Original: T HE D ISCRIMINATIVE J ACKKNIFE : Q UANTIFYING P REDICTIVE U NCERTAINTY VIA H IGHER -O RDER I NFLUENCE F UNCTIONS
Revised: T HE D ISCRIMINATIVE J ACKKNIFE : Q UANTIFYING P REDICTIVE U NCERTAINTY VIA H IGHER -O RDER I NFLUENCE F UNCTIONS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Deep learning models achieve high predictive accuracy in a broad spectrum of tasks, but rigorously quantifying their predictive uncertainty remains challenging.
Revised: Deep learning models achieve high predictive accuracy in a broad spectrum of tasks, but rigorously quantifying their predictive uncertainty remains challenging.

Original: Usable estimates of predictive uncertainty should (1) cover the true prediction target with a high probability, and (2) discriminate between high- and low-conﬁdence prediction instances.
Revised: Usable estimates of predictive uncertainty should (1) cover the true prediction target with a high probability, and (2) discriminate between high- and low-conﬁdence prediction instances.

Original: State-of-the-art methods for uncertainty quantiﬁcation are based predominantly on Bayesian neural networks.
Revised: State-of-the-art methods for uncertainty quantiﬁcation are based predominantly on Bayesian neural networks.

Original: However, Bayesian methods may fall short of (1) and (2) — i.e., Bayesian credible intervals do not guarantee frequentist coverage, and approximate posterior inference may undermine discriminative accuracy.
Revised: However, Bayesian methods may fall short of (1) and (2) — i.e., Bayesian credible intervals do not guarantee frequentist coverage, and approximate posterior inference may undermine discriminative accuracy.

Original: To this end, this paper tackles the following question: can we devise an alternative frequentist approach for uncertainty quantiﬁcation that satisﬁes (1) and (2)?
Revised: To this end, this paper tackles the following question: can we devise an alternative frequentist approach for uncertainty quantiﬁcation that satisﬁes (1) and (2)?

Original: To address this question, we develop the discriminative jackknife (DJ), a formal inference procedure that constructs predictive conﬁdence intervals for a wide range of regression models, is easy to implement, and provides rigorous theoretical guarantees on (1) and (2).
Revised: To address this question, we develop the discriminative jackknife (DJ), a formal inference procedure that constructs predictive conﬁdence intervals for a wide range of regression models, is easy to implement, and provides rigorous theoretical guarantees on (1) and (2).

Original: The DJ procedure uses higher-order inﬂuence functions (HOIFs) of the trained model parameters to construct a jackknife (leave-one-out) estimator of predictive conﬁdence intervals.
Revised: The DJ procedure uses higher-order inﬂuence functions (HOIFs) of the trained model parameters to construct a jackknife (leave-one-out) estimator of predictive conﬁdence intervals.

Original: DJ computes HOIFs using a recursive formula that requires only oracle access to loss gradients and Hessian-vector products, hence it can be applied in a post-hoc fashion without compromising model accuracy or interfering with model training.
Revised: DJ computes HOIFs using a recursive formula that requires only oracle access to loss gradients and Hessian-vector products, hence it can be applied in a post-hoc fashion without compromising model accuracy or interfering with model training.

Original: Experiments demonstrate that DJ performs competitively compared to existing Bayesian and non-Bayesian baselines.
Revised: Experiments demonstrate that DJ performs competitively compared to existing Bayesian and non-Bayesian baselines.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Deep learning models have achieved state-of-the-art performance on a variety of supervised learning tasks, and are becoming increasingly popular in various application domains (LeCun et al.
Revised: Deep learning models have achieved state-of-the-art performance on a variety of supervised learning tasks, and are becoming increasingly popular in various application domains (LeCun et al.

Original: A key question often asked of such models is “ Can we trust this particular model prediction? ”
Revised: A key question often asked of such models is “ Can we trust this particular model prediction? ”

Original: This question is highly relevant in high-stakes applications where the model is used to guide and inform critical decision-making — examples of such applications include: medical decision support, autonomous vehicle control, and ﬁnancial forecasting (Amodei et al.
Revised: This question is highly relevant in high-stakes applications where the model is used to guide and inform critical decision-making — examples of such applications include: medical decision support, autonomous vehicle control, and ﬁnancial forecasting (Amodei et al.

Original: Despite their impressive predictive accuracy, rigorously quantifying the predictive uncertainty of deep learning models is a challenging and yet an unresolved problem (Gal (2016); Ovadia et al.
Revised: Despite their impressive predictive accuracy, rigorously quantifying the predictive uncertainty of deep learning models is a challenging and yet an unresolved problem (Gal (2016); Ovadia et al.

Original: Actionable estimates of predictive uncertainty are ones that (1) cover the true prediction target with high probability, and (2) discriminate between high- and low-conﬁdence predictions.
Revised: Actionable estimates of predictive uncertainty are ones that (1) cover the true prediction target with high probability, and (2) discriminate between high- and low-conﬁdence predictions.

Original: (Figure 1 depicts a pictorial visualization for the coverage and discrimination requirements.)
Revised: (Figure 1 depicts a pictorial visualization for the coverage and discrimination requirements.)

Original: The ﬁrst requirement, frequentist coverage, is especially relevant in applications where predictive uncertainty is incorporated in a decision-theoretic framework (e.g., administering medical treatments (Dusenberry et al.
Revised: The ﬁrst requirement, frequentist coverage, is especially relevant in applications where predictive uncertainty is incorporated in a decision-theoretic framework (e.g., administering medical treatments (Dusenberry et al.

Original: (2019)), or estimating value functions in model-free reinforcement learning (White & White (2010))).
Revised: (2019)), or estimating value functions in model-free reinforcement learning (White & White (2010))).

Original: The second requirement, discrimination, is crucial for auditing model reliability (Schulam & Saria (2019)), detecting dataset shifts and out-of-distribution samples (Barber et al.
Revised: The second requirement, discrimination, is crucial for auditing model reliability (Schulam & Saria (2019)), detecting dataset shifts and out-of-distribution samples (Barber et al.

Original: (2019a)), and actively collecting new training examples for which the model is not conﬁdent (Cohn et al.
Revised: (2019a)), and actively collecting new training examples for which the model is not conﬁdent (Cohn et al.

Original: Existing methods for uncertainty quantiﬁcation are based predominantly on Bayesian neural networks (BNNs), whereby predictive uncertainty is evaluated via posterior credible intervals (Welling & Teh (2011); Hernández-Lobato & Adams (2015); Ritter et al.
Revised: Existing methods for uncertainty quantiﬁcation are based predominantly on Bayesian neural networks (BNNs), whereby predictive uncertainty is evaluated via posterior credible intervals (Welling & Teh (2011); Hernández-Lobato & Adams (2015); Ritter et al.

Original: ; Maddox et al.
Revised: ; Maddox et al.

Original: However,
Revised: However,

Original: [Figure]
Revised: [Figure]

Original: M o d e l
Revised: M o d e l

Original: O u t pu t
Revised: O u t pu t

Original: M o d e l
Revised: M o d e l

Original: O u t pu t
Revised: O u t pu t

Original: M o d e l
Revised: M o d e l

Original: O u t pu t
Revised: O u t pu t

Original: Accurate coverage, no discrimination
Revised: Accurate coverage, no discrimination

Original: No coverage, accurate discrimination
Revised: No coverage, accurate discrimination

Original: Accurate coverage and discrimination
Revised: Accurate coverage and discrimination

Original: BNNs require signiﬁcant modiﬁcations to the training procedure, and exact Bayesian inference is computationally prohibitive in practice.
Revised: BNNs require signiﬁcant modiﬁcations to the training procedure, and exact Bayesian inference is computationally prohibitive in practice.

Original: Approximate dropout-based inference schemes (e.g., Monte
Revised: Approximate dropout-based inference schemes (e.g., Monte

Original: Carlo dropout (Gal & Ghahramani (2016)) and variational dropout (Kingma et al.
Revised: Carlo dropout (Gal & Ghahramani (2016)) and variational dropout (Kingma et al.

Original: (2015))) have been recently proposed as computationally efﬁcient alternatives.
Revised: (2015))) have been recently proposed as computationally efﬁcient alternatives.

