David Rügamer, Sonja Greven
Statistical inference after model selection requires an inference framework that takes the selection into account in order to be valid. Following recent work on selective inference, we derive analytical expressions for inference after likelihood- or test-based model selection for linear models.
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Several strategies have been developed recently to ensure valid inference after model selection; some of these are easy to compute, while others fare better in terms of inferential power. In this paper, we consider a selective inference framework for Gaussian data. We propose a new method for inference through approximate maximum likelihood estimation. Our goal is to: (i) achieve better inferential power with the aid of randomization, (ii) bypass expensive MCMC sampling from exact conditional distributions that are hard to evaluate in closed forms. We construct approximate inference, e.g., p-v…
Selective Inference in Graphical Models via Maximum Likelihood
该研究针对图套索推断边值函数时缺乏统计保证的问题，提出带随机化的选择性推断方法，通过条件最大似然实现有效区间估计与假设检验，效果优于数据拆分。
In multi-center clinical trials, due to various reasons, the individual-level data are strictly restricted to be assessed publicly. Instead, the summarized information is widely available from published results. With the advance of computational technology, it has become very common in data analyses to run on hundreds or thousands of machines simultaneous, with the data distributed across those machines and no longer available in a single central location. How to effectively assemble the summarized clinical data information or information from each machine in parallel computation has become a …
Post-selection inference is a statistical technique for determining salient variables after model or variable selection. Recently, selective inference, a kind of post-selection inference framework, has garnered the attention in the statistics and machine learning communities. By conditioning on a specific variable selection procedure, selective inference can properly control for so-called selective type I error, which is a type I error conditional on a variable selection procedure, without imposing excessive additional computational costs. While selective inference can provide a valid hypothes…
Distributed Selective Inference for Quantile Regression
针对高维分位数回归的后选择推断难题，提出仅需三轮通信的分布式选择性推断框架，通过响应替代策略处理非光滑分位数损失，经模拟与实际数据验证其有限样本性能良好。
It is common to show the confidence intervals or $p$-values of selected features, or predictor variables in regression, but they often involve selection bias. The selective inference approach solves this bias by conditioning on the selection event. Most existing studies of selective inference consider a specific algorithm, such as Lasso, for feature selection, and thus they have difficulties in handling more complicated algorithms. Moreover, existing studies often consider unnecessarily restrictive events, leading to over-conditioning and lower statistical power. Our novel and widely-applicabl…
Know When to Abstain: Optimal Selective Classification with Likelihood Ratios
本文提出基于似然比的最优选择性分类方法，通过 Neyman-Pearson 引理统一并改进了选择性分类在协变量偏移下的性能。
We present a new method for post-selection inference for L1 (lasso)-penalized likelihood models, including generalized regression models. Our approach generalizes the post-selection framework presented in Lee et al (2014). The method provides p-values and confidence intervals that are asymptotically valid, conditional on the inherent selection done by the lasso. We present applications of this work to (regularized) logistic regression, Cox's proportional hazards model and the graphical lasso.
Black-box Selective Inference via Bootstrapping
本文提出利用 bootstrap 反复运行黑箱选择算法以估计选择概率和条件数据分布，从而在无精确选择事件刻画时实现条件选择性推断，并给出理论保证。
Empirical likelihood meta analysis with publication bias correction under Copas-like selection model
本文在Copas类选择模型下提出结合条件似然与边际半参数经验似然的全似然元分析方法，证明其估计量渐近正态且似然比渐近卡方，并通过模拟和实例验证其优于条件似然方法。
Applying standard statistical methods after model selection may yield inefficient estimators and hypothesis tests that fail to achieve nominal type-I error rates. The main issue is the fact that the post-selection distribution of the data differs from the original distribution. In particular, the observed data is constrained to lie in a subset of the original sample space that is determined by the selected model. This often makes the post-selection likelihood of the observed data intractable and maximum likelihood inference difficult. In this work, we get around the intractable likelihood by g…
On Selecting and Conditioning in Multiple Testing and Selective Inference
本文统一考察选择、条件化与误差控制，证明直接在完整假设全域上的多重检验方法功效不低于选择性推断方法，并给出理论框架与案例。
We propose an empirical likelihood ratio test for nonparametric model selection, where the competing models may be nested, nonnested, overlapping, misspecified, or correctly specified. It compares the squared prediction errors of models based on the cross-validation and allows for heteroscedasticity of the errors of models. We develop its asymptotic distributions for comparing additive models and varying-coefficient models and extend it to test significance of variables in additive models with massive data. The method is applicable to model selection among supervised learning models. To facili…
