. There is also an interesting relationship with recent work in adaptive function estimation by Donoho and Johnstone. The lasso idea is quite general and can be applied in a variety of statistical models: extensions to generalized regression models and tree-based models are briefly described.},
 author = {Robert Tibshirani},
 journal = {Journal of the Royal Statistical Society. Series B (Methodological)},
 number = {1},
 pages = {267--288},
 publisher = {[Royal Statistical Society, Wiley]},
 title = {Regression Shrinkage and Selection via the Lasso},
 volume = {58},
 year = {1996}
}



@article{HoerlRidge,
 ISSN = {00401706},
 URL = {http://www.jstor.org/stable/1271436},
 abstract = {In multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory, if not incorrect, if the prediction vectors are not orthogonal. Proposed is an estimation procedure based on adding small positive quantities to the diagonal of X′X. Introduced is the ridge trace, a method for showing in two dimensions the effects of nonorthogonality. It is then shown how to augment X′X to obtain biased estimates with smaller mean square error.},
 author = {Arthur E. Hoerl and Robert W. Kennard},
 journal = {Technometrics},
 number = {1},
 pages = {80--86},
 publisher = {[Taylor & Francis, Ltd., American Statistical Association, American Society for Quality]},
 title = {Ridge Regression: Biased Estimation for Nonorthogonal Problems},
 volume = {42},
 year = {2000}
}

@book{SedrakyanHayk2018AI,
abstract = {This unique collection of new and classical problems provides full coverage of algebraic inequalities. Many of the exercises are presented with detailed author-prepared-solutions, developing creativity and an arsenal of new approaches for solving mathematical problems. Algebraic Inequalities can be considered a continuation of the book Geometric Inequalities: Methods of Proving by the authors. This book can serve teachers, high-school students, and mathematical competitors. It may also be used as supplemental reading, providing readers with new and classical methods for proving algebraic inequalities.},
author = {Sedrakyan, Hayk and Sedrakyan, Nairi},
address = {Cham},
series = {Problem Books in Mathematics},
title = {Algebraic Inequalities},
year = {2018},
copyright = {Springer International Publishing AG, part of Springer Nature 2018},
isbn = {9783319778358},
issn = {0941-3502},
keywords = {Algebraic Geometry ; Inequalities (Mathematics) ; Mathematics ; Mathematics and Statistics ; Number Theory},
language = {eng},
publisher = {Springer International Publishing AG},
}


@book{Nolan2020,
abstract = {This textbook highlights the many practical uses of stable distributions, exploring the theory, numerical algorithms, and statistical methods used to work with stable laws. Because of the author’s accessible and comprehensive approach, readers will be able to understand and use these methods. Both mathematicians and non-mathematicians will find this a valuable resource for more accurately modelling and predicting large values in a number of real-world scenarios. Beginning with an introductory chapter that explains key ideas about stable laws, readers will be prepared for the more advanced topics that appear later. The following chapters present the theory of stable distributions, a wide range of applications, and statistical methods, with the final chapters focusing on regression, signal processing, and related distributions. Each chapter ends with a number of carefully chosen exercises. Links to free software are included as well, where readers can put these methods into practice. Univariate Stable Distributions is ideal for advanced undergraduate or graduate students in mathematics, as well as many other fields, such as statistics, economics, engineering, physics, and more. It will also appeal to researchers in probability theory who seek an authoritative reference on stable distributions.},
author = {Nolan, John (John P.)},
address = {Cham, Switzerland},
booktitle = {Univariate stable distributions : models for heavy tailed data},
edition = {1st ed. 2020.},
isbn = {3-030-52915-0},
keywords = {Multivariate analysis},
language = {eng},
publisher = {Springer},
series = {Springer Series in Operations Research and Financial Engineering},
title = {Univariate stable distributions : models for heavy tailed data },
year = {2020},
}



@article{BhadraJMLR,
  author  = {Anindya Bhadra and Jyotishka Datta and Yunfan Li and Nicholas G. Polson and Brandon Willard},
  title   = {Prediction Risk for the Horseshoe Regression},
  journal = {Journal of Machine Learning Research},
  year    = {2019},
  volume  = {20},
  number  = {78},
  pages   = {1-39},
  url     = {http://jmlr.org/papers/v20/18-321.html}
}