Original: However, Bayesian inference in dropout-based models has been shown to be ill-posed, since the induced posterior distributions in such models do not concentrate asymptotically (Osband (2016);
Revised: However, Bayesian inference in dropout-based models has been shown to be ill-posed, since the induced posterior distributions in such models do not concentrate asymptotically (Osband (2016);

Original: Hron et al.
Revised: Hron et al.

Original: (2017)), which jeopardizes both the coverage and discrimination performance of the resulting credible intervals.
Revised: (2017)), which jeopardizes both the coverage and discrimination performance of the resulting credible intervals.

Original: Moreover, even with exact inference, Bayesian credible intervals generally do not guarantee frequentist coverage (Bayarri & Berger (2004)).
Revised: Moreover, even with exact inference, Bayesian credible intervals generally do not guarantee frequentist coverage (Bayarri & Berger (2004)).

Original: Alternative non-Bayesian methods have been recently developed — mostly based on ad-hoc ensemble approaches (Lakshminarayanan et al.
Revised: Alternative non-Bayesian methods have been recently developed — mostly based on ad-hoc ensemble approaches (Lakshminarayanan et al.

Original: — but formal frequentist methods with rigorous theoretical guarantees are still lacking.
Revised: — but formal frequentist methods with rigorous theoretical guarantees are still lacking.

Original: Summary of Contributions.
Revised: Summary of Contributions.

Original: In this paper, we develop a formal procedure for constructing frequentist (pointwise) conﬁdence intervals on the predictions of a broad class of deep learning models.
Revised: In this paper, we develop a formal procedure for constructing frequentist (pointwise) conﬁdence intervals on the predictions of a broad class of deep learning models.

Original: Our method — which we call the discriminative jackknife (DJ) — is inspired by the classic jackknife leave-one-out (LOO) re-sampling procedure for estimating variability in statistical models (Miller (1974);
Revised: Our method — which we call the discriminative jackknife (DJ) — is inspired by the classic jackknife leave-one-out (LOO) re-sampling procedure for estimating variability in statistical models (Miller (1974);

Original: Efron (1992)).
Revised: Efron (1992)).

Original: In order to ensure both frequentist coverage and discrimination, DJ constructs feature-dependent conﬁdence intervals using the LOO local prediction variance at the input feature, and adjusts the interval width (for a given coverage probability) using the model’s average LOO error residuals.
Revised: In order to ensure both frequentist coverage and discrimination, DJ constructs feature-dependent conﬁdence intervals using the LOO local prediction variance at the input feature, and adjusts the interval width (for a given coverage probability) using the model’s average LOO error residuals.

Original: Whereas the classic jackknife satisﬁes neither the coverage nor the discrimination requirements (Barber et al. (2019b)), DJ provides strong theoretical guarantees on both (i.e., DJ generates conﬁdence intervals resembling those in the rightmost panel of Figure 1 with high probability).
Revised: Whereas the classic jackknife satisﬁes neither the coverage nor the discrimination requirements (Barber et al. (2019b)), DJ provides strong theoretical guarantees on both (i.e., DJ generates conﬁdence intervals resembling those in the rightmost panel of Figure 1 with high probability).

Original: Central to the DJ procedure is the usage of inﬂuence functions — a key concept in robust statistics and variational calculus (Efron (1992); Cook & Weisberg (1982);
Revised: Central to the DJ procedure is the usage of inﬂuence functions — a key concept in robust statistics and variational calculus (Efron (1992); Cook & Weisberg (1982);

Original: Hampel et al.
Revised: Hampel et al.

Original: — in order to estimate the parameters of models trained on LOO versions of the training data, without the need to exhaustively re-train the model for each held-out data point.
Revised: — in order to estimate the parameters of models trained on LOO versions of the training data, without the need to exhaustively re-train the model for each held-out data point.

Original: That is, using the von Mises expansion (Fernholz (2012)) — a variant of Taylor series expansion for statistical functionals — we represent the (counter-factual) model parameters that would have been learned on LOO versions of the training dataset in terms of an inﬁnite series of higher-order inﬂuence functions (HOIFs) for the model parameters trained on the complete dataset.
Revised: That is, using the von Mises expansion (Fernholz (2012)) — a variant of Taylor series expansion for statistical functionals — we represent the (counter-factual) model parameters that would have been learned on LOO versions of the training dataset in terms of an inﬁnite series of higher-order inﬂuence functions (HOIFs) for the model parameters trained on the complete dataset.

Original: We show that a truncated von Mises expansion (with a ﬁnite number of HOIFs) is sufﬁcient to preserve the guarantees on coverage and discrimination.
Revised: We show that a truncated von Mises expansion (with a ﬁnite number of HOIFs) is sufﬁcient to preserve the guarantees on coverage and discrimination.

Original: To enable efﬁcient computation of the von Mises expansion, we derive a recursive formula that links
Revised: To enable efﬁcient computation of the von Mises expansion, we derive a recursive formula that links

Original: HOIFs with lower-order inﬂuence functions and the partial derivatives of the model loss function.
Revised: HOIFs with lower-order inﬂuence functions and the partial derivatives of the model loss function.

Original: Comprehensive experimental evaluation demonstrates that the DJ procedure performs competitively compared to both Bayesian and non-Bayesian methods.
Revised: Comprehensive experimental evaluation demonstrates that the DJ procedure performs competitively compared to both Bayesian and non-Bayesian methods.

Original: In particular, we found that the DJ procedure outperforms Bayesian neural networks with respect to discriminative performance, and provides better frequentist coverage compared to existing non-Bayesian methods, while being signiﬁcantly less computationally expensive than both classes of methods.
Revised: In particular, we found that the DJ procedure outperforms Bayesian neural networks with respect to discriminative performance, and provides better frequentist coverage compared to existing non-Bayesian methods, while being signiﬁcantly less computationally expensive than both classes of methods.

Original:  P RELIMINARIES
Revised:  P RELIMINARIES

Original: Learning Setup.
Revised: Learning Setup.

Original: We consider a standard supervised learning setup with ( x , y ) being a feature-label pair, where the feature x belongs to a d -dimensional feature space X ⊆ R d , and y ∈ R .
Revised: We consider a standard supervised learning setup with ( x , y ) being a feature-label pair, where the feature x belongs to a d -dimensional feature space X ⊆ R d , and y ∈ R .

Original: A model is trained to predict y using a dataset D (cid:44) { ( x i , y i ) } ni =1 of n training examples which are drawn i.i.d from an unknown distribution P .
Revised: A model is trained to predict y using a dataset D (cid:44) { ( x i , y i ) } ni =1 of n training examples which are drawn i.i.d from an unknown distribution P .

Original: Let the learning model be denoted as f ( x ; θ ) :
Revised: Let the learning model be denoted as f ( x ; θ ) :

Original: X → Y , where θ ∈ Θ are the model parameters, and Θ is the parameter space.
Revised: X → Y , where θ ∈ Θ are the model parameters, and Θ is the parameter space.

Original: The learned model parameters ˆ θ ∈ Θ are obtained by solving the optimization problem ˆ θ = arg min θ ∈ Θ L ( D , θ ) , with a loss function
Revised: The learned model parameters ˆ θ ∈ Θ are obtained by solving the optimization problem ˆ θ = arg min θ ∈ Θ L ( D , θ ) , with a loss function

Original: [Equation]
Revised: [Equation]

Original: where we fold in any regularization terms into (cid:96) ( . ) .
Revised: where we fold in any regularization terms into (cid:96) ( . ) .

Original: We do not pose any assumptions on how the loss function in (1) is optimized.
Revised: We do not pose any assumptions on how the loss function in (1) is optimized.

Original: The model f
Revised: The model f

Original: ( x ; θ ) can be any neural network variant (e.g., feed-forward, convolutional, etc).
Revised: ( x ; θ ) can be any neural network variant (e.g., feed-forward, convolutional, etc).

Original: Throughout the paper, we limit our discussion to the regression task; extending our framework to classiﬁcation tasks is an interesting direction for future work.
Revised: Throughout the paper, we limit our discussion to the regression task; extending our framework to classiﬁcation tasks is an interesting direction for future work.

Original: Uncertainty Quantiﬁcation.
Revised: Uncertainty Quantiﬁcation.

