{"title":"The Secret Life of I. J. Good","authors":"S. Zabell","doi":"10.1214/22-sts870","DOIUrl":"https://doi.org/10.1214/22-sts870","url":null,"abstract":"","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46359832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. David John Aldous was born in Exeter U.K. on July 13, 1952. He received a B.A. and Ph.D. in Mathematics in 1973 and 1977, respectively from Cambridge. After spending two years as a research fellow at St. John’s College, Cambridge, he joined the Department of Statistics at the University of California, Berkeley in 1979 where he spent the rest of his academic career until retiring in 2018. He is known for seminal contributions on many topics within probability including weak convergence and tightness, exchangeability, Markov chain mixing times, Poisson clumping heuristic and limit theory for large discrete random structures including random trees, stochastic coagulation and fragmentation systems, models of complex networks and interacting particle systems on such structures. For his contributions to the field, he has received numerous honors and awards including the Rollo David-son prize in 1980, the inaugural Loeve prize in Probability in 1993, and the Brouwer medal in 2021, and being elected as an IMS fellow in 1985, Fellow of the Royal Society in 1994, Fellow of the American Academy of Arts and Sciences in 2004, elected to the National Academy of Sciences (foreign associate) in 2010, ICM plenary speaker in 2010 and AMS fellow in 2012.
{"title":"A Conversation with David J. Aldous","authors":"S. Bhamidi","doi":"10.1214/22-sts849","DOIUrl":"https://doi.org/10.1214/22-sts849","url":null,"abstract":". David John Aldous was born in Exeter U.K. on July 13, 1952. He received a B.A. and Ph.D. in Mathematics in 1973 and 1977, respectively from Cambridge. After spending two years as a research fellow at St. John’s College, Cambridge, he joined the Department of Statistics at the University of California, Berkeley in 1979 where he spent the rest of his academic career until retiring in 2018. He is known for seminal contributions on many topics within probability including weak convergence and tightness, exchangeability, Markov chain mixing times, Poisson clumping heuristic and limit theory for large discrete random structures including random trees, stochastic coagulation and fragmentation systems, models of complex networks and interacting particle systems on such structures. For his contributions to the field, he has received numerous honors and awards including the Rollo David-son prize in 1980, the inaugural Loeve prize in Probability in 1993, and the Brouwer medal in 2021, and being elected as an IMS fellow in 1985, Fellow of the Royal Society in 1994, Fellow of the American Academy of Arts and Sciences in 2004, elected to the National Academy of Sciences (foreign associate) in 2010, ICM plenary speaker in 2010 and AMS fellow in 2012.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44257148","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Accurate diagnosis of disease is of fundamental importance in clinical practice and medical research. Before a medical diagnostic test is routinely used in practice, its ability to distinguish between diseased and nondiseased states must be rigorously assessed. The receiver operating characteristic (ROC) curve is the most popular used tool for evaluating the diagnostic accuracy of continuous-outcome tests. It has been acknowledged that several factors (e.g., subject-specific characteristics such as age and/or gender) can affect the test outcomes and accuracy beyond disease status. Recently, the covariate-adjusted ROC curve has been proposed and successfully applied as a global summary measure of diagnostic accuracy that takes covariate information into account. The aim of this paper is three-fold. First, we motivate the importance of including covariate-information, whenever available, in ROC analysis and, in particular, how the covariate-adjusted ROC curve is an important tool in this context. Second, we review and provide insight on the existing approaches for estimating the covariate-adjusted ROC curve. Third, we develop a highly flexible Bayesian method, based on the combination of a Dirichlet process mixture of additive normal models and the Bayesian bootstrap, for conducting inference about the covariate-adjusted ROC curve. A simulation study is conducted to assess the performance of the different methods and it also demonstrates the ability of our proposed Bayesian model to successfully recover the true covariate-adjusted ROC curve and to produce valid inferences in a variety of complex scenarios. The methods are applied to an endocrine study where the goal is to assess the accuracy of the body mass index, adjusted for age and gender, for detecting clusters of cardiovascular disease risk factors. key words: Classification accuracy; Covariate-adjustment; Decision threshold; Diagnostic test; Dirichlet process (mixture) model; Receiver operating characteristic curve. Vanda Inácio, School of Mathematics, University of Edinburgh, Scotland, UK (vanda.inacio@ed.ac.uk). Maŕıa Xosé RodŕıguezÁlvarez, BCAM-Basque Center for Applied Mathematics & IKERBASQUE, Basque Foundation for Science, Bilbao, Basque Country, Spain (mxrodriguez@bcamath.org).
