Model Confidence Bounds for Variable Selection

Li, Yang; Luo, Yuetian; Ferrari, Davide; Hu, Xiaonan; Qin, Yichen

doi:10.1111/biom.13024

Biometrics

2019

DOI: 10.1111/biom.13024

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Model Confidence Bounds for Variable Selection

Yang Li

Yuetian Luo

Davide Ferrari

et al.

Abstract: In this article, we introduce the concept of model confidence bounds (MCB) for variable selection in the context of nested models. Similarly to the endpoints in the familiar confidence interval for parameter estimation, the MCB identifies two nested models (upper and lower confidence bound models) containing the true model at a given level of confidence. Instead of trusting a single selected model obtained from a given model selection method, the MCB proposes a group of nested models as candidates and the MCB'… Show more

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Cited by 16 publications

(35 citation statements)

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“…Let us first strengthen Theorem 1, the main result of Li et al (), a little bit. Recall that

\hat{r} false(m_{1}, m_{2} false) = B^{- 1} \sum_{b = 1}^{B} I false(m_{1} \subseteq {\overset{m}{true}}^{false(b false)} \subseteq m_{2} false)

and define

\tilde{r} false(m_{1}, m_{2} false) = P (m_{1} \subseteq {\hat{m}}^{(b)} \subseteq m_{2} ‖ Y)

, where Y are the data.…”

mentioning

confidence: 65%

“…[Recall that

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

are defined by program (2) using

\hat{r} false(m_{1}, m_{2} false)

as given in the paper.] We note that inspection of the proof of Theorem 1 in Li et al () reveals that this proof actually seems to be given for

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

rather than for

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

(although the theorem also holds for the latter as will transpire from the subsequent result). We are now ready for the improved version of Theorem 1 in Li et al ().…”

mentioning

confidence: 87%

“…We note that inspection of the proof of Theorem 1 in Li et al () reveals that this proof actually seems to be given for

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

rather than for

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

(although the theorem also holds for the latter as will transpire from the subsequent result). We are now ready for the improved version of Theorem 1 in Li et al (). Proposition Assume: (A.1) (Model selection consistency)

P false(\hat{m} \neq m^{*} false) = o false(1 false)

; (A.2) (Bootstrap validity) For the re‐sampled model

{\overset{m}{true}}^{false(b false)}

, assume

P false({\overset{m}{true}}^{false(b false)} \neq \hat{m} false) = o false(1 false)

.…”

mentioning

confidence: 87%

“…bootstrap sample of size B (note that

\tilde{r} false(m_{1}, m_{2} false) = E false(\hat{r} false(m_{1}, m_{2} false) ‖ Y false)

, and thus is the quantity that

\hat{r} false(m_{1}, m_{2} false)

is trying to approximate with the help of the bootstrap sample). Let

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

now be obtained from the exact bootstrap, i.e.,

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

are defined by program (2) in Li et al (), but with

\tilde{r} false(m_{1}, m_{2} false)

replacing

\hat{r} false(m_{1}, m_{2} false)

everywhere. To exclude trivial cases, we assume that the nominal confidence level satisfies

0 < 1 - α < 1

.…”

mentioning

confidence: 99%

“…Obviously, this in turn also implies

\sum_{m \neq \hat{m}} \tilde{r} false(m, m false) \to 0

in probability. Inspection of (2) in Li et al () (with

\tilde{r} false(m_{1}, m_{2} false)

replacing

\hat{r} false(m_{1}, m_{2} false)

everywhere) then shows that

{\overset{m}{true}}_{L} = {\overset{m}{true}}_{U} = \hat{m}

holds on an event that has probability converging to 1. In view of (A.1), the event where

{\overset{m}{true}}_{U} = {\overset{m}{true}}_{L} = \hat{m} = m^{*}

holds then also has probability converging to 1.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Discussion on “Model Confidence Bounds for Variable Selection” by Yang Li, Yuetian Luo, Davide Ferrari, Xiaonan Hu, and Yichen Qin

2019

View full text Add to dashboard Cite

“…Let us first strengthen Theorem 1, the main result of Li et al (), a little bit. Recall that

\hat{r} false(m_{1}, m_{2} false) = B^{- 1} \sum_{b = 1}^{B} I false(m_{1} \subseteq {\overset{m}{true}}^{false(b false)} \subseteq m_{2} false)

and define

\tilde{r} false(m_{1}, m_{2} false) = P (m_{1} \subseteq {\hat{m}}^{(b)} \subseteq m_{2} ‖ Y)