@article{Lemke2015,
author = {Lemke, Tatjana and Riabiz, Marina and Godsill, S.J.},
year = {2015},
month = {10},
pages = {},
title = {Fully {Bayesian} Inference for $\alpha$-Stable Distributions Using a Poisson Series Representation},
volume = {47},
journal = {Digital Signal Processing},
doi = {10.1016/j.dsp.2015.08.018}
}



@article{GoodmanPollack,
author = {Goodman, Jacob E. and Pollack, Richard},
title = {Multidimensional Sorting},
journal = {SIAM Journal on Computing},
volume = {12},
number = {3},
pages = {484-507},
year = {1983},
doi = {10.1137/0212032},

URL = { 
    
        https://doi.org/10.1137/0212032
    
    

},
eprint = { 
    
        https://doi.org/10.1137/0212032
    
    

}
,
    abstract = { We introduce a process called geometric sorting, which can be applied to an arbitrary configuration of points in d space, and which encodes in compact form the order properties of the configuration, just as the arrangement of a set of numbers in size place encodes its order properties. We give an algorithm for carrying out this sorting procedure in time \$O(n^d \log n)\$, which generalizes the optimum sorting time of \$O(n\log n)\$ for the linear case; In addition, we give an efficient algorithm for determining whether two randomly numbered configurations in \$\mathbb{R}^d \$ have the same order type, using a distinguished family of orderings of each. Finally, we indicate how this new concept of sorting can be applied to problems in pattern recognition, stereochemistry, and cluster analysis. }
}


@article{harding1967number,
  title={The number of partitions of a set of N points in k dimensions induced by hyperplanes},
  author={Harding, Edward Frank},
  journal={Proceedings of the Edinburgh Mathematical Society},
  volume={15},
  number={4},
  pages={285--289},
  year={1967},
  publisher={Cambridge University Press}
}

@MISC {255400,
    TITLE = {$P(X&gt;0,Y&gt;0)$ for a bivariate normal distribution with correlation $\rho$},
    AUTHOR = {Learner (https://math.stackexchange.com/users/48763/learner)},
    HOWPUBLISHED = {Mathematics Stack Exchange},
    NOTE = {URL:https://math.stackexchange.com/q/255400 (version: 2012-12-11)},
    EPRINT = {https://math.stackexchange.com/q/255400},
    URL = {https://math.stackexchange.com/q/255400}
}



@article{DavidF.N.1953Anot,
author = {David, F. N.},
issn = {0006-3444},
journal = {Biometrika},
keywords = {Geometric planes ; Miscellanea ; Recurrence relations},
language = {eng},
number = {3-4},
pages = {458-459},
publisher = {Oxford University Press},
title = {A note on the evaluation of the multivariate normal integral},
volume = {40},
year = {1953},
}





@ARTICLE{GibbsGemanGeman,
  author={Geman, Stuart and Geman, Donald},
  journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, 
  title={Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images}, 
  year={1984},
  volume={PAMI-6},
  number={6},
  pages={721-741},
  doi={10.1109/TPAMI.1984.4767596}}



@MISC{rankone,
    TITLE = {Rank of sum of rank-$1$ matrices},
    AUTHOR = {robjohn (https://math.stackexchange.com/users/13854/robjohn)},
    HOWPUBLISHED = {Mathematics Stack Exchange},
    NOTE = {URL:https://math.stackexchange.com/q/707245 (version: 2014-03-10)},
    EPRINT = {https://math.stackexchange.com/q/707245},
    URL = {https://math.stackexchange.com/q/707245}
}
@article{orthant,
 ISSN = {13697412, 14679868},
 URL = {http://www.jstor.org/stable/20203820},
 abstract = {A new method is introduced for geometrically reconstructing orthant probabilities for non-singular multivariate normal distributions. Orthant probabilities are expressed in terms of those for auto-regressive sequences and an efficient method is developed for numerical approximation of the latter. The approach allows more efficient accurate evaluation of the multivariate normal cumulative distribution function than previously, for many situations where the original distribution arises from a graphical model. An implementation is available as a package for the statistical software R and an application is given to multivariate probit models.},
 author = {Peter Craig},
 journal = {Journal of the Royal Statistical Society. Series B (Statistical Methodology)},
 number = {1},
 pages = {227--243},
 publisher = {[Royal Statistical Society, Wiley]},
 title = {A New Reconstruction of Multivariate Normal Orthant Probabilities},
 urldate = {2023-03-23},
 volume = {70},
 year = {2008}
}