Inspired by sample splitting and the reusable holdout introduced in the field of differential privacy, we consider selective inference with a randomized response. We discuss two major advantages of using a randomized response for model selection. First, the selectively valid tests are more powerful after randomized selection. Second, it allows consistent estimation and weak convergence of selective inference procedures. Under independent sampling, we prove a selective (or privatized) central limit theorem that transfers procedures valid under asymptotic normality without selection to their cor…
To perform inference after model selection, we propose controlling the selective type I error; i.e., the error rate of a test given that it was performed. By doing so, we recover long-run frequency properties among selected hypotheses analogous to those that apply in the classical (non-adaptive) context. Our proposal is closely related to data splitting and has a similar intuitive justification, but is more powerful. Exploiting the classical theory of Lehmann and Scheffé (1955), we derive most powerful unbiased selective tests and confidence intervals for inference in exponential family models…
In this work, we provide a refinement of the selective CLT result of Tian and Taylor (2015), which allows for selective inference in non-parametric settings by adjusting for the asymptotic Gaussian limit for selection. Under some regularity assumptions on the density of the randomization, including heavier tails than Gaussian satisfied by e.g. logistic distribution, we prove the selective CLT holds without any assumptions on the underlying parameter, allowing for rare selection events. We also show a selective CLT result for Gaussian randomization, though the quantitative results are qualitati…
This work is concern with testing the low-dimensional parameters of interest with divergent dimensional data and variable selection for the rest under the sparse case. A consistent test via the partial penalized likelihood approach, called the partial penalized likelihood ratio test statistic is derived, and its asymptotic distributions under the null hypothesis and the local alternatives of order $n^{-1/2}$ are obtained under some regularity conditions. Meanwhile, the oracle property of the partial penalized likelihood estimator also holds. The proposed partial penalized likelihood ratio test…
Selective and marginal selective inference for exceptional groups
针对多群体研究中选择性推断导致置信区间期望宽度无穷大的问题，提出选择调整的经验贝叶斯方法，在保证有限期望宽度的同时实现近似选择性覆盖控制。
A constructive approach to selective risk control
本文提出一种构造性方法，通过迭代控制后选择风险来统一控制选择性风险，并发现BH过程的两个新性质，扩展了其应用。
Although the log-likelihood is widely used in model selection, the log-likelihood ratio has had few applications in this area. We develop a log-likelihood ratio based method for selecting regression models by focusing on the set of models deemed plausible by the likelihood ratio test. We show that when the sample size is large and the significance level of the test is small, there is a high probability that the smallest model in the set is the true model; thus, we select this smallest model. The significance level of the test serves as a parameter for this method. We consider three levels of t…
Online Selective Conformal Prediction with Asymmetric Rules: A Permutation Test Approach
本文提出PEMI框架，用于在线选择性符合预测，解决非对称选取机制下的预测集覆盖问题。
Online Selective Conformal Prediction: Errors and Solutions
针对在线选择性共形推断中现有校准选择策略存在的理论错误，本文提出可保持数据可交换性的新策略，证明其能同时保证选择条件覆盖率与FCR控制，并通过实验验证了有效方法间的权衡关系。
Discovering statistically significant patterns from databases is an important challenging problem. The main obstacle of this problem is in the difficulty of taking into account the selection bias, i.e., the bias arising from the fact that patterns are selected from extremely large number of candidates in databases. In this paper, we introduce a new approach for predictive pattern mining problems that can address the selection bias issue. Our approach is built on a recently popularized statistical inference framework called selective inference. In selective inference, statistical inferences (su…
Conditional selective inference (SI) has been actively studied as a new statistical inference framework for data-driven hypotheses. The basic idea of conditional SI is to make inferences conditional on the selection event characterized by a set of linear and/or quadratic inequalities. Conditional SI has been mainly studied in the context of feature selection such as stepwise feature selection (SFS). The main limitation of the existing conditional SI methods is the loss of power due to over-conditioning, which is required for computational tractability. In this study, we develop a more powerful…
The traditional activity of model selection aims at discovering a single model superior to other candidate models. In the presence of pronounced noise, however, multiple models are often found to explain the same data equally well. To resolve this model selection ambiguity, we introduce the general approach of model selection confidence sets (MSCSs) based on likelihood ratio testing. A MSCS is defined as a list of models statistically indistinguishable from the true model at a user-specified level of confidence, which extends the familiar notion of confidence intervals to the model-selection f…
