diff --git "a/MNAyT4oBgHgl3EQfs_kD/content/tmp_files/load_file.txt" "b/MNAyT4oBgHgl3EQfs_kD/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/MNAyT4oBgHgl3EQfs_kD/content/tmp_files/load_file.txt" @@ -0,0 +1,2300 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf,len=2299 +page_content='Selective Conformal Inference with FCR Control Yajie Baoa, Yuyang Huob, Haojie Rena and Changliang Zoub∗ aSchool of Mathematical Sciences, Shanghai Jiao Tong University Shanghai, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' China bSchool of Statistics and Data Science, Nankai University Tianjin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' China January 3, 2023 Abstract Conformal inference is a popular tool for constructing prediction intervals (PI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We consider here the scenario of post-selection/selective conformal inference, that is PIs are reported only for individuals selected from an unlabeled test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To account for multiplicity, we develop a general split conformal framework to construct selective PIs with the false coverage-statement rate (FCR) control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We first investigate the Benjamini and Yekutieli (2005)’s FCR-adjusted method in the present setting, and show that it is able to achieve FCR control but yields uniformly inflated PIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We then propose a novel solution to the problem, named as Selective COnditional conformal Predictions (SCOP), which entails performing selection procedures on both calibration set and test set and construct marginal conformal PIs on the selected sets by the aid of conditional empirical distribution obtained by the calibration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under a unified framework and exchangeable assumptions, we show that the SCOP can exactly control the FCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' More importantly, we provide non-asymptotic miscoverage bounds for a general class of selection procedures beyond exchangeablity and discuss the conditions under which the SCOP is able to control the FCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' As special cases, the SCOP with quantile-based selection or conformal p-values-based multiple testing procedures enjoys valid coverage guarantee under mild conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Numerical results confirm the effectiveness and robustness of SCOP in FCR control and show that it achieves more narrowed PIs over existing methods in many settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Keywords: Conditional empirical distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Distribution-free;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Non-exchangeable conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Post- selection inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Prediction intervals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Split conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' ∗Corresponding Author: nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='chlzou@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='00584v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='ME] 2 Jan 2023 1 Introduction To improve the prediction performance in modern data, many sophisticated machine learning algorithms including various “black-box” models are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' While often witnessing empirical success, quantifying prediction uncertainty is one of the major issues for interpretable machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Conformal inference (Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 1999, 2005) provides a powerful and flexible tool to quantify the uncertainty of predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Consider a typical setting that we observe one labeled data set Dl = {(Xi, Yi)}2n i=1 and a set of unla- belled/test samples Du = {Xi}2n+m i=2n+1 whose outcomes {Yi}2n+m i=2n+1 are unobserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Generally, suppose all (Xi, Yi) ∈ X × Y are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d from some unknown distribution, and µ(x) := Y | X = x as the prediction model associated with (X, Y ), which is usually estimated by the labeled data Dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For any Xj ∈ Du and a given miscoverage level α, standard conformal prediction methods (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018), yield a prediction interval (PI) with distribution-free coverage guarantee, PIα(Xj), P(Yj ∈ PIα(Xj)) ≥ 1 − α, under independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d) (or exchangeable data) assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' With the development of big data, making predictive inference on all available data (Du) is either unnecessary or inefficient in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For example, in the recruitment decisions, only some selected viable candidates can get into interview processes (Faliagka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Shehu and Saeed, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In the drug discovery trials, researchers select promising ones based on predicting candidates’ activity for further clinical trials (Carracedo-Reboredo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Dara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Related applications also appear in financial investment and scientific discovery (Jin and Candès, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In such problems, the most common way is to select a subset of interest with some rules through some statistical/machine learning algorithms at first, and then perform statistical inference only on the selected samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Formally, letting ˆSu ⊆ {2n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' , 2n + m} be the selected subset, our goal is to construct the PI of Yj for each j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' As pointed by Benjamini and Yekutieli (2005), ignoring the multiplicity in construction of post-selection intervals will result in distorted average coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under the context of post-selection inference in which confidence intervals for multiple selected parameters/variables are being reported, Benjamini and Yekutieli (2005) pioneered the criterion, false coverage-statement rate (FCR), to take account for multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The FCR, an analog of the false discovery rate (FDR), can readily be adapted to the present conformal inference setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' It is defined as the expected ratio of number of reported PIs failing to cover their respective true outcomes to the total number of reported PIs, say FCR := E � |{j ∈ ˆSu : Yj ̸∈ PIj}| max{| ˆSu|, 1} � , (1) where PIj is the PI for the selected sample j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Benjamini