{"title":"The Covariate-Adjusted ROC Curve: The Concept and Its Importance, Review of Inferential Methods, and a New Bayesian Estimator","authors":"Vanda Inácio, M. Rodríguez-Álvarez","doi":"10.1214/21-sts839","DOIUrl":"https://doi.org/10.1214/21-sts839","url":null,"abstract":"Accurate diagnosis of disease is of fundamental importance in clinical practice and medical research. Before a medical diagnostic test is routinely used in practice, its ability to distinguish between diseased and nondiseased states must be rigorously assessed. The receiver operating characteristic (ROC) curve is the most popular used tool for evaluating the diagnostic accuracy of continuous-outcome tests. It has been acknowledged that several factors (e.g., subject-specific characteristics such as age and/or gender) can affect the test outcomes and accuracy beyond disease status. Recently, the covariate-adjusted ROC curve has been proposed and successfully applied as a global summary measure of diagnostic accuracy that takes covariate information into account. The aim of this paper is three-fold. First, we motivate the importance of including covariate-information, whenever available, in ROC analysis and, in particular, how the covariate-adjusted ROC curve is an important tool in this context. Second, we review and provide insight on the existing approaches for estimating the covariate-adjusted ROC curve. Third, we develop a highly flexible Bayesian method, based on the combination of a Dirichlet process mixture of additive normal models and the Bayesian bootstrap, for conducting inference about the covariate-adjusted ROC curve. A simulation study is conducted to assess the performance of the different methods and it also demonstrates the ability of our proposed Bayesian model to successfully recover the true covariate-adjusted ROC curve and to produce valid inferences in a variety of complex scenarios. The methods are applied to an endocrine study where the goal is to assess the accuracy of the body mass index, adjusted for age and gender, for detecting clusters of cardiovascular disease risk factors. key words: Classification accuracy; Covariate-adjustment; Decision threshold; Diagnostic test; Dirichlet process (mixture) model; Receiver operating characteristic curve. Vanda Inácio, School of Mathematics, University of Edinburgh, Scotland, UK (vanda.inacio@ed.ac.uk). Maŕıa Xosé RodŕıguezÁlvarez, BCAM-Basque Center for Applied Mathematics & IKERBASQUE, Basque Foundation for Science, Bilbao, Basque Country, Spain (mxrodriguez@bcamath.org).","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43020500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01Epub Date: 2022-10-13DOI: 10.1214/21-sts835
Seyoon Ko, Hua Zhou, Jin J Zhou, Joong-Ho Won
Technological advances in the past decade, hardware and software alike, have made access to high-performance computing (HPC) easier than ever. We review these advances from a statistical computing perspective. Cloud computing makes access to supercomputers affordable. Deep learning software libraries make programming statistical algorithms easy and enable users to write code once and run it anywhere-from a laptop to a workstation with multiple graphics processing units (GPUs) or a supercomputer in a cloud. Highlighting how these developments benefit statisticians, we review recent optimization algorithms that are useful for high-dimensional models and can harness the power of HPC. Code snippets are provided to demonstrate the ease of programming. We also provide an easy-to-use distributed matrix data structure suitable for HPC. Employing this data structure, we illustrate various statistical applications including large-scale positron emission tomography and ℓ1-regularized Cox regression. Our examples easily scale up to an 8-GPU workstation and a 720-CPU-core cluster in a cloud. As a case in point, we analyze the onset of type-2 diabetes from the UK Biobank with 200,000 subjects and about 500,000 single nucleotide polymorphisms using the HPC ℓ1-regularized Cox regression. Fitting this half-million-variate model takes less than 45 minutes and reconfirms known associations. To our knowledge, this is the first demonstration of the feasibility of penalized regression of survival outcomes at this scale.
