Abstract Direct use of the likelihood function typically produces severely biased estimates when the dimension of the parameter vector is large relative to the effective sample size. With linearly separable data generated from a logistic regression model, the loglikelihood function asymptotes and the maximum likelihood estimator does not exist. We show that an exact analysis for each regression coefficient produces half-infinite confidence sets for some parameters when the data are separable. Such conclusions are not vacuous, but an honest portrayal of the limitations of the data. Finite confidence sets are only achievable when additional, perhaps implicit, assumptions are made. Under a notional double-asymptotic regime in which the dimension of the logistic coefficient vector increases with the sample size, the present paper considers the implications of enforcing a natural constraint on the vector of logistic-transformed probabilities. We derive a relationship between the logistic coefficients and a notional parameter obtained as a probability limit of an ordinary least squares estimator. The latter exists even when the data are separable. Consistency is ascertained under weak conditions on the design matrix.
{"title":"On inference in high-dimensional logistic regression models with separated data","authors":"R M Lewis, H S Battey","doi":"10.1093/biomet/asad065","DOIUrl":"https://doi.org/10.1093/biomet/asad065","url":null,"abstract":"Abstract Direct use of the likelihood function typically produces severely biased estimates when the dimension of the parameter vector is large relative to the effective sample size. With linearly separable data generated from a logistic regression model, the loglikelihood function asymptotes and the maximum likelihood estimator does not exist. We show that an exact analysis for each regression coefficient produces half-infinite confidence sets for some parameters when the data are separable. Such conclusions are not vacuous, but an honest portrayal of the limitations of the data. Finite confidence sets are only achievable when additional, perhaps implicit, assumptions are made. Under a notional double-asymptotic regime in which the dimension of the logistic coefficient vector increases with the sample size, the present paper considers the implications of enforcing a natural constraint on the vector of logistic-transformed probabilities. We derive a relationship between the logistic coefficients and a notional parameter obtained as a probability limit of an ordinary least squares estimator. The latter exists even when the data are separable. Consistency is ascertained under weak conditions on the design matrix.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135975796","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}
Summary Modelling of the dependence structure across heterogeneous data is crucial for Bayesian inference since it directly impacts the borrowing of information. Despite the extensive advances over the last two decades, most available proposals only allow for nonnegative correlations. We derive a new class of dependent nonparametric priors that can induce correlations of any sign, thus introducing a new and more flexible idea of borrowing of information. This is achieved thanks to a novel concept, which we term hyper-tie, and represents a direct and simple measure of dependence. We investigate prior and posterior distributional properties of the model and develop algorithms to perform posterior inference. Illustrative examples on simulated and real data show that our proposal outperforms alternatives in terms of prediction and clustering.
{"title":"Nonparametric priors with full-range borrowing of information","authors":"F Ascolani, B Franzolini, A Lijoi, I Prünster","doi":"10.1093/biomet/asad063","DOIUrl":"https://doi.org/10.1093/biomet/asad063","url":null,"abstract":"Summary Modelling of the dependence structure across heterogeneous data is crucial for Bayesian inference since it directly impacts the borrowing of information. Despite the extensive advances over the last two decades, most available proposals only allow for nonnegative correlations. We derive a new class of dependent nonparametric priors that can induce correlations of any sign, thus introducing a new and more flexible idea of borrowing of information. This is achieved thanks to a novel concept, which we term hyper-tie, and represents a direct and simple measure of dependence. We investigate prior and posterior distributional properties of the model and develop algorithms to perform posterior inference. Illustrative examples on simulated and real data show that our proposal outperforms alternatives in terms of prediction and clustering.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135729762","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}
Summary Likelihood-based inference under nonconvex constraints on model parameters has become increasingly common in biomedical research. In this paper, we establish large-sample properties of the maximum likelihood estimator when the true parameter value lies at the boundary of a nonconvex parameter space. We further derive the asymptotic distribution of the likelihood ratio test statistic under nonconvex constraints on model parameters. A general Monte Carlo procedure for generating the limiting distribution is provided. The theoretical results are demonstrated by five examples in Anderson’s stereotype logistic regression model, genetic association studies, gene-environment interaction tests, cost-constrained linear regression and fairness-constrained linear regression.