and Yekutieli (2005) provided a selection- agnostic method which adjusts the confidence level through multiplying α by a quantity which is related to the proportion of selected candidates over all candidates and then constructed the marginal confidence 2 intervals at the adjusted level for each selected candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We will hereafter call it the FCR-adjusted method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Accordingly, we may expect that the FCR-adjusted PIs enjoy valid FCR control properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, due to the dependence structure among PI| ˆ Su|α/m(Xj)’s, the results in Benjamini and Yekutieli (2005) are not directly applicable in the setting of conformal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Please refer to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 for detailed discussions and rigorous theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We notice that Weinstein and Ramdas (2020) also discussed the selective inference problem under the framework of conformal prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The authors suggested to use the FCR-adjusted method, however, they did not provide theoretical or empirical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' While the FCR-adjusted approach can reach FCR control and is widely used, it is generally known to yield uniformly inflated confidence intervals (Weinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This is because that when calculating the noncoverage probabilities of confidence intervals, the adjusted confidence intervals do not take into account the selection event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Along this line, Weinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2013), Zhao and Cui (2020) and Zhao (2022) further proposed some methods to narrow the adjusted confidence intervals by incorporating more selection information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Among some others, Fithian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2014), Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2016) and Taylor and Tibshirani (2018) proposed constructing conditional confidence intervals for each selected variables and showed that the selective error rate can be controlled given that the selected set is equal to some deterministic subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, those methods either require some tractable conditional distribution assumptions or are only applicable for some given prediction algorithms, such as normality assumptions or LASSO model, which would limit their applicability in the conformal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Fortunately, by the virtue of the availability of Dl, distribution/model-agnostic conditional prediction intervals with theoretical guarantee can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 Our contributions In this paper, we develop a novel conformal framework to construct post-selection prediction intervals while control the FCR, named as Selective COnditional conformal Predictions (SCOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our method stems from the split conformal inference (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Fithian and Lei, 2020), where the labeled data Dl is split into two disjoint parts, one as the training set for obtaining a prediction model ˆµ(X), and the remaining one as the calibration set for estimating the distribution of the discrepancy between the Y and ˆµ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then, the key ingredient of our proposal entails performing a pre-specified selective procedure on both the calibration set and the test set and construct the marginal conformal PIs on the selected sets with the help of conditional empirical distribution obtained by the calibration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The proposed SCOP procedure is model- or distribution-agnostic, in the sense that it could wrap around any prediction algorithms with commonly used selection procedures to construct PIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The main contributions of the paper can be summarized as follows: Firstly, we investigate the FCR-adjusted method in the setting of conformal inference and show that it is able to achieve FCR control under mild conditions, which lays a foundation for our subsequent development of SCOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 3 Secondly, under a unified framework and exchangeable assumptions, we show that the SCOP can exactly control the FCR at the target level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Thirdly, we provide non-asymptotic miscoverage bounds for a general class of selection procedures beyond exchangeablity, termed as ranking-based procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This broadens the scopes of our SCOP in theoretical guarantee and practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To address the non-exchangeability between the the post-selection test set and calibration set, we introduce a virtual post-selection calibration set in our proof, and then quantify the conditional miscoverage gap between the virtual calibration and the real calibration in SCOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This new technique may be of independent interest for conformal prediction for non-exchangeable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Finally, we illustrate the easy coupling of the SCOP with commonly used prediction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Numerical experiments indicate that it yields more accurate FCR control than existing methods, while offers the narrowed prediction intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2 Connections to existing works Post-selection inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Post-selection inference on a large number of variables has attracted considerable research attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Besides the references mentioned before, a relevant direction is the splitting-based strategy for high-dimensional inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The number of variables is firstly reduced to a manageable size using one part of data, while confidence intervals or significance tests can be constructed by computing estimates in a low-dimensional region with the other part of data and selected variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' See Wasserman and Roeder (2009), Rinaldo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2019), Du et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2021) and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' One potential related work is Chen and Bien (2020), in which the authors considered to construct confidence intervals for regression coefficients after removing the potential outliers from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our paradigm differs substantially with those works as we focus on post-selection inference for sample selection rather than variable selection, and