Original: The predictions issued by the trained model (with learned parameters ˆ θ ) are given by f ( x ; ˆ θ ) .
Revised: The predictions issued by the trained model (with learned parameters ˆ θ ) are given by f ( x ; ˆ θ ) .

Original: Our main goal is to obtain estimates for the uncertainty in the model predictions, expressed through a pointwise conﬁdence interval C ( x ; ˆ θ ) deﬁned as follows:
Revised: Our main goal is to obtain estimates for the uncertainty in the model predictions, expressed through a pointwise conﬁdence interval C ( x ; ˆ θ ) deﬁned as follows:

Original: [Equation]
Revised: [Equation]

Original: The extent of uncertainty in the model’s prediction (for a data point with feature x ) is quantiﬁed by the interval width W ( . ) of C ( x ; ˆ θ ) , deﬁned as W ( C ( x ; ˆ θ ))
Revised: The extent of uncertainty in the model’s prediction (for a data point with feature x ) is quantiﬁed by the interval width W ( . ) of C ( x ; ˆ θ ) , deﬁned as W ( C ( x ; ˆ θ ))

Original: (cid:44) f
Revised: (cid:44) f

Original: U ( x ; ˆ θ ) − f L ( x ; ˆ θ ) .
Revised: U ( x ; ˆ θ ) − f L ( x ; ˆ θ ) .

Original: Wider intervals imply less conﬁdence, and vice versa.
Revised: Wider intervals imply less conﬁdence, and vice versa.

Original: For C ( x ; ˆ θ ) to be usable, it has to satisfy the following requirements:
Revised: For C ( x ; ˆ θ ) to be usable, it has to satisfy the following requirements:

Original: (i) Frequentist coverage:
Revised: (i) Frequentist coverage:

Original: The conﬁdence interval C ( x ; ˆ θ ) covers the true prediction target with an arbitrarily prespeciﬁed coverage probability of (1 − α ) , α ∈ (0 , 1) ,
Revised: The conﬁdence interval C ( x ; ˆ θ ) covers the true prediction target with an arbitrarily prespeciﬁed coverage probability of (1 − α ) , α ∈ (0 , 1) ,

Original: i
Revised: i

Original: e.,
Revised: e.,

Original: [Equation]
Revised: [Equation]

Original: where the probability is taken with respect to a (new) test point ( x , y ) as well as with respect to the training data D (Lawless & Fredette (2005)
Revised: where the probability is taken with respect to a (new) test point ( x , y ) as well as with respect to the training data D (Lawless & Fredette (2005)

Original: ; Barber et al.
Revised: ; Barber et al.

Original: (2019b)).
Revised: (2019b)).

Original: (ii) Discrimination: The conﬁdence interval C ( x ; ˆ θ ) is wider (on average) for test points with less accurate predictions (Leonard et al.
Revised: (ii) Discrimination: The conﬁdence interval C ( x ; ˆ θ ) is wider (on average) for test points with less accurate predictions (Leonard et al.

Original: , i
Revised: , i

Original: e., for the test points x , x (cid:48) ∈ X , we have
Revised: e., for the test points x , x (cid:48) ∈ X , we have

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: the expectation E [
Revised: the expectation E [

Original: ] is taken with
Revised: ] is taken with

Original: respect
Revised: respect

Original: to the training data D
Revised: to the training data D

Original: In the next Section, we develop a formal post-hoc procedure that takes the dataset D and the trained model f ( x ; ˆ θ ) as inputs, and evaluates an uncertainty estimate (cid:98) C ( x ; ˆ θ ) that satisﬁes both requirements.
Revised: In the next Section, we develop a formal post-hoc procedure that takes the dataset D and the trained model f ( x ; ˆ θ ) as inputs, and evaluates an uncertainty estimate (cid:98) C ( x ; ˆ θ ) that satisﬁes both requirements.

Original:  T HE D ISCRIMINATIVE J ACKKNIFE
Revised:  T HE D ISCRIMINATIVE J ACKKNIFE

Original: Before presenting our proposed discriminative jackknife (DJ) procedure, we start with a brief recap of the classical jackknife.
Revised: Before presenting our proposed discriminative jackknife (DJ) procedure, we start with a brief recap of the classical jackknife.

Original: The jackknife quantiﬁes predictive uncertainty in terms of the (average) prediction error, which is estimated with a leave-one-out (LOO) construction found by systematically leaving out each data point from the training dataset D , and evaluating the error of the re-trained model on the held-out sample.
Revised: The jackknife quantiﬁes predictive uncertainty in terms of the (average) prediction error, which is estimated with a leave-one-out (LOO) construction found by systematically leaving out each data point from the training dataset D , and evaluating the error of the re-trained model on the held-out sample.

Original: That is, for a target coverage of (1 − α ) , a naïve jackknife estimates the uncertainty in the prediction for a test point x as (Miller (1974); Efron (1992)):
Revised: That is, for a target coverage of (1 − α ) , a naïve jackknife estimates the uncertainty in the prediction for a test point x as (Miller (1974); Efron (1992)):

Original: [Equation]
Revised: [Equation]

Original: where r i = | y i − f ( x i ; ˆ θ − i ) | is the error residual on the i th data point, ˆ θ − i are the parameters of the model re-trained on the dataset D \ { ( x i , y i ) } with the i th point removed, and (cid:98) Q α,n is the (1 − α ) quantile for the empirical distribution of elements of the set R = { r 1 , . . .
Revised: where r i = | y i − f ( x i ; ˆ θ − i ) | is the error residual on the i th data point, ˆ θ − i are the parameters of the model re-trained on the dataset D \ { ( x i , y i ) } with the i th point removed, and (cid:98) Q α,n is the (1 − α ) quantile for the empirical distribution of elements of the set R = { r 1 , . . .

Original: , r n } deﬁned as(cid:98) Q α,n ( R ) (cid:44) the (cid:100) (1 − α ) ( n + 1) (cid:101) -th smallest value of the elements of R .
Revised: , r n } deﬁned as(cid:98) Q α,n ( R ) (cid:44) the (cid:100) (1 − α ) ( n + 1) (cid:101) -th smallest value of the elements of R .

Original: Albeit intuitive, the naïve jackknife estimate in (3) is not guaranteed to achieve the target coverage of (1 − α )
Revised: Albeit intuitive, the naïve jackknife estimate in (3) is not guaranteed to achieve the target coverage of (1 − α )

Original: (Barber et al.
Revised: (Barber et al.

Original: (2019b)).
Revised: (2019b)).

Original: More crucially, the interval width W ( (cid:98) C α,n ( x ; ˆ θ )) is a constant that does not depend on the feature x , which renders discrimination impossible, i.e., when applied, the naïve jackknife would result in conﬁdence intervals resembling the leftmost panel in Figure 1.
Revised: More crucially, the interval width W ( (cid:98) C α,n ( x ; ˆ θ )) is a constant that does not depend on the feature x , which renders discrimination impossible, i.e., when applied, the naïve jackknife would result in conﬁdence intervals resembling the leftmost panel in Figure 1.

Original: 3. E XACT CONSTRUCTION OF THE DJ C ONFIDENCE INTERVALS
Revised: 3. E XACT CONSTRUCTION OF THE DJ C ONFIDENCE INTERVALS

Original: We propose an ameliorated jackknife procedure, the DJ, which addresses the shortcomings of the classical jackknife.
Revised: We propose an ameliorated jackknife procedure, the DJ, which addresses the shortcomings of the classical jackknife.

Original: We start by deﬁning some notation.
Revised: We start by deﬁning some notation.

Original: Let the sets V + n ( x ) and V − n ( x ) be deﬁned as:
Revised: Let the sets V + n ( x ) and V − n ( x ) be deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: , and
Revised: , and

Original: [Equation]
Revised: [Equation]

Original: where v i ( x ) = f ( x ; ˆ θ ) − f ( x ; ˆ θ − i ) , n + = |V + n | , and n − =
Revised: where v i ( x ) = f ( x ; ˆ θ ) − f ( x ; ˆ θ − i ) , n + = |V + n | , and n − =

Original: |V − n | .
Revised: |V − n | .