{"title":"Likelihood-based Inference under Non-Convex Boundary Constraints","authors":"J Y Wang, Z S YE, Y Chen","doi":"10.1093/biomet/asad062","DOIUrl":"https://doi.org/10.1093/biomet/asad062","url":null,"abstract":"Summary Likelihood-based inference under nonconvex constraints on model parameters has become increasingly common in biomedical research. In this paper, we establish large-sample properties of the maximum likelihood estimator when the true parameter value lies at the boundary of a nonconvex parameter space. We further derive the asymptotic distribution of the likelihood ratio test statistic under nonconvex constraints on model parameters. A general Monte Carlo procedure for generating the limiting distribution is provided. The theoretical results are demonstrated by five examples in Anderson’s stereotype logistic regression model, genetic association studies, gene-environment interaction tests, cost-constrained linear regression and fairness-constrained linear regression.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135729962","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}
Summary While the Metropolis Adjusted Langevin Algorithm (MALA) is a popular and widely used Markov chain Monte Carlo method, very few papers derive conditions that ensure its convergence. In particular, to the authors' knowledge, assumptions that are both easy to verify and guarantee geometric convergence, are still missing. In this work, we establish V-uniformly geometric convergence for MALA under mild assumptions about the target distribution. Unlike previous work, we only consider tail and smoothness conditions for the potential associated with the target distribution. These conditions are quite common in the MCMC literature. Finally, we pay special attention to the dependence of the bounds we derive on the step size of the Euler-Maruyama discretization, which corresponds to the proposal Markov kernel of MALA.
{"title":"On geometric convergence for MALA under simple conditions","authors":"Alain Oliviero-Durmus, Éric Moulines","doi":"10.1093/biomet/asad060","DOIUrl":"https://doi.org/10.1093/biomet/asad060","url":null,"abstract":"Summary While the Metropolis Adjusted Langevin Algorithm (MALA) is a popular and widely used Markov chain Monte Carlo method, very few papers derive conditions that ensure its convergence. In particular, to the authors' knowledge, assumptions that are both easy to verify and guarantee geometric convergence, are still missing. In this work, we establish V-uniformly geometric convergence for MALA under mild assumptions about the target distribution. Unlike previous work, we only consider tail and smoothness conditions for the potential associated with the target distribution. These conditions are quite common in the MCMC literature. Finally, we pay special attention to the dependence of the bounds we derive on the step size of the Euler-Maruyama discretization, which corresponds to the proposal Markov kernel of MALA.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135695601","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}
We study the efficient off-policy evaluation of natural stochastic policies, which are defined in terms of deviations from the behavior policy. This is a departure from the literature on off-policy evaluation where most work consider the evaluation of explicitly specified policies. Crucially, offline reinforcement learning with natural stochastic policies can help alleviate issues of weak overlap, lead to policies that build upon current practice, and improve policies' implementability in practice. Compared with the classic case of a pre-specified evaluation policy, when evaluating natural stochastic policies, the efficiency bound, which measures the best-achievable estimation error, is inflated since the evaluation policy itself is unknown. In this paper, we derive the efficiency bounds of two major types of natural stochastic policies: tilting policies and modified treatment policies. We then propose efficient nonparametric estimators that attain the efficiency bounds under very lax conditions. These also enjoy a (partial) double robustness property.