{"title":"High-Performance Statistical Computing in the Computing Environments of the 2020s.","authors":"Seyoon Ko, Hua Zhou, Jin J Zhou, Joong-Ho Won","doi":"10.1214/21-sts835","DOIUrl":"10.1214/21-sts835","url":null,"abstract":"<p><p>Technological advances in the past decade, hardware and software alike, have made access to high-performance computing (HPC) easier than ever. We review these advances from a statistical computing perspective. Cloud computing makes access to supercomputers affordable. Deep learning software libraries make programming statistical algorithms easy and enable users to write code once and run it anywhere-from a laptop to a workstation with multiple graphics processing units (GPUs) or a supercomputer in a cloud. Highlighting how these developments benefit statisticians, we review recent optimization algorithms that are useful for high-dimensional models and can harness the power of HPC. Code snippets are provided to demonstrate the ease of programming. We also provide an easy-to-use distributed matrix data structure suitable for HPC. Employing this data structure, we illustrate various statistical applications including large-scale positron emission tomography and <i>ℓ</i><sub>1</sub>-regularized Cox regression. Our examples easily scale up to an 8-GPU workstation and a 720-CPU-core cluster in a cloud. As a case in point, we analyze the onset of type-2 diabetes from the UK Biobank with 200,000 subjects and about 500,000 single nucleotide polymorphisms using the HPC <i>ℓ</i><sub>1</sub>-regularized Cox regression. Fitting this half-million-variate model takes less than 45 minutes and reconfirms known associations. To our knowledge, this is the first demonstration of the feasibility of penalized regression of survival outcomes at this scale.</p>","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":"37 4","pages":"494-518"},"PeriodicalIF":3.9,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10168006/pdf/nihms-1884249.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9502219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Pawitan and Lee (2021) attempt to show a correspondence between confidence and likelihood, specifically, that “confidence is in fact an extended likelihood” (Pawitan and Lee, 2021, abstract). The word “extended” means that the likelihood function can accommodate unobserved random variables such as random effects and future values; see Bjørnstad (1996) for details. Here we argue that the extended likelihood presented by Pawitan and Lee (2021) is not the correct extended likelihood and does not justify interpreting confidence as likelihood.
Pawitan和Lee（2021）试图展示信心和可能性之间的对应关系，特别是“信心实际上是一种扩展的可能性”（Pawitan and Lee，2021，摘要）。“扩展”一词意味着似然函数可以容纳未观察到的随机变量，如随机效应和未来值；详见Bjørnstad（1996）。在这里，我们认为Pawitan和Lee（2021）提出的扩展可能性不是正确的扩展可能性，也不能证明将置信度解释为可能性是合理的。
{"title":"Comments on Confidence as Likelihood by Pawitan and Lee in Statistical Science, November 2021","authors":"M. Lavine, J. F. Bjørnstad","doi":"10.1214/22-sts862","DOIUrl":"https://doi.org/10.1214/22-sts862","url":null,"abstract":". Pawitan and Lee (2021) attempt to show a correspondence between confidence and likelihood, specifically, that “confidence is in fact an extended likelihood” (Pawitan and Lee, 2021, abstract). The word “extended” means that the likelihood function can accommodate unobserved random variables such as random effects and future values; see Bjørnstad (1996) for details. Here we argue that the extended likelihood presented by Pawitan and Lee (2021) is not the correct extended likelihood and does not justify interpreting confidence as likelihood.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42862716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a seminal paper, Sejdinovic, et al. [49] showed the equivalence of the Hilbert-Schmidt Independence Criterion (HSIC) [20] and a generalization of distance covariance [62]. In this paper the two notions of dependence are unified with a third prominent concept for independence testing, the “global test” introduced in [16]. The new viewpoint provides novel insights into all three test traditions, as well as a unified overall view of the way all three tests contrast with classical association tests. As our main result, a regression perspective on HSIC and generalized distance covariance is obtained, allowing such tests to be used with nuisance covariates or for survival data. Several more examples of cross-fertilization of the three traditions are provided, involving theoretical results and novel methodology. To illustrate the difference between classical statistical tests and the unified HSIC/distance covariance/global tests we investigate the case of association between two categorical variables in depth.