Original: Our DJ procedure estimates the predictive conﬁdence interval for test point x as follows: (cid:98) C α,n ( x ; ˆ θ ) =
Revised: Our DJ procedure estimates the predictive conﬁdence interval for test point x as follows: (cid:98) C α,n ( x ; ˆ θ ) =

Original: [ f L ( x ; ˆ θ ) , f U ( x ; ˆ θ ) ] , where
Revised: [ f L ( x ; ˆ θ ) , f U ( x ; ˆ θ ) ] , where

Original: [Equation]
Revised: [Equation]

Original: Error
Revised: Error

Original: [Equation]
Revised: [Equation]

Original: Local Variability .
Revised: Local Variability .

Original: Unlike the traditional jackknife, the construction of the DJ conﬁdence intervals comprises two terms: marginal prediction error and local prediction variability .
Revised: Unlike the traditional jackknife, the construction of the DJ conﬁdence intervals comprises two terms: marginal prediction error and local prediction variability .

Original: The prediction error term, (cid:98) Q α,n ( R n ) , uses the LOO residuals to estimate the model’s (marginal) generalization error.
Revised: The prediction error term, (cid:98) Q α,n ( R n ) , uses the LOO residuals to estimate the model’s (marginal) generalization error.

Original: The prediction variability terms, (cid:98) Q α,n + ( V + n ( x )) and (cid:98) Q 1 − α,n − ( V − n ( x )) , quantify the extent to which each training data point impacts the value of the prediction at test point x .
Revised: The prediction variability terms, (cid:98) Q α,n + ( V + n ( x )) and (cid:98) Q 1 − α,n − ( V − n ( x )) , quantify the extent to which each training data point impacts the value of the prediction at test point x .

Original: The prediction error is constant, i.e., does not depend on x , hence it only contributes to coverage but does not contribute to discrimination.
Revised: The prediction error is constant, i.e., does not depend on x , hence it only contributes to coverage but does not contribute to discrimination.

Original: On the contrary, the local variability terms depend on the test point feature x , hence they fully determine the discrimination performance.
Revised: On the contrary, the local variability terms depend on the test point feature x , hence they fully determine the discrimination performance.

Original: As we show later in Section 3.4, constructing conﬁdence intervals this way provides strong theoretical guarantees on the achieved frequentist coverage and discrimination.
Revised: As we show later in Section 3.4, constructing conﬁdence intervals this way provides strong theoretical guarantees on the achieved frequentist coverage and discrimination.

Original: Figure 2 illustrates the construction of the DJ conﬁdence intervals in (5).
Revised: Figure 2 illustrates the construction of the DJ conﬁdence intervals in (5).

Original: The conﬁdence intervals are chosen so that the boundaries of the average error and local variability are exceeded by (cid:100)
Revised: The conﬁdence intervals are chosen so that the boundaries of the average error and local variability are exceeded by (cid:100)

Original: ( n + 1)(1 − α ) (cid:101) out of the n LOO samples — these are marked with a star.
Revised: ( n + 1)(1 − α ) (cid:101) out of the n LOO samples — these are marked with a star.

Original: For the average prediction error term, the width of the resulting boundary is the same for any test data point x ∈ X .
Revised: For the average prediction error term, the width of the resulting boundary is the same for any test data point x ∈ X .

Original: For the local prediction variability term, the width of the boundary depends on the speciﬁc value of the feature x , and should get wider for less conﬁdent prediction instances.
Revised: For the local prediction variability term, the width of the boundary depends on the speciﬁc value of the feature x , and should get wider for less conﬁdent prediction instances.

Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: Based on the conﬁdence intervals in (5), it follows that the DJ interval width is given by
Revised: Based on the conﬁdence intervals in (5), it follows that the DJ interval width is given by

Original: [Equation]
Revised: [Equation]

Original: The marginal error and local variability terms in (6) jointly capture two types of uncertainty: epistemic and aleatoric uncertainties (Gal (2016)).
Revised: The marginal error and local variability terms in (6) jointly capture two types of uncertainty: epistemic and aleatoric uncertainties (Gal (2016)).

Original: Epistemic uncertainty measures how well the model is matched to the data, and is reducible as the size of training data increases.
Revised: Epistemic uncertainty measures how well the model is matched to the data, and is reducible as the size of training data increases.

Original: On the contrary, aleatoric uncertainty is the irreducible uncertainty arising from the inherent sources of complexity in the data, such as label noise or hidden features (Malinin & Gales (2018)).
Revised: On the contrary, aleatoric uncertainty is the irreducible uncertainty arising from the inherent sources of complexity in the data, such as label noise or hidden features (Malinin & Gales (2018)).

Original: Consistency of the DJ conﬁdence estimates requires that W ( (cid:98) C α,n ( x ; ˆ θ ))
Revised: Consistency of the DJ conﬁdence estimates requires that W ( (cid:98) C α,n ( x ; ˆ θ ))

Original: → 0 , i.e., the interval width vanishes, as the size of the training data increases ( n → ∞ ).
Revised: → 0 , i.e., the interval width vanishes, as the size of the training data increases ( n → ∞ ).

Original: It follows from (6)
Revised: It follows from (6)

Original: that if there is no aleatoric uncertainty, and if the underlying model is stable (i.e., lim n →∞ v i = 0 ) and consistent (i.e., lim n →∞ r i = 0 ), then the interval width W ( (cid:98) C α,n ( x ; ˆ θ )) vanishes as n grows asymptotically (more training data is collected).
Revised: that if there is no aleatoric uncertainty, and if the underlying model is stable (i.e., lim n →∞ v i = 0 ) and consistent (i.e., lim n →∞ r i = 0 ), then the interval width W ( (cid:98) C α,n ( x ; ˆ θ )) vanishes as n grows asymptotically (more training data is collected).

Original: 3. E FFICIENT I MPLEMENTATION VIA H IGHER -O RDER I NFLUENCE F UNCTIONS
Revised: 3. E FFICIENT I MPLEMENTATION VIA H IGHER -O RDER I NFLUENCE F UNCTIONS

Original: Exact computation of the DJ conﬁdence intervals in (5) requires re-training the model n times in order to collect the “perturbed” LOO parameters { ˆ θ − i } ni =1 .
Revised: Exact computation of the DJ conﬁdence intervals in (5) requires re-training the model n times in order to collect the “perturbed” LOO parameters { ˆ θ − i } ni =1 .

Original: To scale up the DJ procedure, we propose to use inﬂuence functions — a classic tool from robust statistics (Huber & Ronchetti (1981);
Revised: To scale up the DJ procedure, we propose to use inﬂuence functions — a classic tool from robust statistics (Huber & Ronchetti (1981);

Original: Hampel et al.
Revised: Hampel et al.

Original: — in order to recover the parameters { ˆ θ − i } ni =1 on the basis of the trained model f ( x ; ˆ θ ) , without the need for repeated re-training.
Revised: — in order to recover the parameters { ˆ θ − i } ni =1 on the basis of the trained model f ( x ; ˆ θ ) , without the need for repeated re-training.

Original: Through our inﬂuence function-based implementation, the
Revised: Through our inﬂuence function-based implementation, the

Original: DJ can be applied in a post-hoc fashion, requiring only knowledge of the model loss gradients.
Revised: DJ can be applied in a post-hoc fashion, requiring only knowledge of the model loss gradients.

Original: Inﬂuence functions enable efﬁcient computation of the effect of a training data point ( x i , y i ) on ˆ θ .
Revised: Inﬂuence functions enable efﬁcient computation of the effect of a training data point ( x i , y i ) on ˆ θ .

Original: This is achieved by evaluating the change in ˆ θ if ( x i , y i ) was up-weighted 2 by some small (cid:15) , resulting in a new parameter ˆ θ i,(cid:15) (cid:44) arg min θ ∈ Θ L ( D , θ ) + (cid:15) · (cid:96) ( y i , f ( x i ; θ )) .
Revised: This is achieved by evaluating the change in ˆ θ if ( x i , y i ) was up-weighted 2 by some small (cid:15) , resulting in a new parameter ˆ θ i,(cid:15) (cid:44) arg min θ ∈ Θ L ( D , θ ) + (cid:15) · (cid:96) ( y i , f ( x i ; θ )) .