{"title":"Efficient Evaluation of Natural Stochastic Policies in Offline Reinforcement Learning","authors":"Nathan Kallus, Masatoshi Uehara","doi":"10.1093/biomet/asad059","DOIUrl":"https://doi.org/10.1093/biomet/asad059","url":null,"abstract":"We study the efficient off-policy evaluation of natural stochastic policies, which are defined in terms of deviations from the behavior policy. This is a departure from the literature on off-policy evaluation where most work consider the evaluation of explicitly specified policies. Crucially, offline reinforcement learning with natural stochastic policies can help alleviate issues of weak overlap, lead to policies that build upon current practice, and improve policies' implementability in practice. Compared with the classic case of a pre-specified evaluation policy, when evaluating natural stochastic policies, the efficiency bound, which measures the best-achievable estimation error, is inflated since the evaluation policy itself is unknown. In this paper, we derive the efficiency bounds of two major types of natural stochastic policies: tilting policies and modified treatment policies. We then propose efficient nonparametric estimators that attain the efficiency bounds under very lax conditions. These also enjoy a (partial) double robustness property.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135471800","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}
Abstract While model selection is a well-studied topic in parametric and nonparametric regression or density estimation, selection of possibly high-dimensional nuisance parameters in semiparametric problems is far less developed. In this paper, we propose a selective machine learning framework for making inferences about a finite-dimensional functional defined on a semiparametric model, when the latter admits a doubly robust estimating function and several candidate machine learning algorithms are available for estimating the nuisance parameters. We introduce a new selection criterion aimed at bias reduction in estimating the functional of interest based on a novel definition of pseudo-risk inspired by the double robustness property. Intuitively, the proposed criterion selects a pair of learners with the smallest pseudo-risk, so that the estimated functional is least sensitive to perturbations of a nuisance parameter. We establish an oracle property for a multi-fold cross-validation version of the new selection criterion which states that our empirical criterion performs nearly as well as an oracle with a priori knowledge of the pseudo-risk for each pair of candidate learners. Finally, we apply the approach to model selection of a semiparametric estimator of average treatment effect given an ensemble of candidate machine learners to account for confounding in an observational study which we illustrate in simulations and a data application.
{"title":"Selective machine learning of doubly robust functionals","authors":"Y Cui, E Tchetgen Tchetgen","doi":"10.1093/biomet/asad055","DOIUrl":"https://doi.org/10.1093/biomet/asad055","url":null,"abstract":"Abstract While model selection is a well-studied topic in parametric and nonparametric regression or density estimation, selection of possibly high-dimensional nuisance parameters in semiparametric problems is far less developed. In this paper, we propose a selective machine learning framework for making inferences about a finite-dimensional functional defined on a semiparametric model, when the latter admits a doubly robust estimating function and several candidate machine learning algorithms are available for estimating the nuisance parameters. We introduce a new selection criterion aimed at bias reduction in estimating the functional of interest based on a novel definition of pseudo-risk inspired by the double robustness property. Intuitively, the proposed criterion selects a pair of learners with the smallest pseudo-risk, so that the estimated functional is least sensitive to perturbations of a nuisance parameter. We establish an oracle property for a multi-fold cross-validation version of the new selection criterion which states that our empirical criterion performs nearly as well as an oracle with a priori knowledge of the pseudo-risk for each pair of candidate learners. Finally, we apply the approach to model selection of a semiparametric estimator of average treatment effect given an ensemble of candidate machine learners to account for confounding in an observational study which we illustrate in simulations and a data application.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134960403","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}
Summary In this paper, we develop an eigenvector-assisted estimation framework for a collection of signal-plus-noise matrix models arising in high-dimensional statistics and many applications. The framework is built upon a novel asymptotically unbiased estimating equation using the leading eigenvectors of the data matrix. However, the estimator obtained by directly solving the estimating equation could be numerically unstable in practice and lacks robustness against model misspecification. We propose to use the quasi-posterior distribution by exponentiating a criterion function whose maximizer coincides with the estimating equation estimator. The proposed framework can incorporate heteroskedastic variance information but does not require the complete specification of the sampling distribution and is also robust to the potential misspecification of the distribution of the noise matrix. Computationally, the quasi-posterior distribution can be obtained via a Markov Chain Monte Carlo sampler, which exhibits superior numerical stability than some of the existing optimization-based estimators and is straightforward for uncertainty quantification. Under mild regularity conditions, we establish the large sample properties of the quasi-posterior distributions. In particular, the quasi-posterior credible sets have the correct frequentist nominal coverage probability provided that the criterion function is carefully selected. The validity and usefulness of the proposed framework are demonstrated through the analysis of synthetic datasets and the real-world ENZYMES network datasets.