existing works on variable selection is difficult to extend to the present problem due to the requirements on model or distribution assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Conformal prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The building block of our SCOP is the conformal inference framework, which has been well studied in many settings, including non-parametric regression (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2013), quantile regression (Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2019), high-dimensional regression (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018) and classification (Sadinle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2020), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' More comprehensive reviews can be found in Shafer and Vovk (2008), Zeni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2020) and Angelopoulos and Bates (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Conventionally, conformal PIs enjoy distribution-free marginal coverage guarantee with the assumption that the data are exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, the exchangeability may be violated in practice and would be more severe in the post-selection conformal inference because the selection procedure might be determined by either the labelled data or the test data, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In such situations, one particularly difficult issue is that the selected set ˆSu is random and has a complex dependence structure to the labelled data and test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Some conformal inference beyond exchangeability has attracted attention 4 (Tibshirani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Candès et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In particular, Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2022) proposed a general framework to implement conformal inference when the algorithms cannot treat data exchangeable and theoretically displayed the coverage deviations compared from exchangeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, how to decouple the dependence to achieve FCR control in the present framework remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Taking a different but related perspective from multiple-testing, Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2021) proposed a method to construct conformal p-values with data splitting and apply it to detect outliers with finite-sample FDR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2022) extended that method and proposed a Jackknife implementation combined with automatic model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Jin and Candès (2022) considered a scenario that one aims to select some individuals of interest from the test sample and proposed a conformal p-value based method to control the FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Those existing works are not concerned about the construction of PIs, which differs with our focus essentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='3 Organization and notations The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We introduce the FCR-adjusted prediction and SCOP for valid FCR control in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Section 3 presents the theoretical properties of SCOP for ranking-based procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Numerical results and real-data examples are presented in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Section 5 concludes the paper, and the technical proofs are relegated to the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For a positive integer n, we use [n] to denote the index set {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let A = {Ai : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', n} be a set of n real numbers, and S ⊆ [n] be an index subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We use AS (ℓ) to denote the ℓ-th smallest value in {Ai : i ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We use 1 {·} to denote the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For a real random sequences Xn and an non-negative real deterministic sequence an, we write Xn = Op(an) if for any ϵ > 0, there exists some constant C > 0 such that P(|Xn| > Can) ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In our paper, the notations with subscript c or u refer to depending on the calibration set or the test set respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 2 Selective conditional conformal prediction Denote the index sets for the labelled data Dl and the test data Du as L = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' , 2n} and U = {2n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' , 2n + m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The main prediction method studied in this paper is built upon the split conformal framework (Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018), which is also called “inductive conformal prediction”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' That is we randomly split Dl into two disjoint parts, the training set Dt and the calibration set Dc with n samples respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We can firstly train a prediction model ˆµ(X) on the Dl, and then compute the empirical quantiles of the residuals Ri = |Yi − ˆµ(Xi)| on the calibration set Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For Xj ∈ Du, the (1 − α)-marginal conformal PI is PIM j = � ˆµ(Xj) − QC(1 − α), ˆµ(Xj) + QC(1 − α) � , (2) where QC(1 − α) is the ⌈(1 − α)(n + 1)⌉-st smallest value in RC = {Ri = |Yi − ˆµ(Xi)| : i ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (or more generally, exchangeable) assumption on Dc ∪ {(Xj, Yj)}, the marginal PI in (2) enjoys the coverage guarantee, P � Yj /∈ PIM j � ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 5 Suppose g : X → R be one plausible score function, which can be user-specified or estimated by the training data Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' A particular selection procedure S can be applied to g(Xi) for i ∈ U to find the samples of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For simplicity, denote Ti = g(Xi) and those Xi’s with smaller values of Ti tend to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Denote the selected set as ˆSu = {i ∈ U : Ti ≤ ˆτ}, where ˆτ is the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Different selective procedures S can be chosen from different perspectives, and we summarize the selection threshold ˆτ into three types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (Fixed threshold) The ˆτ is user-specified or independent of the whole data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For example, ˆτ = t, where t is either known as a priori or could possibly be obtained from an independent process of Dc ∪ Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (Self-driven threshold) The ˆτ is only dependent on the scores {Ti : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This type includes the Top-K which