Original: The rate of change in ˆ θ due to an inﬁnitesimal perturbation (cid:15) in data point i is give by the (ﬁrst-order) inﬂuence function, deﬁned as
Revised: The rate of change in ˆ θ due to an inﬁnitesimal perturbation (cid:15) in data point i is give by the (ﬁrst-order) inﬂuence function, deﬁned as

Original: [Equation]
Revised: [Equation]

Original: Note that the model parameter ˆ θ is a statistical functional of the data distribution P .
Revised: Note that the model parameter ˆ θ is a statistical functional of the data distribution P .

Original: Perturbing the i th training point is equivalent to perturbing P to create a new distribution P i,(cid:15) =
Revised: Perturbing the i th training point is equivalent to perturbing P to create a new distribution P i,(cid:15) =

Original: (1 − (cid:15) )
Revised: (1 − (cid:15) )

Original: P + (cid:15) ∆( x i , y i ) , where ∆( x i , y i ) denotes the Dirac distribution in the point ( x i , y i ) .
Revised: P + (cid:15) ∆( x i , y i ) , where ∆( x i , y i ) denotes the Dirac distribution in the point ( x i , y i ) .

Original: In this sense, the inﬂuence function in (7) operationalizes the concept of derivatives to statistical functionals, i.e., it quantiﬁes the derivative of the learned model parameters ˆ θ with respect to the data distribution P .
Revised: In this sense, the inﬂuence function in (7) operationalizes the concept of derivatives to statistical functionals, i.e., it quantiﬁes the derivative of the learned model parameters ˆ θ with respect to the data distribution P .

Original: By recognizing that inﬂuence functions are the “derivatives” of the model parameters ˆ θ with respect to the data distribution P , we can use a Taylor-type expansion to represent the counter-factual model parameter ˆ θ i,(cid:15) (that would have been learned from a dataset with the i th data point up-weighted) in terms of the true model parameter ˆ θ (learned from the true training dataset D ) as follows:
Revised: By recognizing that inﬂuence functions are the “derivatives” of the model parameters ˆ θ with respect to the data distribution P , we can use a Taylor-type expansion to represent the counter-factual model parameter ˆ θ i,(cid:15) (that would have been learned from a dataset with the i th data point up-weighted) in terms of the true model parameter ˆ θ (learned from the true training dataset D ) as follows:

Original: [Equation]
Revised: [Equation]

Original: where I ( k ) θ ( x i , y i ) is the k th order inﬂuence function, deﬁned as I (k ) θ ( x i , y i ) =
Revised: where I ( k ) θ ( x i , y i ) is the k th order inﬂuence function, deﬁned as I (

Original: ∂ ˆ θ i,(cid:15) /∂

Original: [Equation]
Revised: [Equation]

Original: k | [Equation] =0 . The expansion in (8), known as the von Mises expansion (Fernholz (2012)), is a distributional analog of the Taylor expansion for a statistical functional.
Revised: The expansion in (8), known as the von Mises expansion (Fernholz (2012)), is a distributional analog of the Taylor expansion for a statistical functional.

Original: If all of the higher-order inﬂuence functions (HOIFs) in (8) exist, then we can recover ˆ θ i,(cid:15) without re-training the model on the perturbed training dataset.
Revised: If all of the higher-order inﬂuence functions (HOIFs) in (8) exist, then we can recover ˆ θ i,(cid:15) without re-training the model on the perturbed training dataset.

Original: In Section 3.3, we will derive a recursive formula for efﬁciently computing the HOIFs in (8).
Revised: In Section 3.3, we will derive a recursive formula for efﬁciently computing the HOIFs in (8).

Original: Since exact reconstruction of ˆ θ i,(cid:15) requires an inﬁnite number of HOIFs, we can only approximate ˆ θ i,(cid:15) by including a sufﬁcient number of HOIF terms from the von Mises expansion.
Revised: Since exact reconstruction of ˆ θ i,(cid:15) requires an inﬁnite number of HOIFs, we can only approximate ˆ θ i,(cid:15) by including a sufﬁcient number of HOIF terms from the von Mises expansion.

Original: We discuss the impact of this approximation on the coverage and discrimination performance of the DJ in Section 3.4.
Revised: We discuss the impact of this approximation on the coverage and discrimination performance of the DJ in Section 3.4.

Original: Note that the LOO model parameters { ˆ θ − i } ni =1 (required for the construction of the DJ conﬁdence intervals) can be obtained by setting (cid:15) =
Revised: Note that the LOO model parameters { ˆ θ − i } ni =1 (required for the construction of the DJ conﬁdence intervals) can be obtained by setting (cid:15) =

Original: − 1 /n , i.e., ˆ θ − i
Revised: − 1 /n , i.e., ˆ θ − i

Original: =
Revised: =

Original: ˆ θ
Revised: ˆ θ

Original: i, − 1 n , since removing a training point is equivalent to up-weighting it by − 1 /n in the loss function
Revised: i, − 1 n , since removing a training point is equivalent to up-weighting it by − 1 /n in the loss function

Original: L ( D ; θ ) .
Revised: L ( D ; θ ) .

Original: Thus, by setting (cid:15) = − 1 /n and selecting a prespeciﬁed number of HOIF terms m for obtaining the approximate
Revised: Thus, by setting (cid:15) = − 1 /n and selecting a prespeciﬁed number of HOIF terms m for obtaining the approximate

Original: LOO parameters ˆ θ ( m ) − i , the DJ conﬁdence intervals can be computed using the steps in Algorithm 1.
Revised: LOO parameters ˆ θ ( m ) − i , the DJ conﬁdence intervals can be computed using the steps in Algorithm 1.

Original: [Table]
Revised: [Table]

Original: [Table]

Original: +
Revised: =

Original: [Table]
Revised: [Table]

Original: Q α,n
Revised: ←

Original: [Table]
Revised: [Table]

Original: R n ) +
Revised: Y i − f (

Original: [Table]
Revised: [Table]

Original: Q α,
Revised: i

Original: [Table]
Revised: [Table]

Original: + ( 
Revised: ˆ θ ( m ) − i ) (cid:12)(cid:12) 4: v i ( x ) ← f ( x

Original: [Table]
Revised: [Table]

Revised: ˆ θ ) − f ( x

Revised: [Table]

Revised: ˆ θ ( m ) − i )

Revised: [Table]

Revised: end 6: f L ( x

Revised: [Table]

Revised: θ ) ← f ( x ; ˆ θ ) − (cid:98) Q α,n ( R n ) − (cid:98) Q 1 − α,n − ( V − n ( x ))

Revised: [Table]

Revised: f U ( x ; ˆ θ ) ← f ( x ; ˆ θ ) + (cid:98) Q α,n ( R n ) + (cid:98) Q α,n + (

Original: 3. C OMPUTING H IGHER -O RDER I NFLUENCE F UNCTIONS
Revised: 3. C OMPUTING H IGHER -O RDER I NFLUENCE F UNCTIONS

Original: How can we compute the inﬂuence functions of the von Mises expansion in (8)?
Revised: How can we compute the inﬂuence functions of the von Mises expansion in (8)?

Original: The recent work on model interpretability in (Koh & Liang (2017)) has studied the use of inﬂuence functions to quantify the impact of each training data point on a model’s predictions.
Revised: The recent work on model interpretability in (Koh & Liang (2017)) has studied the use of inﬂuence functions to quantify the impact of each training data point on a model’s predictions.

Original: In this work, ﬁrst-order inﬂuence functions were computed — using the classical result in (Cook & Weisberg (1982)) — as follows:
Revised: In this work, ﬁrst-order inﬂuence functions were computed — using the classical result in (Cook & Weisberg (1982)) — as follows:

Original: [Equation]
Revised: [Equation]

Original: where H θ (cid:44) ∇ 2 θ (cid:96) ( y, f ( x , θ )) is the Hessian of the loss function, which is assumed to be positive deﬁnite.
Revised: where H θ (cid:44) ∇ 2 θ (cid:96) ( y, f ( x , θ )) is the Hessian of the loss function, which is assumed to be positive deﬁnite.

Original: However, the expression in (9) is only limited to ﬁrst-order inﬂuence functions, and to the best of our knowledge, no similar expressions exist for HOIFs.
Revised: However, the expression in (9) is only limited to ﬁrst-order inﬂuence functions.