{"title":"An eigenvector-assisted estimation framework for signal-plus-noise matrix models","authors":"Fangzheng Xie, Dingbo Wu","doi":"10.1093/biomet/asad058","DOIUrl":"https://doi.org/10.1093/biomet/asad058","url":null,"abstract":"Summary In this paper, we develop an eigenvector-assisted estimation framework for a collection of signal-plus-noise matrix models arising in high-dimensional statistics and many applications. The framework is built upon a novel asymptotically unbiased estimating equation using the leading eigenvectors of the data matrix. However, the estimator obtained by directly solving the estimating equation could be numerically unstable in practice and lacks robustness against model misspecification. We propose to use the quasi-posterior distribution by exponentiating a criterion function whose maximizer coincides with the estimating equation estimator. The proposed framework can incorporate heteroskedastic variance information but does not require the complete specification of the sampling distribution and is also robust to the potential misspecification of the distribution of the noise matrix. Computationally, the quasi-posterior distribution can be obtained via a Markov Chain Monte Carlo sampler, which exhibits superior numerical stability than some of the existing optimization-based estimators and is straightforward for uncertainty quantification. Under mild regularity conditions, we establish the large sample properties of the quasi-posterior distributions. In particular, the quasi-posterior credible sets have the correct frequentist nominal coverage probability provided that the criterion function is carefully selected. The validity and usefulness of the proposed framework are demonstrated through the analysis of synthetic datasets and the real-world ENZYMES network datasets.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135060566","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}
Summary We study how to combine p-values and e-values, and design multiple testing procedures where both p-values and e-values are available for every hypothesis. Our results provide a new perspective on multiple testing with data-driven weights: while standard weighted multiple testing methods require the weights to deterministically add up to the number of hypotheses being tested, we show that this normalization is not required when the weights are e-values that are independent of the p-values. Such e-values can be obtained in meta-analysis where a primary dataset is used to compute p-values, and an independent secondary dataset is used to compute e-values. Going beyond meta-analysis, we showcase settings wherein independent e-values and p-values can be constructed on a single dataset itself. Our procedures can result in a substantial increase in power, especially if the non-null hypotheses have e-values much larger than one.
{"title":"E-values as unnormalized weights in multiple testing","authors":"Nikolaos Ignatiadis, Ruodu Wang, Aaditya Ramdas","doi":"10.1093/biomet/asad057","DOIUrl":"https://doi.org/10.1093/biomet/asad057","url":null,"abstract":"Summary We study how to combine p-values and e-values, and design multiple testing procedures where both p-values and e-values are available for every hypothesis. Our results provide a new perspective on multiple testing with data-driven weights: while standard weighted multiple testing methods require the weights to deterministically add up to the number of hypotheses being tested, we show that this normalization is not required when the weights are e-values that are independent of the p-values. Such e-values can be obtained in meta-analysis where a primary dataset is used to compute p-values, and an independent secondary dataset is used to compute e-values. Going beyond meta-analysis, we showcase settings wherein independent e-values and p-values can be constructed on a single dataset itself. Our procedures can result in a substantial increase in power, especially if the non-null hypotheses have e-values much larger than one.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135436744","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}
Summary As highlighted in Dawid (2000) and Pearl & Mackenzie (2018), deducing the causes of given effects is a more challenging problem than evaluating the effects of causes in causal inference. Lu et al. (2023) proposed an approach for deducing causes of a single effect variable based on posterior causal effects. In many applications, there are multiple effect variables, and thus they can be used simultaneously to more accurately deduce the causes. To retrospectively deduce causes from multiple effects, we propose multivariate posterior total, intervention and direct causal effects conditional on the observed evidence. We describe the assumptions of no-confounding and monotonicity, under which we prove identifiability of the multivariate posterior causal effects and provide their identification equations. The proposed approach can be applied for causal attributions, medical diagnosis, blame and responsibility in various studies with multiple effect or outcome variables. Two examples are used to illustrate the proposed approach.