choose the first K individuals, and the quantile of Ti values in the test set which a given proportion of individuals with smallest Ti values in the test set, respectively (Fithian and Lei, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (Calibration-assisted selection) The ˆτ relies on the calibration set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For example, ˆτ is some quantile of true response Yi in calibration set, or the quantile based on both calibration and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In particular, one may employ some multiple testing procedures to achieve error rate control, such like FDR control based on the Benjamini–Hochberg (BH) procedure (Benjamini and Hochberg, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Consequently, the {Ti : i ∈ C} is required to approximate the distribution of {Ti : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our goal is to construct conformal PIs for the selected subset ˆSu with the FCR control at α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 Adjusted conformal prediction We firstly adapt the Benjamini and Yekutieli (2005)’s FCR-adjusted method to the present setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Define Mj min := min y � | ˆSTj←y u | : j ∈ ˆSTj←y u � , where ˆSTj←y u denotes the selected subset when replacing Tj with value y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The FCR-adjusted conformal PIs are amount to marginally constructing larger 1 − α × Mj min/m PIs instead of 1 − α level in (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', PIAD j = � ˆµ(Xj) − QC(1 − α × Mj min), ˆµ(Xj) + QC(1 − α × Mj min) � , j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (3) Notice that given ˆµ(·) and ˆSu, PIAD j ’s are not independent of each other because they all rely on the empirical quantile obtained from Dc, and therefore the proofs in Benjamini and Yekutieli (2005) are not readily extended to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following result demonstrates that the FCR-adjusted approach can successfully control the FCR for any selection threshold that is independent of the calibration set given training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose that given Dt, {Ti : i ∈ C ∪ U} are independent random variables and the selection threshold ˆτ is independent of Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then the FCR value of the FCR-adjusted method in (3) satisfies FCRAD ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For many plausible selection rules such as fixed-threshold selection, the Mj min can be replaced by the cardinality of the selected subset | �Su|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In practice, for ease of computation, one may prefer to use this 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='0 Ri density Marginal Conditional Test Figure 1: The densities of Ri for i ∈ Dc (in blue), Ri for i ∈ ˆSc (in green) and the density of Rj for j ∈ ˆSu (in red), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' There are 2n = 400 labeled data and m = 200 test data generated from a linear model with heterogeneous noise, where the details of the model are in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The selection rule is ˆS = {k : ˆµ(Xk) ≤ −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' simplification, even though it does not have a theoretical guarantee for many data-dependent selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The FCR-adjusted method is known to be quite conservative (Weinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2013), because it does not incorporate the selection event into the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Take the Top-K selection as an intuitive example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The selected set ˆSu is fixed with | ˆSu| = K and the FCR can be written as FCR = 1 K � j∈U P � j ∈ ˆSu, Yj ̸∈ PIj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (4) Since the marginal PIAD j reaches the 1 − αK/m confidence level for any fixed K, the FCR-adjusted method achieves the FCR control via FCRAD = 1 K � j∈U P � j ∈ ˆSu, Yj ̸∈ PIAD j � ≤ 1 K � j∈U P � Yj ̸∈ PIAD j � ≤ α, where the first inequality might be rather loose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' A simple yet effective remedy is to use conditional calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2 Selective conditional conformal prediction (SCOP) We start by making a decomposition of the FCR according to the contribution of each sample in the selected set ˆSu, given as P � Yj ̸∈ PIj |j ∈ ˆSu � P � j ∈ ˆSu � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Notice that the FCR can naturally be controlled at level α if the conditional control satisfies P(Yj ̸∈ PIj |j ∈ ˆSu) ≤ α, which sheds light on the construction of conditional conformal PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In the regime of conformal inference, the conditional uncertainty of |Yj − ˆµ(Xj)| given j ∈ ˆSu can be reliably approximated using the calibration set Dc, enabling us to construct model/distribution-agnostic conditional 7 PIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To be specific, we conduct the selective algorithm S on the fitted score values {Ti = g(Xi) : i ∈ C} and obtain the post-selection calibration set ˆSc = {i ∈ C : Ti ≤ ˆτ} with the same threshold ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Notice that ˆSc is formed via the same selection criterion with ˆSu, and thus we utilize the residuals Ri for i ∈ ˆSc to approximately characterize the conditional uncertainty of Rj for j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To visualize the effect, we consider a linear model with heterogeneous noise, where we use ordinary least-squares for predictions and select ˆSu = {j ∈ U : ˆµ(Xj) ≤ −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In Figure 1, we display the densities of Ri for i ∈ Dc, Ri for i ∈ ˆSc and the density of Rj for j ∈ ˆSu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The selection procedure significantly distorts the distribution of residuals, but the conditional uncertainty on ˆSu can be well approximated by that on ˆSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The conditional conformal PI for j ∈ ˆSu can accordingly be constructed as PISCOP j = � ˆµ(Xj) − Q ˆ Sc(1 − α), ˆµ(Xj) + Q ˆ Sc(1 − α) � , (5) where Q ˆ Sc(1 − α) is the ⌈(1 − α)(| ˆSc| + 1)⌉-st smallest value in R ˆ Sc = {Ri : i ∈ ˆSc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We refer this procedure as Selective COnditional conformal Prediction (SCOP) and summarize it in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following theorem shows that SCOP can control the FCR at α for exchangeable selective procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Further, if the selection scores Ti are continuous (or almost surely distinct), we can obtain a lower bound for the FCR value, guaranteeing that the SCOP is nearly exact in O(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose {Ti : i ∈ C ∪ U} are exchangeable random variables, and the threshold ˆτ is also exchangeable with