Original: In the following Theorem 3 , we derive a recursive formula for computingall HOIFs of the model parameters θ .
Revised: In the following Theorem 3 , we derive a recursive formula for computing HOIFs of the model parameters θ .

Original: Theorem 1.
Revised: Theorem 1.

Original: Let (cid:96) ( . ) be a differentiable function.
Revised: Let (cid:96) ( . ) be a differentiable function with a positive deﬁnite Hessian matrix H θ .

Original: Then the ( k + 1) -th order inﬂuence function of θ exists for all ( x , y ) ∈
Revised: If (cid:96) ( . ) is locally convex in the neighborhood of θ , then the ( k + 1) -th order inﬂuence function of θ exists for all ( x , y ) ∈

Original: X × Y , and for all k ≥ 1 we have
Revised: X × Y , and for all k ≥ 1 we have

Original: I ( k +1) θ

Original: [Equation]
Revised: [Equation]

Original: where a m is a constant, and g m is a polynomial function, ∀ m ∈ { 1 , . .
Revised: where a m is a constant, and g m is a polynomial function, ∀ m ∈ { 1 , . .

Original: , m } .
Revised: , m } .

Original: [Equation]
Revised: [Equation]

Revised: Similar expressions have been arrived at in (Giordano et al.

Revised: ; Debruyne et al.

Original: We give the precise deﬁnition of the constant term a m ( . ) and the function g m ( . ) — in terms of Bell polynomials — in Appendix B. A key consequence of Theorem 1 is that to compute all HOIFs, we need to evaluate the inverse of the Hessian matrix H − 1 θ only once, hence evaluating HOIFs scales linearly with the number of training examples.
Revised: We give the precise deﬁnition of the constant term a m ( . ) and the function g m ( . ) — in terms of Bell polynomials — in Appendix B. A key consequence of Theorem 1 is that to compute all HOIFs, we need to evaluate the inverse of the Hessian matrix H − 1 θ only once, hence evaluating HOIFs scales linearly with the number of training examples.

Original: With automatic differentiation tools at our disposal (Paszke et al.
Revised: With automatic differentiation tools at our disposal (Paszke et al.

Original: , computing the loss gradients (arguments of g m ( . ) ) can be done efﬁciently.
Revised: , computing the loss gradients (arguments of g m ( . ) ) can be done efﬁciently.

Original: As it is the case in ﬁrst-order inﬂuence functions, inverting H θ is the bottleneck operation in evaluating
Revised: As it is the case in ﬁrst-order inﬂuence functions, inverting H θ is the bottleneck operation in evaluating

Original: HOIFs, hence much of the machinery involved in (Koh & Liang (2017)) can be imported into our framework.
Revised: HOIFs, hence much of the machinery involved in (Koh & Liang (2017)) can be imported into our framework.

Original: With p parameters in θ , inverting H θ requires O ( np 2 + p 3 ) operations, which can be cumbersome for deep learning models with large number of parameters.
Revised: With p parameters in θ , inverting H θ requires O ( np 2 + p 3 ) operations, which can be cumbersome for deep learning models with large number of parameters.

Original: Following (Koh & Liang (2017)), we avoid explicitly computing H θ , and instead use implicit Hessian-vector products to efﬁciently approximate the Hessian products in (9).
Revised: Following (Koh & Liang (2017)), we avoid explicitly computing H θ , and instead use implicit Hessian-vector products to efﬁciently approximate the Hessian products in (9).

Original: For models with a very large p , we can use the stochastic estimation approach in (Agarwal et al.
Revised: For models with a very large p , we can use the stochastic estimation approach in (Agarwal et al.

Original: to estimate H − 1 θ with O ( p ) complexity.
Revised: to estimate H − 1 θ with O ( p ) complexity.

Original: 3. T HEORETICAL G UARANTEES
Revised: 3. T HEORETICAL G UARANTEES

Original: Now that we have developed a practical algorithm for implementing the DJ, we conclude this Section by revisiting the performance requirements in Section 2.
Revised: Now that we have developed a practical algorithm for implementing the DJ, we conclude this Section by revisiting the performance requirements in Section 2.

Original: In the following Theorem, we show that the approximate DJ conﬁdence intervals satisfy the desired coverage and discrimination requirements.
Revised: In the following Theorem, we show that the approximate DJ conﬁdence intervals satisfy the desired coverage and discrimination requirements.

Original: Theorem 2.
Revised: Theorem 2.

Original: Let (cid:98) C ( m ) α,n ( x ; ˆ θ ) be the approximate DJ conﬁdence interval evaluated with m HOIFs using Algorithm 1.
Revised: Let (cid:98) C ( m ) α,n ( x ; ˆ θ ) be the approximate DJ conﬁdence interval evaluated with m HOIFs using Algorithm 1.

Original: Then, for any model f ( x ; ˆ θ ) , the coverage rate achieved by the DJ is
Revised: Then, for any model f ( x ; ˆ θ ) , the coverage rate achieved by the DJ is

Original: [Equation]
Revised: [Equation]

Original:  If the model is stable, i.e., P { | f ( x ; ˆ θ ) − f ( x ; ˆ θ − i ) | ≤ ε } ≥ 1 − δ , ∀ i , for some ε and δ ∈ (0 , 1) (Bousquet & Elisseeff (2002)), then ∀ x , x (cid:48) ∈ X , y, y (cid:48) ∈ Y , and for a sufﬁciently large n , we have
Revised: ≥ 1 − α. If the model is stable, i.e., P { | f ( x ; ˆ θ ) − f ( x ; ˆ θ − i ) | ≤ ε } ≥ 1 − δ , ∀ i , for some ε and δ ∈ (0 , 1) (Bousquet & Elisseeff (2002)), then ∀ x , x (cid:48) ∈ X , y, y (cid:48) ∈ Y , and for a sufﬁciently large n , we have

Original: E [ W ( (cid:98) C ( ∞ ) α,n ( x ; ˆ θ )) ] ≥ E [ W ( (cid:98) C ( ∞ ) α,n ( x (cid:48) ; ˆ θ )) ] ⇔ E [ (cid:96) ( y, f ( x ; ˆ θ )) ] ≥ E [ (cid:96) ( y (cid:48) , f ( x (cid:48) ; ˆ θ )) ] .
Revised: E [ W ( (cid:98) C ( ∞ ) α,n ( x ; ˆ θ )) ] ≥ E [ W ( (cid:98) C ( ∞ ) α,n ( x (cid:48) ; ˆ θ )) ] ⇔ E [ (cid:96) ( y, f ( x ; ˆ θ )) ] ≥ E [ (cid:96) ( y (cid:48) , f ( x (cid:48) ; ˆ θ )) ] .

Original: (cid:3)
Revised: (cid:3)

Original: Theorem 2 provides a very strong guarantee on the frequentist coverage of the DJ conﬁdence intervals.
Revised: Theorem 2 provides a very strong guarantee on the frequentist coverage of the DJ conﬁdence intervals.

Original: That is, without any assumptions, the exact DJ interval (cid:98) C ( ∞ ) α,n ( x ; ˆ θ ) has a worst case coverage that fulﬁlls the target coverage rate of (1 − α ) .
Revised: That is, without any assumptions, the exact DJ interval (cid:98) C ( ∞ ) α,n ( x ; ˆ θ ) has a worst case coverage that fulﬁlls the target coverage rate of (1 − α ) .

Original: (The approximate DJ intervals, (cid:98) C ( m ) α,n ( x ; ˆ θ ) , achieve the target coverage in the limit of m → ∞ .)
Revised: (The approximate DJ intervals, (cid:98) C ( m ) α,n ( x ; ˆ θ ) , achieve the target coverage in the limit of m → ∞ .)

Original: In Section 5, we show through empirical evaluation that in practice — even with a ﬁnite number of HOIFs — the DJ intervals will achieve the (1 − α ) coverage.
Revised: In Section 5, we show through empirical evaluation that in practice — even with a ﬁnite number of HOIFs — the DJ intervals will achieve the (1 − α ) coverage.