{"title":"A Regression Perspective on Generalized Distance Covariance and the Hilbert–Schmidt Independence Criterion","authors":"Dominic Edelmann, J. Goeman","doi":"10.1214/21-sts841","DOIUrl":"https://doi.org/10.1214/21-sts841","url":null,"abstract":"In a seminal paper, Sejdinovic, et al. [49] showed the equivalence of the Hilbert-Schmidt Independence Criterion (HSIC) [20] and a generalization of distance covariance [62]. In this paper the two notions of dependence are unified with a third prominent concept for independence testing, the “global test” introduced in [16]. The new viewpoint provides novel insights into all three test traditions, as well as a unified overall view of the way all three tests contrast with classical association tests. As our main result, a regression perspective on HSIC and generalized distance covariance is obtained, allowing such tests to be used with nuisance covariates or for survival data. Several more examples of cross-fertilization of the three traditions are provided, involving theoretical results and novel methodology. To illustrate the difference between classical statistical tests and the unified HSIC/distance covariance/global tests we investigate the case of association between two categorical variables in depth.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46642491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of constructing a reasonably simple yet well behaved confidence interval for a binomial parameter p is old but still fascinating and surprisingly complex. During the last century many alternatives to the poorly behaved standard Wald interval have been suggested. It seems though that the Wald interval is still much in use in spite of many efforts over the years through publications to point out its deficiencies. This paper constitutes yet another attempt to provide an alternative and it builds on a special case of a general technique for adjusted intervals primarily based on Wald type statistics. The main idea is to construct an approximate pivot with uncorrelated, or nearly uncorrelated, components. The resulting (AN) Andersson-Nerman interval, as well as a modification thereof, is compared with the well renowned Wilson and AC (Agresti-Coull) intervals and the subsequent discussion will in itself hopefully shed some new light on this seemingly elementary interval estimation situation. Generally, an alternative to the Wald interval is to be judged not only by performance, its expression should also indicate why we will obtain a better behaved interval. It is argued that the well-behaved AN interval meets this requirement.
{"title":"Approximate Confidence Intervals for a Binomial p—Once Again","authors":"Per Gösta Andersson","doi":"10.1214/21-sts837","DOIUrl":"https://doi.org/10.1214/21-sts837","url":null,"abstract":"The problem of constructing a reasonably simple yet well behaved confidence interval for a binomial parameter p is old but still fascinating and surprisingly complex. During the last century many alternatives to the poorly behaved standard Wald interval have been suggested. It seems though that the Wald interval is still much in use in spite of many efforts over the years through publications to point out its deficiencies. This paper constitutes yet another attempt to provide an alternative and it builds on a special case of a general technique for adjusted intervals primarily based on Wald type statistics. The main idea is to construct an approximate pivot with uncorrelated, or nearly uncorrelated, components. The resulting (AN) Andersson-Nerman interval, as well as a modification thereof, is compared with the well renowned Wilson and AC (Agresti-Coull) intervals and the subsequent discussion will in itself hopefully shed some new light on this seemingly elementary interval estimation situation. Generally, an alternative to the Wald interval is to be judged not only by performance, its expression should also indicate why we will obtain a better behaved interval. It is argued that the well-behaved AN interval meets this requirement.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41959851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose well-calibrated null preference priors for use with one-sided hypothesis tests, such that resulting Bayesian and frequentist inferences agree. Null preference priors mean that they have nearly 100% of their prior belief in the null hypothesis, and well-calibrated priors mean that the resulting posterior beliefs in the alternative hypothesis are not overconfident. This formulation expands the class of problems giving Bayes-frequentist agreement to include problems involving discrete distributions such as binomial and negative binomial oneand two-sample exact (i.e., valid) tests. When applicable, these priors give posterior belief in the null hypothesis that is a valid p-value, and the null preference prior emphasizes that large p-values may simply represent insufficient data to overturn prior belief. This formulation gives a Bayesian interpretation of some common frequentist tests, as well as more intuitively explaining lesser known and less straightforward confidence intervals for two-sample tests.