respective to the {Ti : i ∈ C ∪ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For each j ∈ U, the conditional miscoverage probability is bounded by P � Yj ̸∈ PISCOP j |j ∈ ˆSu � ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (6) Further, the FCR value of the SCOP algorithm is controlled at FCRSCOP ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In addition, if Ti follows a continuous distribution for i ∈ C ∪ U and P(| ˆSu| > 0) = 1, we also have P � Yj ̸∈ PISCOP j |j ∈ ˆSu � ≥ α − 1 n + 1 and FCRSCOP ≥ α − 1 n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under the exchangeable assumption, the FCR results actually match the marginal miscoverage results of original conformal PIs (Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This theorem relies on exchangeability in two ways, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', the fitted selection score {Ti : i ∈ C ∪ U} are exchangeable and the selection threshold ˆτ is assumed to keep the same value by swapping Tj and Tk for any j, k ∈ C ∪ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The former one is commonly used in conformal inference and holds easily when the data are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d given Dt (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The later one imposes restrictions on the selection procedures and can be fulfilled with some practical thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The simplest case is the fixed threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Another popular example is that ˆτ is some quantile of {Ti : i ∈ C ∪ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, many selection procedures may be excluded, such as the Top-K selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In such cases, the threshold ˆτ is only determined by the test data U, which does not treat the data points from calibration and test sets symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We will next explore the effectiveness of the proposed SCOP for more general selection procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 8 Algorithm 1 Selective COnditional conformal Prediction (SCOP) Input: Labeled set Dl, test set Du, selection procedure S, target FCR level α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 1 (Splitting and training) Split Dl into training set Dt and calibration set Dc with equal size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Fit prediction model ˆµ(·) and score function g (if needed) on the training set Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 2 (Selection) Compute the scores: TC = {Ti = g(Xi) : i ∈ C} and TU = {Ti = g(Xi) : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Apply the selective procedure S to TC ∪ TU and obtain the threshold value ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Obtain the post-selection subsets: ˆSu = {i ∈ U : Ti ≤ ˆτ} and ˆSc = {i ∈ C : Ti ≤ ˆτ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 3 (Calibration) Compute residuals: RSc = {Ri = |Yi − ˆµ(Xi)| : i ∈ ˆSc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Find the ⌈(1−α)(| ˆSc|+1)⌉-st smallest value of RSc, Q ˆ Sc(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 4 (Construction) Construct PI for each j ∈ ˆSu as PISCOP j = [ˆµ(Xj)−Q ˆ Sc(1−α), ˆµ(Xj) +Q ˆ Sc(1−α)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Output: Prediction intervals {PISCOP j : j ∈ ˆSu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In predictive inference, several works considered to approximately construct the conditional PI (Chernozhukov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2021), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', P (Yj /∈ PI(Xj)|Xj = x) ≤ α, for any x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (7) However, it is well known that achieving “fully" conditional validity in (7) is impossible in distribution-free regime (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Foygel Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our conditional miscoverage control in (6) is a weaker guarantee compared with (7), since we only consider the selection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For more discussion about these two conditional guarantees, we refer to Appendix B in Weinstein and Ramdas (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The SCOP can leverage the post-selection calibration set to approximate the selective conditional distribution of residuals, which contributes to achieve a better conditional coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In addition, the conditional calibration of SCOP provides an anti-conservative lower bound for FCR value in the continuous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 3 Ranking-based selection In this section, we consider a general class of selection procedures named ranking-based selection and discuss the conditions under which the proposed SCOP is able to control the FCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2, we discuss the FCR control for the self-driven selection procedures and calibration-assisted ones, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='3, we demonstrate the effectiveness of the SCOP procedure when the selection procedures based on conformal p-values are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We begin with some general assumptions and notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For simplicity, we suppose Ti ∈ [0, 1] and the selection algorithm S conducted on {Ti : i ∈ C ∪ U} outputs a ranking threshold ˆκ ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Say, we have the selection threshold ˆτ = TU (ˆκ) as the ˆκ-th smallest value in TU = {Tj : j ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then the selected subset of the 9 test set can be rewritten as ˆSu = � j ∈ U : Tj ≤ TU (ˆκ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (8) The ranking-based procedure in (8) incorporates many practical examples, such as Top-K selection, quantile- based selection, step-up procedures (Fithian and Lei, 2020) and the well-known BH procedure1 (Benjamini and Hochberg, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' With the ranking based selection, we have | ˆSu| = ˆκ, which is usually random and coupled to each test sample Xj ∈ Du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To decouple the dependence, we introduce Lemma 1 to control the FCR through conditioning on the leave-one-out data set Du,−j, which is the test set Du without the sample j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Denote EDu,−j[·] and PDu,−j(·) as the conditional expectation and probability given Du,−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let ˆκj←tu be the ranking threshold obtained from the selection algorithm S by replacing Tj with some deterministic value tu ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose | ˆSu| > 0 almost surely and ˆκ = ˆκj←tu holds for any j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' If the conditional false coverage probability satisfies ���PDu,−j � Yj ̸∈ PIj ��j ∈ ˆSu � − α ��� ≤ ∆(Du,−j), (9) where ∆(Du,−j) only depends on the data