Original: With further assumption on algorithmic stability (Bousquet & Elisseeff (2002)), Theorem 2 states that if a large number of HOIFs are involved in constructing the approximate DJ intervals, then the discrimination condition will be satisﬁed if we have a sufﬁciently large training sample.
Revised: With further assumption on algorithmic stability (Bousquet & Elisseeff (2002)), Theorem 2 states that if a large number of HOIFs are involved in constructing the approximate DJ intervals, then the discrimination condition will be satisﬁed if we have a sufﬁciently large training sample.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: The DJ is a post-hoc method that operates on deep learning models after being trained.
Revised: The DJ is a post-hoc method that operates on deep learning models after being trained.

Original: Post-hoc methods are advantageous in that they neither compromise the model accuracy, nor require any modiﬁcations in model structure or training.
Revised: Post-hoc methods are advantageous in that they neither compromise the model accuracy, nor require any modiﬁcations in model structure or training.

Original: (To implement Algorithm 1 (Section 3.3), we only need access to the model’s loss gradients and Hessian-vector products.)
Revised: (To implement Algorithm 1 (Section 3.3), we only need access to the model’s loss gradients and Hessian-vector products.)

Original: It is worth noting that, while our primary motivation is to estimate uncertainty in deep learning models, our method is applicable to any machine learning model for which HOIFs are accessible.
Revised: It is worth noting that, while our primary motivation is to estimate uncertainty in deep learning models, our method is applicable to any machine learning model for which HOIFs are accessible.

Original: Additionally, the exact DJ procedure can be applied to any model via exhaustive leave-one-out re-training, even when gradients are inaccessible.
Revised: Additionally, the exact DJ procedure can be applied to any model via exhaustive leave-one-out re-training, even when gradients are inaccessible.

Original: Post-hoc methods have been traditionally underexplored since existing approaches, such as calibration via temperature scaling (Platt et al.
Revised: Post-hoc methods have been traditionally underexplored since existing approaches, such as calibration via temperature scaling (Platt et al.

Original: (1999)), are known to under-perform compared to built-in methods (Ovadia et al.
Revised: (1999)), are known to under-perform compared to built-in methods (Ovadia et al.

Original: However, recent works have revived post-hoc approaches using ideas based on bootstrapping (Schulam & Saria (2019)), jackknife resampling (Barber et al.
Revised: However, recent works have revived post-hoc approaches using ideas based on bootstrapping (Schulam & Saria (2019)), jackknife resampling (Barber et al.

Original: (2019b)
Revised: (2019b)

Original: ; Giordano et al.
Revised: ; Giordano et al.

Original: and cross-validation (Vovk et al.
Revised: and cross-validation (Vovk et al.

Original: ; Barber et al. (2019a)).
Revised: ; Barber et al. (2019a)).

Original: An overview of the different classes of post
Revised: An overview of the different classes of post

Original: [Table]
Revised: [Table]

Original: hoc and built-in methods proposed in recent literature is provided in Table 1.
Revised: hoc and built-in methods proposed in recent literature is provided in Table 1.

Original: [Table]
Revised: [Table]

Original: As we discussed earlier, Bayesianism has been the dominant approach to uncertainty quantiﬁcation in deep learning (Welling & Teh (2011); Hernández-Lobato & Adams (2015); Ritter et al.
Revised: As we discussed earlier, Bayesianism has been the dominant approach to uncertainty quantiﬁcation in deep learning (Welling & Teh (2011); Hernández-Lobato & Adams (2015); Ritter et al.

Original: Maddox et al.
Revised: Maddox et al.

Original: By their very nature, Bayesian models cannot be applied in a post-hoc fashion since they require specifying priors over model parameters, which leads to major modiﬁcations in the inference algorithms.
Revised: By their very nature, Bayesian models cannot be applied in a post-hoc fashion since they require specifying priors over model parameters, which leads to major modiﬁcations in the inference algorithms.

Original: While Bayesian models provide a formal framework for uncertainty estimation, posterior credible intervals do not guarantee frequentist coverage, and more crucially, the achieved coverage can be very sensitive to hyper-parameter tuning (Bayarri & Berger (2004)).
Revised: While Bayesian models provide a formal framework for uncertainty estimation, posterior credible intervals do not guarantee frequentist coverage, and more crucially, the achieved coverage can be very sensitive to hyper-parameter tuning (Bayarri & Berger (2004)).

Original: Moreover, exact
Revised: Moreover, exact

Original: Bayesian inference is often computationally prohibitive, and alternative approximations — e.g., (Gal & Ghahramani (2016)) and (Kingma et al.
Revised: Bayesian inference is often computationally prohibitive, and alternative approximations — e.g., (Gal & Ghahramani (2016)) and (Kingma et al.

Original: — may induced posterior distributions that do not concentrate asymptotically (Osband (2016);
Revised: — may induced posterior distributions that do not concentrate asymptotically (Osband (2016);

Original: Hron et al.
Revised: Hron et al.

Original: Deep ensembles (Lakshminarayanan et al.
Revised: Deep ensembles (Lakshminarayanan et al.

Original: (2017)) are currently considered the most competitive (nonBayesian) benchmark for uncertainty quantiﬁcation (Ovadia et al.
Revised: (2017)) are currently considered the most competitive (nonBayesian) benchmark for uncertainty quantiﬁcation (Ovadia et al.

Original: This method repeatedly retrains the model on sub-samples of the data (using adversarial training), and then estimates uncertainty through the variance of the aggregate predictions.
Revised: This method repeatedly retrains the model on sub-samples of the data (using adversarial training), and then estimates uncertainty through the variance of the aggregate predictions.

Original: A similar bootstrapping approach, developed in (Schulam & Saria (2019)), uses the model’s Hessian and loss gradients to create an ensemble without re-training.
Revised: A similar bootstrapping approach, developed in (Schulam & Saria (2019)), uses the model’s Hessian and loss gradients to create an ensemble without re-training.

Original: While these methods may perform favorably in terms of discrimination, they are likely to undercover, since they only consider local variability terms akin to those in (5).
Revised: While these methods may perform favorably in terms of discrimination, they are likely to undercover, since they only consider local variability terms akin to those in (5).

Original: Additionally, ensemble methods do not provide theoretical guarantees on coverage and discrimination.
Revised: Additionally, ensemble methods do not provide theoretical guarantees on coverage and discrimination.

Original: The (inﬁnitesimal) jackknife method was previously applied for quantifying the predictive uncertainty in random forest models (Wager et al.
Revised: The (inﬁnitesimal) jackknife method was previously applied for quantifying the predictive uncertainty in random forest models (Wager et al.

Original: (2014); Mentch & Hooker (2016); Wager & Athey (2018)).
Revised: (2014); Mentch & Hooker (2016); Wager & Athey (2018)).

Original: In these works, however, the developed jackknife estimators are bespokely tailored to bagging predictors, and cannot be straightforwardly extended to deep neural networks.
Revised: In these works, however, the developed jackknife estimators are bespokely tailored to bagging predictors, and cannot be straightforwardly extended to deep neural networks.

Original: More recently, general-purpose jackknife estimators were developed in (Barber et al.
Revised: More recently, general-purpose jackknife estimators were developed in (Barber et al.

Original: (2019b)),
Revised: (2019b)),

Original: where two exhaustive leave-oneout procedures: the jackknife+ and the jackknife-minmax where shown to have assumption-free worst-case coverage guarantees of (1 − 2 α ) and (1 − α ) , respectively.
Revised: where two exhaustive leave-oneout procedures: the jackknife+ and the jackknife-minmax where shown to have assumption-free worst-case coverage guarantees of (1 − 2 α ) and (1 − α ) , respectively.

Original: Our work improves on these results in two ways.
Revised: Our work improves on these results in two ways.

Original: First, the DJ achieves the target coverage rate of (1 − α ) with a strictly narrower interval width compared to the jackknife-minmax (proof is provided in Appendix D).
Revised: First, the DJ achieves the target coverage rate of (1 − α ) with a strictly narrower interval width compared to the jackknife-minmax (proof is provided in Appendix D).

Original: Second, our proposed HOIF-based implementation of the jackknife alleviates the need for exhaustive leave-one-out re-training, which enables scaling up any jackknife-based method to large datasets.
Revised: Second, our proposed HOIF-based implementation of the jackknife alleviates the need for exhaustive leave-one-out re-training, which enables scaling up any jackknife-based method to large datasets.