We consider the problem of selecting covariates in spatial linear models with Gaussian process errors. Penalized maximum likelihood estimation (PMLE) that enables simultaneous variable selection and parameter estimation is developed and, for ease of computation, PMLE is approximated by one-step sparse estimation (OSE). To further improve computational efficiency, particularly with large sample sizes, we propose penalized maximum covariance-tapered likelihood estimation (PMLE$_{\mathrm{T}}$) and its one-step sparse estimation (OSE$_{\mathrm{T}}$). General forms of penalty functions with an emph…
Mixture of Experts (MoE) are successful models for modeling heterogeneous data in many statistical learning problems including regression, clustering and classification. Generally fitted by maximum likelihood estimation via the well-known EM algorithm, their application to high-dimensional problems is still therefore challenging. We consider the problem of fitting and feature selection in MoE models, and propose a regularized maximum likelihood estimation approach that encourages sparse solutions for heterogeneous regression data models with potentially high-dimensional predictors. Unlike stat…
Exact Selective Inference with Randomization
本文提出一种闭式枢轴量，将随机化下的精确选择性推断简化为二元截断高斯分布，虽牺牲部分功效，但产生更窄置信区间，并在模拟和HIV数据上验证权衡。
A number of discrete time, finite population size models in genetics describing the dynamics of allele frequencies are known to converge (subject to suitable scaling) to a diffusion process in the infinite population limit, termed the Wright-Fisher diffusion. In this article we show that the diffusion is ergodic uniformly in the selection and mutation parameters, and that the measures induced by the solution to the stochastic differential equation are uniformly locally asymptotically normal. Subsequently these two results are used to analyse the statistical properties of the Maximum Likelihood…
A Two-step Metropolis Hastings Method for Bayesian Empirical Likelihood Computation with Application to Quantile Regression and Bayesian Model Selection
本文提出一种两步Metropolis-Hastings算法，用于解决贝叶斯经验似然（BayesEL）后验分布采样困难的问题，可应用于分位数回归与贝叶斯模型选择等场景。
In this technical report, we consider conditional density estimation with a maximum likelihood approach. Under weak assumptions, we obtain a theoretical bound for a Kullback-Leibler type loss for a single model maximum likelihood estimate. We use a penalized model selection technique to select a best model within a collection. We give a general condition on penalty choice that leads to oracle type inequality for the resulting estimate. This construction is applied to two examples of partition-based conditional density models, models in which the conditional density depends only in a piecewise …
In conventional supervised pattern recognition tasks, model selection is typically accomplished by minimizing the classification error rate on a set of so-called development data, subject to ground-truth labeling by human experts or some other means. In the context of speech processing systems and other large-scale practical applications, however, such labeled development data are typically costly and difficult to obtain. This article proposes an alternative semi-supervised framework for likelihood-based model selection that leverages unlabeled data by using trained classifiers representing ea…
A new family of penalty functions, adaptive to likelihood, is introduced for model selection in general regression models. It arises naturally through assuming certain types of prior distribution on the regression parameters. To study stability properties of the penalized maximum likelihood estimator, two types of asymptotic stability are defined. Theoretical properties, including the parameter estimation consistency, model selection consistency, and asymptotic stability, are established under suitable regularity conditions. An efficient coordinate-descent algorithm is proposed. Simulation res…
This is an up-to-date introduction to, and overview of, marginal likelihood computation for model selection and hypothesis testing. Computing normalizing constants of probability models (or ratio of constants) is a fundamental issue in many applications in statistics, applied mathematics, signal processing and machine learning. This article provides a comprehensive study of the state-of-the-art of the topic. We highlight limitations, benefits, connections and differences among the different techniques. Problems and possible solutions with the use of improper priors are also described. Some of …
Selective machine learning of doubly robust functionals
本文提出一种选择性机器学习框架，基于双重稳健性质定义伪风险并选择最优学习器对，用于半参数模型中有限维泛函的推断，并应用于平均处理效应的估计，理论证明其oracle性质，模拟和实证验证了有效性。