set Du,−j, then we have | FCR −α| ≤ E � � 1 | ˆSu| � j∈ ˆ Su ∆(Du,−j) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The leave-one-out technique often appears in the literature about FDR control under dependence (Heesen and Janssen, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Fithian and Lei, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The key is to decompose the FCR into the summation of conditional miscoverage probability of each candidate given other test samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='e, FCR = E � �� j∈U 1 ˆκj←tu PDu,−j � Yj ̸∈ PIj |j ∈ ˆSu � PDu,−j � j ∈ ˆSu � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We may regard the term ∆(Du,−j) in (9) as the individual FCR contribution of the j-th candidate in ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The detailed proof of Lemma 1 is deferred to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Next, we introduce two universal assumptions to find how the conditional false coverage probability in (9) holds and further control the FCR of SCOP with the ranking-based selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Denote the cumulative distribution functions (CDF) of Ri and Ti as FT (·) and FR(·), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let F(R,T )(·, ·) be the joint CDF of (Ri, Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The score function g depends only on the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose {Ti : i ∈ C ∪ U} and {Ri : i ∈ C ∪ U} are both i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' continuous random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' There exists some ρ ∈ (0, 1) such that d drF(R,T ) � F −1 R (r), F −1 T (t) � ≥ ρt, holds for any t ∈ (0, 1) and r ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 1The BH procedure is also an example of step-up procedures, see Fithian and Lei (2020) 10 Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' There exists some deterministic value tu ∈ [0, 1] such that ˆκj←tu = ˆκ holds for any j ∈ ˆSu, and ˆκj←tu ≤ ˆκ + Iu holds for any j ∈ U \\ ˆSu and some positive integer Iu ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To facilitate our technical development, we impose mild distributional assumption on the joint CDF of (Ri, Ti) in Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' It is worth noticing that the same selected set ˆSu and ˆSc can be obtained if one applies the ranking-based selection procedure to the transformed scores {FT (Ti) : i ∈ U} instead of the scores {Ti : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Also, the transformed residuals {FR(Ri) : i ∈ C ∪ U} keep the original order as the residuals {Ri : i ∈ C ∪ U} in the conformal coverage control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Therefore, without loss of generality, we can assume that Ti i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' ∼ Unif([0, 1]) and Ri i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' ∼ Unif([0, 1]) for i ∈ C ∪ U in the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Then the condition on CDF in Assumption 1 will reduce to d drF(R,T )(r, t) ≥ ρt which appears quite weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For Assumption 2, we can verify that ˆκ = ˆκj←tu under the event {j ∈ ˆSu} in many cases, such as the quantile-based selection procedure and BH procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For the selection procedures with fixed ranking threshold, such as quantile-based selection and Top-K selection, Assumption 2 is clearly satisfied with tu = 0 and Iu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For the BH procedure based on the conformal p-values, taking tu as 0 for each j ∈ ˆSu leads a smaller p-value pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' By the property of the BH procedure, for j ∈ ˆSu, assigning pj to a smaller value will not change the rejection set (Fithian and Lei, 2020), and hence we have κj←0 = ˆκ for j ∈ ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='3, we also show that Iu = Op(log m) for the BH procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' From now on, we write ˆκ(j) = ˆκj←tu for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 FCR control for self-driven selection When the self-driven selection procedures are used, the samples in the selected calibration set ˆSc and the selected test set ˆSu are not exchangeable, but the original samples from the calibration set and the test set are exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following theorem provides delicate bounds for the conditional miscoverage gap of the SCOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under Assumptions 1 and 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' for any absolute constant C ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' if 8C log n/(nTU\\{j} (ˆκ(j)) ) ≤ 1 holds almost surely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' the conditional miscoverage probability can be bounded by PDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � Yj ̸∈ PISCOP j ���j ∈ ˆSu � ≤ α + ∆(Du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' and PDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � Yj ̸∈ PISCOP j ���j ∈ ˆSu � ≥ α − 2∆(Du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' where ∆(Du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j) = 8C log n ρTU\\{j} (ˆκ(j)−Iu) � �6C log n nTU\\{j} (ˆκ(j)) + TU\\{j} (ˆκ(j)) − TU\\{j} (ˆκ(j)−1) TU\\{j} (ˆκ(j)) � � + 2 � TU\\{j} (ˆκ(j)) − TU\\{j} (ˆκ(j)−Iu) � TU\\{j} (ˆκ(j)−Iu) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (10) Our theorem is closely connected to Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2022)’s Theorem 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Both theorems involve assessing how the deviations from the “idealized” exchangability would affect the actual miscoverage level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' However, 11 the interpretations are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Barber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2022) showed that under assumption that the test and calibration samples are non-exchangeable, the miscoverage gap can be bounded by an error term regarding the total variation between the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Whereas in SCOP the deviation comes from the possible violation of the similarity between the distributions of {Ri : i ∈ ˆSc} and {Rj : j ∈ ˆSu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The technical difficulty in proving Theorem 2 lies in coping with the dependence of ˆSc and ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To address this problem, we introduce virtual post-selection test set and calibration set, ˆS(j) u = � i ∈ U : Ti ≤ TU\\{j} (ˆκ(j)) � and ˆS(j) c = � i ∈ C : Ti ≤ TU\\{j} (ˆκ(j)) � respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We denote the corresponding virtual conformal PI constructed by ˆS(j) c as PIj( ˆS(j) c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For clarity, we rewrite the real conformal PI constructed by ˆSc in Algorithm 1 as PIj( ˆSc) ≡ PISCOP