[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-126533-en":3,"doc-seo-126533-105":31,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},126533,962085662650,"Jiven","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Parametrization Cookbook - A set of Bijective Parametrizations for using Machine Learning methods in Statistical Inference - overview","The paper presents a method to convert constrained statistical inference problems into unconstrained formulations, enabling the use of modern computational techniques including automatic differentiation, GPU acceleration, and stochastic gradients with mini-batches. The proposed parametrizations are bijective and even diffeomorphisms, preserving key statistical properties, especially identifiability. A cookbook-style set of recipes and a Python package are provided to implement transformations via numpy, JAX, and PyTorch with an expressive interface.","arXiv :2301 .08297v1 [ stat .CO] 19 Jan 2023  \nParametrization Cookbook  \nA set of Bijective Parametrizations for using Machine Learning methods in Statistical Inference  \nJean-Benoist Leger Université de technologie de Compiègne, CNRS, Heudiasyc,  \nCompiègne, France  \nUniversité Paris-Saclay, AgroParisTech, INRAE, UMR MIA  \nParis-Saclay, Palaiseau, France  \nAbstract  \nWe present in this paper a way to transform a constrained statistical inference problem into an unconstrained one in order to be able to use modern computational methods, such as those based on automatic di􀀘erentiation, GPU computing, stochastic gradients with mini-batch.  \nUnlike the parametrizations classically used in Machine Learning, the parametrizations introduced here are all bijective and are even di􀀘eomorphisms, thus allowing to keep the important properties from a statistical inference point of view, 􀀛rst of all identi􀀛ability.  \nThis cookbook presents a set of recipes to use to transform a constrained problem into a unconstrained one.  \nFor an easy use of parametrizations, this paper is at the same time a cookbook, anda Python package allowing the use of parametrizations with numpy, but also JAX and PyTorch, as well as a high level and expressive interface allowing to easily describe a parametrization to transform a di􀀝cult problem of statistical inference into an easier problem addressable with modern optimization tools.  \nKeywords: computational statistics; parametrizations  \nTable of contents  \n1 Introduction 3  \n2 De􀀛nitions of parametrizations 4  \n2.1 Notations ....................................... 4  \n2.1.1 General notations .............................. 4  \n2.1.2 Vectors notations .............................. 5  \n2.1.3 Matrix notations .............................. 6  \n2.2 Parametrization of scalars ............................. 6  \n2.2.1 R and forms like ( 􀀀1 ; a) or (a; +1 ) ................. 6  \n2.2.2 (0 ; 1) and forms like (a; b) ......................... 10  \n2.3 Parametrization of vectors ............................. 12  \n2.3.1 Simplex Sn ................................. 12  \n2.3.2 Sphere S n .................................. 18  \n2.3.3 Half sphere HS n .............................. 22  \n2.3.4 Ball Bn ................................... 25  \n2.4 Parametrization of matrices ............................ 31  \n2.4.1 Diagonal matrices ............................. 31  \n2.4.2 Symmetric matrices ............................ 32  \n2.4.3 Diagonal positive de􀀛nite matrices .................... 33  \n2.4.4 Symmetric positive de􀀛nite matrices ................... 34  \n2.4.5 Correlation matrices ............................ 37  \n3 Python package parametrization_cookbook 39  \n3.1 High-level modules: an easy way to de􀀛ne and use a parametrization ..... 39  \n3.1.1 Common API of all parametrization classes ............... 40  \n3.1.2 Parametrization of scalars ......................... 41  \n3.1.3 Parametrization of vectors ......................... 43  \n3.1.4 Parametrization of matrices ........................ 44  \n3.1.5 Parametrization of Cartesian product ................... 46  \n3.1.6 De􀀛nition of custom parametrization ................... 48  \n3.2 Low-level module: access to elementary functions ................ 50  \n3.2.1 Parametrization of scalars ......................... 50  \n3.2.2 Parametrization of vectors ......................... 51  \n3.2.3 Parametrization of matrices ........................ 51  \n3.2.4 Vectorization ................................ 52  \n4 Examples 54  \n4.1 High-level modules: inference of parameters ofa Multivariate Student distribution 54  \n4.1.1 Introduction ................................. 54  \n4.1.2 With JAX .................................. 56  \n4.1.3 With PyTorch ................................ 60  \n4.2 Low-level modules: inference of parameters of a Gumbel distribution ..... 65  \n4.2.1 Introduction ................................. 65  \n4.2.2 With JAX .................................. 66  \n4.2.3 With PyTorch","cbCaiseFjTxTqI2B","https://ap.wps.com/l/cbCaiseFjTxTqI2B","pdf",1332883,2,1,73,"English","en",105,"# 1 Introduction\n# 2 Definitions of parametrizations\n## 2.1 Notations\n## 2.2 Parametrization of scalars\n## 2.3 Parametrization of vectors\n## 2.4 Parametrization of matrices\n# 3 Python package parametrization_cookbook\n## 3.1 High-level modules\n## 3.2 Low-level module\n# 4 Examples\n## 4.1 Multivariate Student distribution\n## 4.2 Gumbel distribution","[{\"question\":\"How does the paper turn a constrained inference problem into an unconstrained one?\",\"answer\":\"It introduces bijective parametrizations that map constrained parameter spaces to an unconstrained space, allowing standard unconstrained optimization and inference workflows.\"},{\"question\":\"Why are bijective (and diffeomorphic) parametrizations important here?\",\"answer\":\"Bijectivity, and the stronger diffeomorphism property, help preserve essential statistical properties such as identifiability while enabling differentiable optimization.\"},{\"question\":\"What software tools are provided to use the parametrizations in practice?\",\"answer\":\"The paper provides a Python package and recipes with support for numpy, JAX, and PyTorch, together with a high-level interface to define parametrizations efficiently.\"}]","Parametrization Cookbook - A set of Bijective Parametrizations for using Machine Learning methods in Statistical Inference - overview | 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does the paper turn a constrained inference problem into an unconstrained one?","Question",{"text":76,"@type":77},"It introduces bijective parametrizations that map constrained parameter spaces to an unconstrained space, allowing standard unconstrained optimization and inference workflows.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why are bijective (and diffeomorphic) parametrizations important here?",{"text":81,"@type":77},"Bijectivity, and the stronger diffeomorphism property, help preserve essential statistical properties such as identifiability while enabling differentiable optimization.",{"name":83,"@type":74,"acceptedAnswer":84},"What software tools are provided to use the parametrizations in practice?",{"text":85,"@type":77},"The paper provides a Python package and recipes with support for numpy, JAX, and PyTorch, together with a high-level interface to define parametrizations 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