We study the law of the iterated logarithm (LIL) for the maximum likelihood estimation of the parameters (as a convex optimization problem) in the generalized linear models with independent or weakly dependent ($ρ$-mixing, $m$-dependent) responses under mild conditions. The LIL is useful to derive the asymptotic bounds for the discrepancy between the empirical process of the log-likelihood function and the true log-likelihood. As the application of the LIL, the strong consistency of some penalized likelihood based model selection criteria can be shown. Under some regularity conditions, the mod…
Robust Likelihood Ratio Tests for Incomplete Economic Models
针对含多重均衡等的不完全经济模型，用Huber--Strassen理论构建有限样本极小极大似然比检验，给出重复实验中最小不利对的结果，还为单侧备择假设提供条件，蒙特卡洛实验展示检验效果。
We propose new model selection criteria based on generalized ridge estimators dominating the maximum likelihood estimator under the squared risk and the Kullback-Leibler risk in multivariate linear regression. Our model selection criteria have the following favorite properties: consistency, unbiasedness, uniformly minimum variance. Consistency is proven under an asymptotic structure $\frac{p}{n}\to c$ where $n$ is the sample size and $p$ is the parameter dimension of the response variables. In particular, our proposed class of estimators dominates the maximum likelihood estimator under the squ…
Advancing Information Integration through Empirical Likelihood: Selective Reviews and a New Idea
本文综述了仅利用汇总信息、基于经验似然的信息整合方法，并提出一个计算更便利且应用范围更广的新框架，通过数值评估验证其性能并讨论扩展方向。
The method of generalized estimating equations (GEE) is popular in the biostatistics literature for analyzing longitudinal binary and count data. It assumes a generalized linear model (GLM) for the outcome variable, and a working correlation among repeated measurements. In this paper, we introduce a viable competitor: the weighted scores method for GLM margins. We weight the univariate score equations using a working discretized multivariate normal model that is a proper multivariate model. Since the weighted scores method is a parametric method based on likelihood, we propose composite likeli…
Composite likelihood has shown promise in settings where the number of parameters $p$ is large due to its ability to break down complex models into simpler components, thus enabling inference even when the full likelihood is not tractable. Although there are a number of ways to formulate a valid composite likelihood in the finite-$p$ setting, there does not seem to exist agreement on how to construct composite likelihoods that are comp utationally efficient and statistically sound when $p$ is allowed to diverge. This article introduces a method to select sparse composite likelihoods by minimiz…
Conventional likelihood-based information criteria for model selection rely on the distribution assumption of data. However, for complex data that are increasingly available in many scientific fields, the specification of their underlying distribution turns out to be challenging, and the existing criteria may be limited and are not general enough to handle a variety of model selection problems. Here, we propose a robust and consistent model selection criterion based upon the empirical likelihood function which is data-driven. In particular, this framework adopts plug-in estimators that can be …
Integrative analyses based on statistically relevant associations between genomics and a wealth of intermediary phenotypes (such as imaging) provide vital insights into their clinical relevance in terms of the disease mechanisms. Estimates for uncertainty in the resulting integrative models are however unreliable unless inference accounts for the selection of these associations with accuracy. In this article, we develop selection-aware Bayesian methods which: (i) counteract the impact of model selection bias through a "selection-aware posterior" in a flexible class of integrative Bayesian mode…
This work addresses the problem of conducting valid inference for additive and linear mixed models after model selection. One possible solution to overcome overconfident inference results after model selection is selective inference, which constitutes a post-selection inference framework, yielding valid inference statements by conditioning on the selection event. We extend recent work on selective inference to the class of additive and linear mixed models for any type of model selection mechanism that can be expressed as a function of the outcome variable (and potentially on covariates on whic…
Grouping structures arise naturally in many statistical modeling problems. Several methods have been proposed for variable selection that respect grouping structure in variables. Examples include the group LASSO and several concave group selection methods. In this article, we give a selective review of group selection concerning methodological developments, theoretical properties and computational algorithms. We pay particular attention to group selection methods involving concave penalties. We address both group selection and bi-level selection methods. We describe several applications of the…