@article{van2017convolutional,
  title={Convolutional gaussian processes},
  author={Van der Wilk, Mark and Rasmussen, Carl Edward and Hensman, James},
  journal={Advances in Neural Information Processing Systems},
  volume={30},
  year={2017}
}


@article{chib,
 ISSN = {00031305},
 URL = {http://www.jstor.org/stable/2684568},
 abstract = {We provide a detailed, introductory exposition of the Metropolis-Hastings algorithm, a powerful Markov chain method to simulate multivariate distributions. A simple, intuitive derivation of this method is given along with guidance on implementation. Also discussed are two applications of the algorithm, one for implementing acceptance-rejection sampling when a blanketing function is not available and the other for implementing the algorithm with block-at-a-time scans. In the latter situation, many different algorithms, including the Gibbs sampler, are shown to be special cases of the Metropolis-Hastings algorithm. The methods are illustrated with examples.},
 author = {Siddhartha Chib and Edward Greenberg},
 journal = {The American Statistician},
 number = {4},
 pages = {327--335},
 publisher = {[American Statistical Association, Taylor & Francis, Ltd.]},
 title = {Understanding the {Metropolis-Hastings} Algorithm},
 urldate = {2023-03-27},
 volume = {49},
 year = {1995}
}

@book{BickelDoksum,
author = {Bickel, Peter J.},
address = {Upper Saddle River, N.J},
booktitle = {Mathematical statistics : basic ideas and selected topics},
edition = {2nd ed.},
isbn = {013850363X},
keywords = {Mathematical statistics},
language = {eng},
lccn = {00031377},
publisher = {Prentice Hall},
series = {Texts in statistical science series},
title = {Mathematical statistics : basic ideas and selected topics },
year = {2001},
}

@Manual{brnn,
    title = {brnn: Bayesian Regularization for Feed-Forward Neural Networks},
    author = {Paulino Perez Rodriguez and Daniel Gianola},
    year = {2022},
    note = {R package version 0.9.2},
    url = {https://CRAN.R-project.org/package=brnn},
  }


@Article{tgp,
    title = {Categorical Inputs, Sensitivity Analysis, Optimization and Importance Tempering with {tgp} Version 2, an {R} Package for Treed {Gaussian} Process Models},
    author = {Robert B. Gramacy and Matthew Taddy},
    journal = {Journal of Statistical Software},
    year = {2010},
    volume = {33},
    number = {6},
    pages = {1--48},
    url = {https://www.jstatsoft.org/v33/i06/},
    doi = {10.18637/jss.v033.i06},
  }

@Article{mlegp,
    author = {Garrett M Dancik and Karin S Dorman},
    title = {{mlegp}: statistical analysis for computer models of biological systems using {R}},
    journal = {Bioinformatics},
    year = {2008},
    volume = {24},
    number = {17},
    pages = {1966}
  }




@article{BretzGenz,
author = {Alan Genz and Frank Bretz},
title = {Comparison of Methods for the Computation of Multivariate t Probabilities},
journal = {Journal of Computational and Graphical Statistics},
volume = {11},
number = {4},
pages = {950-971},
year  = {2002},
publisher = {Taylor & Francis},
doi = {10.1198/106186002394},
URL = { https://doi.org/10.1198/106186002394},
eprint = { https://doi.org/10.1198/106186002394}
}

