BernoulliPub Date : 2021-05-28DOI: 10.3150/22-bej1484
P. Hoff
{"title":"Bayes-optimal prediction with frequentist coverage control","authors":"P. Hoff","doi":"10.3150/22-bej1484","DOIUrl":"https://doi.org/10.3150/22-bej1484","url":null,"abstract":"This article illustrates how indirect or prior information can be optimally used to construct a prediction region that maintains a target frequentist coverage rate. If the indirect information is accurate, the volume of the prediction region is lower on average than that of other regions with the same coverage rate. Even if the indirect information is inaccurate, the resulting region still maintains the target coverage rate. Such a prediction region can be constructed for models that have a complete sufficient statistic, which includes many widely-used parametric and nonparametric models. Particular examples include a Bayes-optimal conformal prediction procedure that maintains a constant coverage rate across distributions in a nonparametric model, as well as a prediction procedure for the normal linear regression model that can utilize a regularizing prior distribution, yet maintain a frequentist coverage rate that is constant as a function of the model parameters and explanatory variables. No results in this article rely on asymptotic approximations.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44566909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BernoulliPub Date : 2021-05-25DOI: 10.3150/22-bej1578
B. Eltzner, Pernille Hansen, S. Huckemann, S. Sommer
{"title":"Diffusion means in geometric spaces","authors":"B. Eltzner, Pernille Hansen, S. Huckemann, S. Sommer","doi":"10.3150/22-bej1578","DOIUrl":"https://doi.org/10.3150/22-bej1578","url":null,"abstract":"We introduce a location statistic for distributions on non-linear geometric spaces, the diffusion mean, serving as an extension and an alternative to the Fr'echet mean. The diffusion mean arises as the generalization of Gaussian maximum likelihood analysis to non-linear spaces by maximizing the likelihood of a Brownian motion. The diffusion mean depends on a time parameter $t$, which admits the interpretation of the allowed variance of the diffusion. The diffusion $t$-mean of a distribution $X$ is the most likely origin of a Brownian motion at time $t$, given the end-point distribution $X$. We give a detailed description of the asymptotic behavior of the diffusion estimator and provide sufficient conditions for the diffusion estimator to be strongly consistent. Particularly, we present a smeary central limit theorem for diffusion means and we show that joint estimation of the mean and diffusion variance rules out smeariness in all directions simultaneously in general situations. Furthermore, we investigate properties of the diffusion mean for distributions on the sphere $mathbb S^n$. Experimentally, we consider simulated data and data from magnetic pole reversals, all indicating similar or improved convergence rate compared to the Fr'echet mean. Here, we additionally estimate $t$ and consider its effects on smeariness and uniqueness of the diffusion mean for distributions on the sphere.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45207548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BernoulliPub Date : 2021-05-23DOI: 10.3150/22-bej1581
Xinjie Du, M. Tang
{"title":"Hypothesis testing for equality of latent positions in random graphs","authors":"Xinjie Du, M. Tang","doi":"10.3150/22-bej1581","DOIUrl":"https://doi.org/10.3150/22-bej1581","url":null,"abstract":"We consider the hypothesis testing problem that two vertices $i$ and $j$ of a generalized random dot product graph have the same latent positions, possibly up to scaling. Special cases of this hypothesis test include testing whether two vertices in a stochastic block model or degree-corrected stochastic block model graph have the same block membership vectors, or testing whether two vertices in a popularity adjusted block model have the same community assignment. We propose several test statistics based on the empirical Mahalanobis distances between the $i$th and $j$th rows of either the adjacency or the normalized Laplacian spectral embedding of the graph. We show that, under mild conditions, these test statistics have limiting chi-square distributions under both the null and local alternative hypothesis, and we derived explicit expressions for the non-centrality parameters under the local alternative. Using these limit results, we address the model selection problems including choosing between the standard stochastic block model and its degree-corrected variant, and choosing between the ER model and stochastic block model. The effectiveness of our proposed tests are illustrated via both simulation studies and real data applications.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48847113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BernoulliPub Date : 2021-05-01DOI: 10.3150/20-BEJ1297
Ma Yi-An, Niladri S. Chatterji, Xiang Cheng, Nicolas, Flammarion, P. Bartlett, Michael I. Jordan
{"title":"Is there an analog of Nesterov acceleration for gradient-based MCMC?","authors":"Ma Yi-An, Niladri S. Chatterji, Xiang Cheng, Nicolas, Flammarion, P. Bartlett, Michael I. Jordan","doi":"10.3150/20-BEJ1297","DOIUrl":"https://doi.org/10.3150/20-BEJ1297","url":null,"abstract":"We formulate gradient-based Markov chain Monte Carlo (MCMC) sampling as optimization on the space of probability measures, with Kullback–Leibler (KL) divergence as the objective functional. We show that an underdamped form of the Langevin algorithm performs accelerated gradient descent in this metric. To characterize the convergence of the algorithm, we construct a Lyapunov functional and exploit hypocoercivity of the underdamped Langevin algorithm. As an application, we show that accelerated rates can be obtained for a class of nonconvex functions with the Langevin algorithm.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"27 1","pages":"1942-1992"},"PeriodicalIF":1.5,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49618849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BernoulliPub Date : 2021-05-01DOI: 10.3150/20-BEJ1280
Michael Law, Y. Ritov
{"title":"Inference without compatibility: Using exponential weighting for inference on a parameter of a linear model","authors":"Michael Law, Y. Ritov","doi":"10.3150/20-BEJ1280","DOIUrl":"https://doi.org/10.3150/20-BEJ1280","url":null,"abstract":"We consider hypotheses testing problems for three parameters in high-dimensional linear models with minimal sparsity assumptions of their type but without any compatibility conditions. Under this framework, we construct the first n-consistent estimators for low-dimensional coefficients, the signal strength, and the noise level. We support our results using numerical simulations and provide comparisons with other estimators.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"27 1","pages":"1467-1495"},"PeriodicalIF":1.5,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45541126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BernoulliPub Date : 2021-05-01DOI: 10.3150/20-BEJ1295