{"title":"Retrospective causal inference with multiple effect variables","authors":"Wei Li, Zitong Lu, Jinzhu Jia, Min Xie, Zhi Geng","doi":"10.1093/biomet/asad056","DOIUrl":"https://doi.org/10.1093/biomet/asad056","url":null,"abstract":"Summary As highlighted in Dawid (2000) and Pearl & Mackenzie (2018), deducing the causes of given effects is a more challenging problem than evaluating the effects of causes in causal inference. Lu et al. (2023) proposed an approach for deducing causes of a single effect variable based on posterior causal effects. In many applications, there are multiple effect variables, and thus they can be used simultaneously to more accurately deduce the causes. To retrospectively deduce causes from multiple effects, we propose multivariate posterior total, intervention and direct causal effects conditional on the observed evidence. We describe the assumptions of no-confounding and monotonicity, under which we prove identifiability of the multivariate posterior causal effects and provide their identification equations. The proposed approach can be applied for causal attributions, medical diagnosis, blame and responsibility in various studies with multiple effect or outcome variables. Two examples are used to illustrate the proposed approach.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135552839","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}
Summary The accurate estimation of prediction errors in time series is an important problem, which has immediate implications for the accuracy of prediction intervals as well as the quality of a number of widely used time series model selection criteria such as the Akaike information criterion. Except for simple cases, however, it is difficult or even impossible to obtain exact analytical expressions for one-step and multi-step predictions. This may be one of the reasons that, unlike in the independent case (see Efron, 2004), up to now there has been no fully established methodology for time series prediction error estimation. Starting from an approximation to the bias-variance decomposition of the squared prediction error, a method for accurate estimation of prediction errors in both univariate and multivariate stationary time series is developed in this article. In particular, several estimates are derived for a general class of predictors that includes most of the popular linear, nonlinear, parametric and nonparametric time series models used in practice, with causal invertible autoregressive moving average and nonparametric autoregressive processes discussed as lead examples. Simulations demonstrate that the proposed estimators perform quite well in finite samples. The estimates may also be used for model selection when the purpose of modelling is prediction.
{"title":"Estimation of prediction error in time series","authors":"Alexander Aue, Prabir Burman","doi":"10.1093/biomet/asad053","DOIUrl":"https://doi.org/10.1093/biomet/asad053","url":null,"abstract":"Summary The accurate estimation of prediction errors in time series is an important problem, which has immediate implications for the accuracy of prediction intervals as well as the quality of a number of widely used time series model selection criteria such as the Akaike information criterion. Except for simple cases, however, it is difficult or even impossible to obtain exact analytical expressions for one-step and multi-step predictions. This may be one of the reasons that, unlike in the independent case (see Efron, 2004), up to now there has been no fully established methodology for time series prediction error estimation. Starting from an approximation to the bias-variance decomposition of the squared prediction error, a method for accurate estimation of prediction errors in both univariate and multivariate stationary time series is developed in this article. In particular, several estimates are derived for a general class of predictors that includes most of the popular linear, nonlinear, parametric and nonparametric time series models used in practice, with causal invertible autoregressive moving average and nonparametric autoregressive processes discussed as lead examples. Simulations demonstrate that the proposed estimators perform quite well in finite samples. The estimates may also be used for model selection when the purpose of modelling is prediction.","PeriodicalId":9001,"journal":{"name":"Biometrika","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136108242","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}
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