j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Notice that, the threshold TU\\{j} (ˆκ(j)) and the virtual selected calibration set ˆS(j) c are independent of the test candidate j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Therefore, the test candidate j and the calibration candidate k are ex- changeable in the set ˆS(j) c ∪{j} under the selection conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' It remains to control two conditional miscoverage gaps: PDu,−j � j ̸∈ PIj( ˆS(j) c )|j ∈ ˆS(j) u � − α and PDu,−j � j ̸∈ PIj( ˆSc)|j ∈ ˆSu � − PDu,−j � j ̸∈ PIj( ˆS(j) c )|j ∈ ˆS(j) u � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' the former can be bounded as in conventional conformal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Our theorem shows that a tight control of the deviation term ∆(Du,−j) in (10) leads to effective FCR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Next we carefully interpret the bound and present more explicit settings in which the FCR achieves or is very close to the nominal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Observe that controlling ∆(Du,−j) actually boils down to establishing the upper bound of the difference TU\\{j} (ˆκ(j)) − TU\\{j} (ˆκ(j)−Iu) and the lower bound of the denominator TU\\{j} (ˆκ(j)−Iu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To guarantee that the denominator will stay away from 0, we impose the following assumption on ˆκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The ranking threshold satisfies ˆκ ≥ γm for some γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The lower bound on ˆκ in Assumption 3 is mild and reasonable, since the FCR control will be extremely difficult when | ˆSu|/n = ˆκ/n = o(1) for a small level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Applying the well-known representation of spacing between consecutive order statistics (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1) to {Ti}i∈U\\{j}, together with Assumption 3, we can obtain the following FCR control result for self-driven selection procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under Assumptions 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' If γ > Iu/m, the FCR value of SCOP with self-driven selection procedures can be controlled at FCRSCOP = α + O � log2(n ∨ m) ργ(γ − Iu/m) �Iu m + 1 n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In the asymptotic regime, FCRSCOP is exact if Iu = o(m), that is lim(n,m)→∞ FCRSCOP = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Recalling that for quantile-based selection and Top-K selection, we have Iu = 0, and thus Theorem 3 guarantees the FCR of SCOP with such selection procedures can attain the target level in a nearly optimal rate (up to a logarithmic factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2 FCR control for calibration-assisted selection For calibration-assisted selective procedures, the analysis is more complex because the ranking threshold ˆκ depends also on the calibration set Dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' It implies that a more tractable ranking threshold is needed to decouple the dependence on the selected samples and the calibration samples simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' That is for any j ∈ ˆSu and k ∈ C, let ˆκ(j,k) ≡ ˆκj←tu,k←tc be the ranking threshold by replacing Tj with tu and Tk with tc simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The virtual post-selection calibration set is further defined as ˆS(j,k) c = � i ∈ C \\ {k} : Ti ≤ TU\\{j} (ˆκ(j,k)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following assumption, an analog of Assumption 2, is imposed to restrict the change in the ranking threshold after replacing one calibration score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' There exists some tc ∈ R and some positive integer Ic ≤ m such that ˆκ ≤ ˆκk←tc ≤ ˆκ + Ic holds for any k ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The following theorem is parallel with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under Assumptions 1-4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' for the calibration-assisted selection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' the conditional miscoverage probability of SCOP satisfies PDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � Yj ̸∈ PISCOP j ��j ∈ ˆSu � ≤ α + EDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � max k ∆(D(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' and PDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � Yj ̸∈ PISCOP j ��j ∈ ˆSu � ≥ α − 2EDu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='−j � max k ∆(D(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' where ∆(D(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) := 2TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � R ˆ S(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c (U (j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) − R ˆ S(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c (L(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � � TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)−Iu−Ic) �2 + 4 � TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) − TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)−Iu−Ic) � TU\\{j} (ˆκ(j)) + d(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) | ˆS(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c | + 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (11) with d(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) = � i∈ ˆ S(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c 1 � TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)−Ic−1) < Ti ≤ TU\\{j} (ˆκ(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' U (j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) = ⌈(1 − α)(| ˆS(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c | + 2)⌉ + d(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) and L(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) = ⌈(1 − α)(| ˆS(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k) c | + 2 − d(j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='k))⌉ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We can see that all the terms in (11) are independent of the samples j ∈ ˆSu and k ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The quantity d(j,k) measures the size difference between the real calibration set ˆSc and the virtual calibration set ˆS(j,k) c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The term R ˆ S(j,k) c (U (j,k)) − R ˆ S(j,k) c (L(j,k)) represents the largest possible distance of the corresponding quantiles in ˆSc and ˆS(j,k) c , which can be bounded in ˆS(j,k) c conditional on the data set Du,−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The remaining parts in maxk ∆(D(j,k)) rely on the difference between thresholds from TU\\{j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Equipped with the conditional miscoverage gap in Theorem 4, we can obtain the FCR control results of SCOP with calibration-assisted selection in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Under Assumptions 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' If