J. Jakubowski, Maciej Wiśniewolski
{"title":"A convolution formula for the local time of an Itô diffusion reflecting at 0 and a generalized Stroock–Williams equation","authors":"J. Jakubowski, Maciej Wiśniewolski","doi":"10.3150/20-BEJ1295","DOIUrl":"https://doi.org/10.3150/20-BEJ1295","url":null,"abstract":"A new probabilistic insight into the structure of local time is presented. A convolution formula for the local time at 0 of Ito diffusions reflecting at 0 is obtained. A simple integro-differential equation for the cumulative distribution function of the local time is given. Finally, a probabilistic representation of a generalized Stroock–Williams equation is presented.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"27 1","pages":"1870-1898"},"PeriodicalIF":1.5,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44477593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BernoulliPub Date : 2021-05-01DOI: 10.3150/20-BEJ1302
F. Nicolussi, M. Cazzaro
{"title":"Context-specific independencies in stratified chain regression graphical models","authors":"F. Nicolussi, M. Cazzaro","doi":"10.3150/20-BEJ1302","DOIUrl":"https://doi.org/10.3150/20-BEJ1302","url":null,"abstract":"Graphical models are a useful tool with increasing diffusion. In the categorical variable framework, they provide important visual support to understand the relationships among the considered variables. Besides, particular chain graphical models are suitable to represent multivariate regression models. However, the associated parameterization, such as marginal log-linear models, is often difficult to interpret when the number of variables increases because of a large number of parameters involved. On the contrary, conditional and marginal independencies reduce the number of parameters needed to represent the joint probability distribution of the variables. In compliance with the parsimonious principle, it is worthwhile to consider also the so-called context-specific independencies, which are conditional independencies holding for particular values of the variables in the conditioning set. In this work, we propose a particular chain graphical model able to represent these context-specific independencies through labeled arcs. We provide also the Markov properties able to describe marginal, conditional, and context-specific independencies from this new chain graph. Finally, we show the results in an application to a real data set.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":"27 1","pages":"2091-2116"},"PeriodicalIF":1.5,"publicationDate":"2021-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48491136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BernoulliPub Date : 2021-04-29DOI: 10.3150/21-bej1355
Hongyuan Cao, W. Wu
{"title":"Testing and estimation for clustered signals","authors":"Hongyuan Cao, W. Wu","doi":"10.3150/21-bej1355","DOIUrl":"https://doi.org/10.3150/21-bej1355","url":null,"abstract":"We propose a change-point detection method for large scale multiple testing problems with data having clustered signals. Unlike the classic change-point setup, the signals can vary in size within a cluster. The clustering structure on the signals enables us to effectively delineate the boundaries between signal and non-signal segments. New test statistics are proposed for observations from one and/or multiple realizations. Their asymptotic distributions are derived. We also study the associated variance estimation problem. We allow the variances to be heteroscedastic in the multiple realization case, which substantially expands the applicability of the proposed method. Simulation studies demonstrate that the proposed approach has a favorable performance. Our procedure is applied to {an array based Comparative Genomic Hybridization (aCGH)} dataset.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45778390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BernoulliPub Date : 2021-04-16DOI: 10.3150/22-bej1468
K. Heine, D. Burrows
{"title":"Multilevel bootstrap particle filter","authors":"K. Heine, D. Burrows","doi":"10.3150/22-bej1468","DOIUrl":"https://doi.org/10.3150/22-bej1468","url":null,"abstract":"We consider situations where the applicability of sequential Monte Carlo particle filters is compromised due to the expensive evaluation of the particle weights. To alleviate this problem, we propose a new particle filter algorithm based on the multilevel approach. We show that the resulting multilevel bootstrap particle filter (MLBPF) retains the strong law of large numbers as well as the central limit theorem of classical particle filters under mild conditions. Our numerical experiments demonstrate up to 85% reduction in computation time compared to the classical bootstrap particle filter, in certain settings. While it should be acknowledged that this reduction is highly application dependent, and a similar gain should not be expected for all applications across the board, we believe that this substantial improvement in certain settings makes MLBPF an important addition to the family of sequential Monte Carlo methods.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48653058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
BernoulliPub Date : 2021-04-12DOI: 10.3150/23-bej1584
Martin Minchev, Mladen Savov
{"title":"Asymptotics for densities of exponential functionals of subordinators","authors":"Martin Minchev, Mladen Savov","doi":"10.3150/23-bej1584","DOIUrl":"https://doi.org/10.3150/23-bej1584","url":null,"abstract":"In this paper we derive non-classical Tauberian asymptotic at infinity for the tail, the density and the derivatives thereof of a large class of exponential functionals of subordinators. More precisely, we consider the case when the L'evy measure of the subordinator satisfies the well-known and mild condition of positive increase. This is achieved via a convoluted application of the saddle point method to the Mellin transform of these exponential functionals which is given in terms of Bernstein-gamma functions. To apply the saddle point method we improved the Stirling type of asymptotic for Bernstein-gamma functions and the latter is of interest beyond this paper as the Bernstein-gamma functions are applicable in different settings especially through their asymptotic behaviour in the complex plane. As an application we have derived the asymptotic of the density and its derivatives for all exponential functionals of non-decreasing, potentially compound Poisson processes which turns out to be precisely as that of an exponentially distributed random variable. We show further that a large class of densities are even analytic in a cone of the complex plane.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48362925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}