@inproceedings{pytorch,
author = {Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and Desmaison, Alban and Kopf, Andreas and Yang, Edward and DeVito, Zachary and Raison, Martin and Tejani, Alykhan and Chilamkurthy, Sasank and Steiner, Benoit and Fang, Lu and Bai, Junjie and Chintala, Soumith},
booktitle = {Advances in Neural Information Processing Systems 32},
editor = {Wallach, H. and Larochelle, H. and Beygelzimer, A. and d'Alché-Buc, F. and Fox, E. and Garnett, R.},
pages = {8024--8035},
publisher = {Curran Associates, Inc.},
title = {{PyTorch: An Imperative Style, High-Performance Deep Learning Library}},
url = {http://papers.neurips.cc/paper/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf},
year = {2019}
}

@article{blnn,
title = {BLNN: An R package for training neural networks using Bayesian inference},
journal = {SoftwareX},
volume = {11},
pages = {100432},
year = {2020},
issn = {2352-7110},
doi = {https://doi.org/10.1016/j.softx.2020.100432},
url = {https://www.sciencedirect.com/science/article/pii/S235271101930322X},
author = {Taysseer Sharaf and Theren Williams and Abdallah Chehade and Keshav Pokhrel},
keywords = {R Bayesian neural networks, HMC sampling, No-U-Turn, Evidence procedure, Bayesian learning, Statistical learning},
abstract = {The Bayesian Learning for Neural Networks (BLNN) package coalesces the predictive power of neural networks with a breadth of Bayesian sampling techniques for the first time in R. BLNN offers users Hamiltonian Monte Carlo (HMC) and No-U-Turn (NUTS) sampling algorithms with dual averaging for posterior weight generation. A robust implementation of hyper-parameters and optional re-estimation through the evidence procedure gives BLNN high predictive precision. BLNN is compatible with RStan diagnostic tool ShinyStan. BLNN can be used in a wide range of applications which are based on developing statistical models such as multiple linear and logistic regression, classification, and survival analysis.}
}


@article{diggle1998model,
  title={Model-based geostatistics},
  author={Diggle, Peter J and Tawn, Jonathan A and Moyeed, Rana A},
  journal={Journal of the Royal Statistical Society Series C: Applied Statistics},
  volume={47},
  number={3},
  pages={299--350},
  year={1998},
  publisher={Oxford University Press}
}

@inproceedings{bickel1967some,
  title={Some contributions to the theory of order statistics},
  author={Bickel, Peter J},
  booktitle={Proc. Fifth Berkeley Sympos. Math. Statist. and Probability},
  volume={1},
  pages={575--591},
  year={1967}
}


@inproceedings{williams1996,
 author = {Williams, Christopher},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {M.C. Mozer and M. Jordan and T. Petsche},
 pages = {},
 publisher = {MIT Press},
 title = {Computing with Infinite Networks},
 url = {https://proceedings.neurips.cc/paper_files/paper/1996/file/ae5e3ce40e0404a45ecacaaf05e5f735-Paper.pdf},
 volume = {9},
 year = {1996}
}


@InProceedings{pmlr-v202-yang23k,
  title = 	 {A theory of representation learning gives a deep generalisation of kernel methods},
  author =       {Yang, Adam X. and Robeyns, Maxime and Milsom, Edward and Anson, Ben and Schoots, Nandi and Aitchison, Laurence},
  booktitle = 	 {Proceedings of the 40th International Conference on Machine Learning},
  pages = 	 {39380--39415},
  year = 	 {2023},
  editor = 	 {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan},
  volume = 	 {202},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {23--29 Jul},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v202/yang23k/yang23k.pdf},
  url = 	 {https://proceedings.mlr.press/v202/yang23k.html},
  abstract = 	 {The successes of modern deep machine learning methods are founded on their ability to transform inputs across multiple layers to build good high-level representations. It is therefore critical to understand this process of representation learning. However, standard theoretical approaches (formally NNGPs) involving infinite width limits eliminate representation learning. We therefore develop a new infinite width limit, the Bayesian representation learning limit, that exhibits representation learning mirroring that in finite-width models, yet at the same time, retains some of the simplicity of standard infinite-width limits. In particular, we show that Deep Gaussian processes (DGPs) in the Bayesian representation learning limit have exactly multivariate Gaussian posteriors, and the posterior covariances can be obtained by optimizing an interpretable objective combining a log-likelihood to improve performance with a series of KL-divergences which keep the posteriors close to the prior. We confirm these results experimentally in wide but finite DGPs. Next, we introduce the possibility of using this limit and objective as a flexible, deep generalisation of kernel methods, that we call deep kernel machines (DKMs). Like most naive kernel methods, DKMs scale cubically in the number of datapoints. We therefore use methods from the Gaussian process inducing point literature to develop a sparse DKM that scales linearly in the number of datapoints. Finally, we extend these approaches to NNs (which have non-Gaussian posteriors) in the Appendices.}
}