γ > (Ic + Iu)/m, the FCR value of SCOP for calibration-assisted selection can be controlled at FCRSCOP = α + O � log2(n ∨ m) ρ(γ − (Ic + Iu)/m)2 �Ic + Iu m + 1 n �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 13 Similar to the results with self-driven selection, the SCOP can control the FCR around the target value with a small gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To decouple the dependence between the ranking threshold and calibration set, an addition term Ic/m appears in Theorem 5, regarding to the effect of replacing one calibration sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' If Ic ∨Iu = o(m), then we can take m = exp{o(n 1 2 )} and have lim(n,m)→∞ FCRSCOP = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='3 Prediction-oriented selection with conformal p-values We discuss the implementation of the SCOP with a special calibration-assisted selection procedure, the selection via multiple testing based on conformal p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The concept of the conformal p-value was proposed by Vovk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Similar to the conformal PI, the conformal p-values enjoy model/distribution-free properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Recently, there exists some works to apply conformal p-values to implement sample selection from a multiple-testing perspective, such as Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2021) and Jin and Candès (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In particular, Jin and Candès (2022) investigated the prediction-oriented selection problem, aiming to select samples whose unobserved outcomes exceed some specified values while control the proportion of falsely selected units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' This problem can be viewed as the following multiple hypothesis tests: for i ∈ U and some b0 ∈ R, H0,i : Yi ≥ b0 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' H1,i : Yi < b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' By choosing a monotone function g0 : R+ → [0, 1], one could take the score function as g(x) = g0(ˆµ(x) − b0) and compute the conformity scores as {Ti = g(Xi) : i ∈ C ∪ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Denote the null set of calibration samples as C0 = {i ∈ C : Yi ≥ b0} and its size as n0 = |C0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Given the conformity scores {Ti : i ∈ C0} in the calibration set, the conformal p-value for each test data point can be calculated by2 pj := p(Xj) = 1 + |{i ∈ C0 : Ti ≤ Tj}| n0 + 1 , for j ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (12) To control FDR at the target level β ∈ (0, 1), we may deploy BH procedure to {pj : j ∈ U} and obtain the rejection set ˆSu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let pU (1) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' ≤ pU (m) be order statistics of conformal p-values in the test set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For any i ∈ U, it holds that {pi ≤ pU (ˆκ)} = {Ti ≤ TU (ˆκ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='1 indicates that using the conformal p-values in (12) to obtain ˆSu is equivalent to using the conformity scores in TU with the same ranking threshold ˆκ, that is ˆSu = {i ∈ U : pi ≤ pU (ˆκ)} ≡ {i ∈ U : Ti ≤ TU (ˆκ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Further, we can also obtain the post-selection calibration set by ˆSc = {i ∈ C : Ti ≤ TU (ˆκ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Therefore, we can frame the BH procedures with conformal p-values as a calibration-assisted selection in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 2Under our continuous assumption, we present the form of conformal p-value without ties in the conformity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For the tie-breaking form, please refer to Bates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' (2021) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' 14 Algorithm 2 SCOP under selection with conformal p-values Input: Training data Dt, calibration data Dc, test data Du, threshold sequence {δ(r) : r ∈ [m]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 1 Fit prediction model ˆµ(·) and score function g(·) on Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Compute the score values TC = {Ti = g(Xi) : i ∈ C} and TU = {Ti = g(Xi) : i ∈ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 2 Compute the conformal p-values {pi : i ∈ U} according to (12) based on DC0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Apply the BH procedure with target level β to TU and obtain (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Obtain the post-selection subsets: ˆSu = {i ∈ U : Ti ≤ TU (ˆκ)} and ˆSc = {i ∈ C : Ti ≤ TU (ˆκ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 3 Compute residuals: RSc = {Ri = |Yi − ˆµ(Xi)| : i ∈ ˆSc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Find the ⌈(1 − α)(| ˆSc| + 1)⌉-st smallest value of RSc, denoted by Q ˆ Sc(1 − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Step 4 Construct PI for each j ∈ ˆSu as PIj = [ˆµ(Xj) − Q ˆ Sc(1 − α), ˆµ(Xj) + Q ˆ Sc(1 − α)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Output: {PIj : j ∈ ˆSu}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To study the FCR control with selection procedures based on conformal p-values, we consider a more general class of step-up procedures introduced by Fithian and Lei (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let 0 ≤ δ(1) ≤ · · · ≤ δ(m) ≤ 1 denote an increasing sequence of thresholds, we choose the ranking threshold for step-up procedures as ˆκ = max � r : pU (r) ≤ δ(r) � , (13) where pU (r) is the rth-smallest conformal p-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Specially, the BH procedure takes δ(r) = rβ/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' We summarize the SCOP with the step-up selection procedures in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' To adapt Assumptions 2 and 4, we can simply take ˆκ(j) = ˆκj←0 and ˆκ(k) = ˆκk←1 by replacing Tj with 0 for j ∈ U and Tk with 1 for k ∈ C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' From Lemma 1 in Fithian and Lei (2020), we have ˆκ(j) = ˆκ for any j ∈ ˆSu in the step-up procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' The next proposition characterizes the magnitudes of ˆκ(j) − ˆκ for any j ̸∈ ˆSu and ˆκ(k) − ˆκ for any k ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Suppose {Xi : i ∈ C ∪ U} are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' continuous random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' Let Ω(r) = {ℓ ∈ [n0] : δ(r) < ℓ+1 n0+1 ≤ δ(r + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For step-up procedures (13) using conformal p-values defined in (12) and any absolute constant C > 1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' For any j ∈ ˆSu, we have ˆκ(j) = ˆκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNAyT4oBgHgl3EQfs_kD/content/2301.00584v1.pdf'} +page_content=' In addition, for any j ∈ U \\ ˆSu, ˆκ(j) − ˆκ ≤ 12C log m + 8Cm log m n0 + 1 max ⌈γm⌉