{"title":"Interpreting p-Values and Confidence Intervals Using Well-Calibrated Null Preference Priors","authors":"M. Fay, M. Proschan, E. Brittain, R. Tiwari","doi":"10.1214/21-sts833","DOIUrl":"https://doi.org/10.1214/21-sts833","url":null,"abstract":"We propose well-calibrated null preference priors for use with one-sided hypothesis tests, such that resulting Bayesian and frequentist inferences agree. Null preference priors mean that they have nearly 100% of their prior belief in the null hypothesis, and well-calibrated priors mean that the resulting posterior beliefs in the alternative hypothesis are not overconfident. This formulation expands the class of problems giving Bayes-frequentist agreement to include problems involving discrete distributions such as binomial and negative binomial oneand two-sample exact (i.e., valid) tests. When applicable, these priors give posterior belief in the null hypothesis that is a valid p-value, and the null preference prior emphasizes that large p-values may simply represent insufficient data to overturn prior belief. This formulation gives a Bayesian interpretation of some common frequentist tests, as well as more intuitively explaining lesser known and less straightforward confidence intervals for two-sample tests.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41886684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Berti, E. Dreassi, F. Leisen, P. Rigo, L. Pratelli
Given a sequence $X=(X_1,X_2,ldots)$ of random observations, a Bayesian forecaster aims to predict $X_{n+1}$ based on $(X_1,ldots,X_n)$ for each $nge 0$. To this end, in principle, she only needs to select a collection $sigma=(sigma_0,sigma_1,ldots)$, called ``strategy"in what follows, where $sigma_0(cdot)=P(X_1incdot)$ is the marginal distribution of $X_1$ and $sigma_n(cdot)=P(X_{n+1}incdotmid X_1,ldots,X_n)$ the $n$-th predictive distribution. Because of the Ionescu-Tulcea theorem, $sigma$ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence $X$ is determined. The strategies concern generalized P'olya urns, random change points, covariates and stationary sequences.
{"title":"A Probabilistic View on Predictive Constructions for Bayesian Learning","authors":"P. Berti, E. Dreassi, F. Leisen, P. Rigo, L. Pratelli","doi":"10.1214/23-sts884","DOIUrl":"https://doi.org/10.1214/23-sts884","url":null,"abstract":"Given a sequence $X=(X_1,X_2,ldots)$ of random observations, a Bayesian forecaster aims to predict $X_{n+1}$ based on $(X_1,ldots,X_n)$ for each $nge 0$. To this end, in principle, she only needs to select a collection $sigma=(sigma_0,sigma_1,ldots)$, called ``strategy\"in what follows, where $sigma_0(cdot)=P(X_1incdot)$ is the marginal distribution of $X_1$ and $sigma_n(cdot)=P(X_{n+1}incdotmid X_1,ldots,X_n)$ the $n$-th predictive distribution. Because of the Ionescu-Tulcea theorem, $sigma$ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence $X$ is determined. The strategies concern generalized P'olya urns, random change points, covariates and stationary sequences.","PeriodicalId":51172,"journal":{"name":"Statistical Science","volume":" ","pages":""},"PeriodicalIF":5.7,"publicationDate":"2022-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48914546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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