@InProceedings{pmlr-v139-aitchison21a,
  title = 	 {Deep Kernel Processes},
  author =       {Aitchison, Laurence and Yang, Adam and Ober, Sebastian W},
  booktitle = 	 {Proceedings of the 38th International Conference on Machine Learning},
  pages = 	 {130--140},
  year = 	 {2021},
  editor = 	 {Meila, Marina and Zhang, Tong},
  volume = 	 {139},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {18--24 Jul},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v139/aitchison21a/aitchison21a.pdf},
  url = 	 {https://proceedings.mlr.press/v139/aitchison21a.html},
  abstract = 	 {We define deep kernel processes in which positive definite Gram matrices are progressively transformed by nonlinear kernel functions and by sampling from (inverse) Wishart distributions. Remarkably, we find that deep Gaussian processes (DGPs), Bayesian neural networks (BNNs), infinite BNNs, and infinite BNNs with bottlenecks can all be written as deep kernel processes. For DGPs the equivalence arises because the Gram matrix formed by the inner product of features is Wishart distributed, and as we show, standard isotropic kernels can be written entirely in terms of this Gram matrix — we do not need knowledge of the underlying features. We define a tractable deep kernel process, the deep inverse Wishart process, and give a doubly-stochastic inducing-point variational inference scheme that operates on the Gram matrices, not on the features, as in DGPs. We show that the deep inverse Wishart process gives superior performance to DGPs and infinite BNNs on fully-connected baselines.}
}


@inproceedings{
matthews2018,
title={Gaussian Process Behaviour in Wide Deep Neural Networks},
author={Alexander G. de G. Matthews and Jiri Hron and Mark Rowland and Richard E. Turner and Zoubin Ghahramani},
booktitle={International Conference on Learning Representations},
year={2018},
url={https://openreview.net/forum?id=H1-nGgWC-},
}


@inproceedings{
novak2019bayesian,
title={Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes},
author={Roman Novak and Lechao Xiao and Yasaman Bahri and Jaehoon Lee and Greg Yang and Daniel A. Abolafia and Jeffrey Pennington and Jascha Sohl-dickstein},
booktitle={International Conference on Learning Representations},
year={2019},
url={https://openreview.net/forum?id=B1g30j0qF7},
}

@inproceedings{fortuin2022bayesian,
  title={{B}ayesian Neural Network Priors Revisited},
  author={Fortuin, Vincent and Garriga-Alonso, Adri{\`a} and Ober, Sebastian W and Wenzel, Florian and R{\"a}tsch, Gunnar and Turner, Richard E and van der Wilk, Mark and Aitchison, Laurence},
  booktitle={International Conference on Learning Representations},
  year={2022},
  url={https://openreview.net/forum?id=xkjqJYqRJy},
  arxiv={https://arxiv.org/abs/2102.06571}
}


@misc{caron2023overparameterised,
      title={Over-parameterised Shallow Neural Networks with Asymmetrical Node Scaling: Global Convergence Guarantees and Feature Learning}, 
      author={Francois Caron and Fadhel Ayed and Paul Jung and Hoil Lee and Juho Lee and Hongseok Yang},
      year={2023},
      eprint={2302.01002},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@article{Fortuin2022Review,
author = {Fortuin, Vincent},
title = {Priors in Bayesian Deep Learning: A Review},
journal = {International Statistical Review},
volume = {90},
number = {3},
pages = {563-591},
keywords = {Bayesian deep learning, Bayesian learning, deep learning, priors},
doi = {https://doi.org/10.1111/insr.12502},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/insr.12502},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1111/insr.12502},
abstract = {Summary While the choice of prior is one of the most critical parts of the Bayesian inference workflow, recent Bayesian deep learning models have often fallen back on vague priors, such as standard Gaussians. In this review, we highlight the importance of prior choices for Bayesian deep learning and present an overview of different priors that have been proposed for (deep) Gaussian processes, variational autoencoders and Bayesian neural networks. We also outline different methods of learning priors for these models from data. We hope to motivate practitioners in Bayesian deep learning to think more carefully about the prior specification for their models and to provide them with some inspiration in this regard.},
year = {2022}
}