Challenging research in various fields has driven a wide range of methodological advances in variable selection for regression models with high-dimensional predictors. In comparison, selection of nonlinear functions in models with additive predictors has been considered only more recently. Several competing suggestions have been developed at about the same time and often do not refer to each other. This article provides a state-of-the-art review on function selection, focusing on penalized likelihood and Bayesian concepts, relating various approaches to each other in a unified framework. In an…
Selective Inference for Sparse Graphs via Neighborhood Selection
针对稀疏图邻域选择估计缺乏不确定性度量的问题，提出基于外部随机化的选择性推断方法，精确调整边选择事件，实现有效推断，提升功效与精度。
Maximum Likelihood Criterion for Non-nested Model Selection
针对非嵌套候选模型，提出基于最大似然值的模型选择准则，该准则不惩罚参数数量，并证明其一致性。
Selective inference using randomized group lasso estimators for general models
本文针对一般分布与损失函数下的组 lasso 估计量开发了基于随机化的选择性推断方法，通过构建选择后似然进行参数估计与置信区域推断，并在真实数据与模拟中验证了其有效性。
Selective inference (post-selection inference) is a methodology that has attracted much attention in recent years in the fields of statistics and machine learning. Naive inference based on data that are also used for model selection tends to show an overestimation, and so the selective inference conditions the event that the model was selected. In this paper, we develop selective inference in propensity score analysis with a semiparametric approach, which has become a standard tool in causal inference. Specifically, for the most basic causal inference model in which the causal effect can be wr…
Automatic selection of hyper-parameters via the use of softened profile likelihood
本文提出了一种扩展方法，用于自动选择高维数据的超参数，通过软化轮廓似然在scree图中识别肘部，并在弹性网络、支持向量机和神经网络中验证了该方法，同时进行了小规模模拟研究以检验假设的稳健性。
Empirical likelihood is a popular nonparametric statistical tool that does not require any distributional assumptions. In this paper, we explore the possibility of conducting variable selection via Bayesian empirical likelihood. We show theoretically that when the prior distribution satisfies certain mild conditions, the corresponding Bayesian empirical likelihood estimators are posteriorly consistent and variable selection consistent. As special cases, we show the prior of Bayesian empirical likelihood LASSO and SCAD satisfies such conditions and thus can identify the non-zero elements of the…
We develop tools to do valid post-selective inference for a family of model selection procedures, including choosing a model via cross-validated Lasso. The tools apply universally when the following random vectors are jointly asymptotically multivariate Gaussian: 1. the vector composed of each model's quality value evaluated under certain model selection criteria (e.g. cross-validation errors across folds, AIC, prediction errors etc.) 2. the test statistics from which we make inference on the parameters; it is worth noting that the parameters here are chosen after model selection methods are p…
Selective inference is easier with p-values
本文提出选择性支配p值框架，证明常用p值均满足该性质，并借此简化对获胜者推断和排名验证等选择性推断问题，同时推广出组合p值方法的选择性变体。
The regularization approach for variable selection was well developed for a completely observed data set in the past two decades. In the presence of missing values, this approach needs to be tailored to different missing data mechanisms. In this paper, we focus on a flexible and generally applicable missing data mechanism, which contains both ignorable and nonignorable missing data mechanism assumptions. We show how the regularization approach for variable selection can be adapted to the situation under this missing data mechanism. The computational and theoretical properties for variable sele…
In a Gaussian graphical model, the conditional independence between two variables are characterized by the corresponding zero entries in the inverse covariance matrix. Maximum likelihood method using the smoothly clipped absolute deviation (SCAD) penalty (Fan and Li, 2001) and the adaptive LASSO penalty (Zou, 2006) have been proposed in literature. In this article, we establish the result that using Bayesian information criterion (BIC) to select the tuning parameter in penalized likelihood estimation with both types of penalties can lead to consistent graphical model selection. We compare the …
Modeled along the truncated approach in Panigrahi (2016), selection-adjusted inference in a Bayesian regime is based on a selective posterior. Such a posterior is determined together by a generative model imposed on data and the selection event that enforces a truncation on the assumed law. The effective difference between the selective posterior and the usual Bayesian framework is reflected in the use of a truncated likelihood. The normalizer of the truncated law in the adjusted framework is the probability of the selection event; this is typically intractable and it leads to the computationa…
Accounting for selection biases in population analyses: equivalence of the in-likelihood and post-processing approaches
本文证明群体研究中直接将选择效应纳入似然与先推断观测分布再事后消除选择效应两种方法在适当假设下等价，并澄清后一种方法不会产生所谓的潜在偏差。
Flexible Selective Inference with Flow-based Transport Maps
本文提出一种利用归一化流学习传输映射的方法，用于在缺乏解析描述的选择事件下近似条件分布，从而实现灵活的选择性推断，提供有效 p 值、置信集及闭式条件密度函数。
Selective Inference in Growth Curve Models
该研究提出一种数据分裂框架，用于增长曲线模型中的事后选择推断，通过高斯噪声分离选择与推断信息，提供有效推断并提高效率，应用实例验证其有效性。
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