, where Y are the data.…”

mentioning

confidence: 65%

“…[Recall that

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

are defined by program (2) using

\hat{r} false(m_{1}, m_{2} false)

as given in the paper.] We note that inspection of the proof of Theorem 1 in Li et al () reveals that this proof actually seems to be given for

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

rather than for

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

(although the theorem also holds for the latter as will transpire from the subsequent result). We are now ready for the improved version of Theorem 1 in Li et al ().…”

mentioning

confidence: 87%

“…We note that inspection of the proof of Theorem 1 in Li et al () reveals that this proof actually seems to be given for

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

rather than for

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

P false(\hat{m} \neq m^{*} false) = o false(1 false)

; (A.2) (Bootstrap validity) For the re‐sampled model

{\overset{m}{true}}^{false(b false)}

, assume

P false({\overset{m}{true}}^{false(b false)} \neq \hat{m} false) = o false(1 false)

.…”

mentioning

confidence: 87%

“…bootstrap sample of size B (note that

\tilde{r} false(m_{1}, m_{2} false) = E false(\hat{r} false(m_{1}, m_{2} false) ‖ Y false)

, and thus is the quantity that

\hat{r} false(m_{1}, m_{2} false)

is trying to approximate with the help of the bootstrap sample). Let

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

now be obtained from the exact bootstrap, i.e.,

{\overset{m}{true}}_{L}

and

{\overset{m}{true}}_{U}

are defined by program (2) in Li et al (), but with

\tilde{r} false(m_{1}, m_{2} false)

replacing

\hat{r} false(m_{1}, m_{2} false)

everywhere. To exclude trivial cases, we assume that the nominal confidence level satisfies

0 < 1 - α < 1

.…”

mentioning

confidence: 99%

“…Obviously, this in turn also implies

\sum_{m \neq \hat{m}} \tilde{r} false(m, m false) \to 0

in probability. Inspection of (2) in Li et al () (with

\tilde{r} false(m_{1}, m_{2} false)

replacing

\hat{r} false(m_{1}, m_{2} false)

everywhere) then shows that

{\overset{m}{true}}_{L} = {\overset{m}{true}}_{U} = \hat{m}

holds on an event that has probability converging to 1. In view of (A.1), the event where

{\overset{m}{true}}_{U} = {\overset{m}{true}}_{L} = \hat{m} = m^{*}

holds then also has probability converging to 1.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Discussion on “Model Confidence Bounds for Variable Selection” by Yang Li, Yuetian Luo, Davide Ferrari, Xiaonan Hu, and Yichen Qin

2019

View full text Add to dashboard Cite

Sparsified simultaneous confidence intervals for high-dimensional linear models

Zhu,

Qin,

Wang

2024

Metrika

View full text Add to dashboard Cite

Statistical inference of the high-dimensional regression coefficients is challenging because the uncertainty introduced by the model selection procedure is hard to account for. Currently, the inference of the model and the inference of the coefficients are separately sought. A critical question remains unsettled; that is, is it possible to embed the inference of the model into the simultaneous inference of the coefficients? If so, then how to properly design a simultaneous inference tool with desired properties? To this end, we propose a notion of simultaneous confidence intervals called the sparsified simultaneous confidence intervals (SSCI). Our intervals are sparse in the sense that some of the intervals’ upper and lower bounds are shrunken to zero (i.e., [0, 0]), indicating the unimportance of the corresponding covariates. These covariates should be excluded from the final model. The rest of the intervals, either containing zero (e.g., $$[-1,1]$$ [ - 1 , 1 ] or [0, 1]) or not containing zero (e.g., [2, 3]), indicate the plausible and significant covariates, respectively. The SSCI intuitively suggests a lower-bound model with significant covariates only and an upper-bound model with plausible and significant covariates. The proposed method can be coupled with various selection procedures, making it ideal for comparing their uncertainty. For the proposed method, we establish desirable asymptotic properties, develop intuitive graphical tools for visualization, and justify its superior performance through simulation and real data analysis.

show abstract

Confidence graphs for graphical model selection

Wang

Qin

2021

Stat Comput

View full text Add to dashboard Cite

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Model Confidence Bounds for Variable Selection

Cited by 16 publications

References 35 publications

Discussion on “Model Confidence Bounds for Variable Selection” by Yang Li, Yuetian Luo, Davide Ferrari, Xiaonan Hu, and Yichen Qin

Discussion on “Model Confidence Bounds for Variable Selection” by Yang Li, Yuetian Luo, Davide Ferrari, Xiaonan Hu, and Yichen Qin

Sparsified simultaneous confidence intervals for high-dimensional linear models

Confidence graphs for graphical model selection

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