@misc{lee2023deep,
      title={Deep neural networks with dependent weights: Gaussian Process mixture limit, heavy tails, sparsity and compressibility}, 
      author={Hoil Lee and Fadhel Ayed and Paul Jung and Juho Lee and Hongseok Yang and François Caron},
      year={2023},
      eprint={2205.08187},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@inproceedings{yang2021tensor,
      title={Tensor Programs {I}: Wide Feedforward or Recurrent Neural Networks of Any Architecture are {G}aussian Processes}, 
      author={Greg Yang},
   booktitle = {Advances in Neural Information Processing Systems},
 editor = {H. Wallach and H. Larochelle and A. Beygelzimer and F. d\textquotesingle Alch\'{e}-Buc and E. Fox and R. Garnett},
 pages = {},
 publisher = {Curran Associates, Inc.},
url = {https://proceedings.neurips.cc/paper_files/paper/2019/file/5e69fda38cda2060819766569fd93aa5-Paper.pdf},
 volume = {32},
 year = {2019}
}

@misc{bracale2022infinitechannel,
   title={Infinite-channel deep stable convolutional neural networks}, 
      author={Daniele Bracale and Stefano Favaro and Sandra Fortini and Stefano Peluchetti},
      year={2022},
      eprint={2102.03739},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@article{
bordino2022infinitely,
title={Infinitely wide limits for deep Stable neural networks: sub-linear, linear and super-linear activation functions},
author={Alberto Bordino and Stefano Favaro and Sandra Fortini},
journal={Transactions on Machine Learning Research},
issn={2835-8856},
year={2022},
url={https://openreview.net/forum?id=A5tIluhDW6},
note={}
}


@misc{yang2023theory,
      title={A theory of representation learning gives a deep generalisation of kernel methods}, 
      author={Adam X. Yang and Maxime Robeyns and Edward Milsom and Ben Anson and Nandi Schoots and Laurence Aitchison},
      year={2023},
      eprint={2108.13097},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}

@misc{aitchison2021deep,
      title={Deep kernel processes}, 
      author={Laurence Aitchison and Adam X. Yang and Sebastian W. Ober},
      year={2021},
      eprint={2010.01590},
      archivePrefix={arXiv},
      primaryClass={stat.ML}
}


@article{IChengTzuKuang,
title = {Building real estate valuation models with comparative approach through case-based reasoning},
journal = {Applied Soft Computing},
volume = {65},
pages = {260-271},
year = {2018},
issn = {1568-4946},
doi = {https://doi.org/10.1016/j.asoc.2018.01.029},
url = {https://www.sciencedirect.com/science/article/pii/S1568494618300358},
author = {I-Cheng Yeh and Tzu-Kuang Hsu},
keywords = {Quantitative, Comparative approach, Case-based reasoning, Real estate, Appraisal},
abstract = {The purpose of this study is to propose an innovative real estate valuation approach called Quantitative Comparative Approach, which can estimate correction coefficients to overcome the shortcomings of subjective decisions of correction coefficients of traditional comparative approach. The principle is to assume that the price per unit area of real estate is the average price per unit area of the particular circle of housing supply and demand multiplied by the product of several dimensionless adjustment coefficients of factors. The single regression models of these adjustment coefficients can be built with the stepwise decomposition regression analysis. Then the adjustment coefficients of comparative cases and target case can be estimated with these single regression models, and finally the correction coefficients can be estimated by dividing the adjustment coefficients of target case by those of comparative case. The empirical samples are collected from four circles of supply and demand, and are divided into four data sets. The empirical results show that the Quantitative Comparative Approach is more accurate than the two classical Hedonic price approaches, multivariate regression analysis and neural networks.}
}