Original: [Figure]
Revised: [Figure]

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: Our DJ procedure (Algorithm 1) was implemented on top of PyTorch models, where we computed the loss gradients and Hessian required for evaluating the HOIFs of a model’s parameters (Theorem 1) using autograd (Maclaurin et al.
Revised: Our DJ procedure (Algorithm 1) was implemented on top of PyTorch models, where we computed the loss gradients and Hessian required for evaluating the HOIFs of a model’s parameters (Theorem 1) using autograd (Maclaurin et al.

Original: The only hyper-parameter involved in our method is the number of HOIFs m — this was tuned by optimizing the evaluation metrics (listed below) on the training samples.
Revised: The only hyper-parameter involved in our method is the number of HOIFs m — this was tuned by optimizing the evaluation metrics (listed below) on the training samples.

Original: Code snippets for the DJ procedure are provided in the supplementary appendix.
Revised: Code snippets for the DJ procedure are provided in the supplementary appendix.

Original: Baselines.
Revised: Baselines.

Original: We compared our DJ procedure with 6 state-of-the-art baseline methods.
Revised: We compared our DJ procedure with 6 state-of-the-art baseline methods.

Original: These included 3 built-in Bayesian methods: (1) Monte Carlo Dropout (MCDP) (Gal & Ghahramani (2016)), (2)
Revised: These included 3 built-in Bayesian methods: (1) Monte Carlo Dropout (MCDP) (Gal & Ghahramani (2016)), (2)

Original: Probabilistic backpropagation (PBP) (Hernández-Lobato & Adams (2015)), and (3) Bayesian neural networks with inference via stochastic gradient Langevin dynamics (BNN-SGLD)
Revised: Probabilistic backpropagation (PBP) (Hernández-Lobato & Adams (2015)), and (3) Bayesian neural networks with inference via stochastic gradient Langevin dynamics (BNN-SGLD)

Original: (Welling & Teh (2011)).
Revised: (Welling & Teh (2011)).

Original: In addition, we considered deep ensembles (DE) (Lakshminarayanan et al.
Revised: In addition, we considered deep ensembles (DE) (Lakshminarayanan et al.

Original: (2017)), which is deemed the most competitive built-in frequentist method (Ovadia et al.
Revised: (2017)), which is deemed the most competitive built-in frequentist method (Ovadia et al.

Original: We also includedpost-hoc methods: resampling uncertainty estimation (RUE) (Schulam & Saria (2019)) and crossconformal predictive distributions (CCP) (Vovk et al.
Revised: We also includedpost-hoc methods: resampling uncertainty estimation (RUE) (Schulam & Saria (2019)) and crossconformal predictive distributions (CCP) (Vovk et al.

Original: For a target coverage of (1 − α ) , uncertainty estimates were obtained by setting the posterior quantile functions to (1 − α ) (for Bayesian methods), or obtaining the (1 − α ) percentile point of a normal distribution (for frequentist methods).
Revised: For a target coverage of (1 − α ) , uncertainty estimates were obtained by setting the posterior quantile functions to (1 − α ) (for Bayesian methods), or obtaining the (1 − α ) percentile point of a normal distribution (for frequentist methods).

Original: We implemented all baselines in PyTorch (details are provided in the supplementary appendix).
Revised: We implemented all baselines in PyTorch (details are provided in the supplementary appendix).

Original: Evaluation metrics.
Revised: Evaluation metrics.

Original: In all experiments, we used the mean squared error (MSE) as the loss L ( D , ˆ θ ) for the trained model f ( x ; θ ) .
Revised: In all experiments, we used the mean squared error (MSE) as the loss L ( D , ˆ θ ) for the trained model f ( x ; θ ) .

Original: To ensure a fair comparison, the hyper-parameters of the model f ( x ; θ ) were the same for all baselines.
Revised: To ensure a fair comparison, the hyper-parameters of the model f ( x ; θ ) were the same for all baselines.

Original: In each experiment, an uncertainty estimate (cid:98) C α,n ( x ; θ ) is obtained from a training sample, and then coverage and discrimination are evaluated on a test sample.
Revised: In each experiment, an uncertainty estimate (cid:98) C α,n ( x ; θ ) is obtained from a training sample, and then coverage and discrimination are evaluated on a test sample.

Original: To evaluate empirical coverage probability, we compute the fraction of test samples for which y resides in (cid:98) C α,n ( x ; ˆ θ ) .
Revised: To evaluate empirical coverage probability, we compute the fraction of test samples for which y resides in (cid:98) C α,n ( x ; ˆ θ ) .

Original: Discrimination was evaluated as follows.
Revised: Discrimination was evaluated as follows.

Original: For each baseline method, we evaluate the interval width W ( (cid:98) C α,n ( x ; θ )) for all test points.
Revised: For each baseline method, we evaluate the interval width W ( (cid:98) C α,n ( x ; θ )) for all test points.

Original: For a given error threshold E , we use the interval width to detect whether the test prediction f ( x ; θ ) is a high-conﬁdence, i.e., (cid:96) ( y, f ( x ; θ )) ≤ E , or a low-conﬁdence prediction, i.e., (cid:96) ( y, f ( x ; θ )) >
Revised: For a given error threshold E , we use the interval width to detect whether the test prediction f ( x ; θ ) is a high-conﬁdence, i.e., (cid:96) ( y, f ( x ; θ )) ≤ E , or a low-conﬁdence prediction, i.e., (cid:96) ( y, f ( x ; θ )) >

Original: E .
Revised: E .

Original: We use the area under the ROC curve (AUC-ROC) to evaluate the accuracy of classifying high- and low-conﬁdence predictions using W ( (cid:98) C α,n ( x ; θ )) .
Revised: We use the area under the ROC curve (AUC-ROC) to evaluate the accuracy of classifying high- and low-conﬁdence predictions using W ( (cid:98) C α,n ( x ; θ )) .

Original: 5. S YNTHETIC D ATA
Revised: 5. S YNTHETIC D ATA

Original: Qualitative assessment.
Revised: Qualitative assessment.

Original: First, we qualitatively assess the DJ conﬁdence intervals using the synthetic data model in (Hernández-Lobato & Adams (2015)).
Revised: First, we qualitatively assess the DJ conﬁdence intervals using the synthetic data model in (Hernández-Lobato & Adams (2015)).

Original: This dataset is based on the one-dimensional model:
Revised: This dataset is based on the one-dimensional model:

Original: y = x 3 + (cid:15), where (cid:15) ∼ N (0 , 3 2 ) .
Revised: y = x 3 + (cid:15), where (cid:15) ∼ N (0 , 3 2 ) .

Original: We ﬁt a feed-forward neural network model (with the same architecture as (Hernández-Lobato & Adams (2015))) on 50 training examples drawn from a uniform distribution on X =
Revised: We ﬁt a feed-forward neural network model (with the same architecture as (Hernández-Lobato & Adams (2015))) on 50 training examples drawn from a uniform distribution on X =

Original: [ − 4 , 4] , and evaluate conﬁdence intervals on its predictions with a target coverage of 90 % .
Revised: [ − 4 , 4] , and evaluate conﬁdence intervals on its predictions with a target coverage of 90 % .

Original: The conﬁdence intervals constructed by all baselines are visualized in Figure 3.
Revised: The conﬁdence intervals constructed by all baselines are visualized in Figure 3.

Original: As we can see in Figure 3, Bayesian methods tend to undercover the target function — i.e., PBP overestimated model conﬁdence, whereas BNN-SGLD failed to recognize that the region | x | >should be assigned higher uncertainty since it contains no training data.
Revised: As we can see in Figure 3, Bayesian methods tend to undercover the target function — i.e., PBP overestimated model conﬁdence, whereas BNN-SGLD failed to recognize that the region | x | >should be assigned higher uncertainty since it contains no training data.

Original: For MCDP, we can see that higher uncertainty is assigned to the region | x | > 4 , but the posterior does not concentrate around the training data, and hence credible intervals are poorly calibrated.
Revised: For MCDP, we can see that higher uncertainty is assigned to the region | x | > 4 , but the posterior does not concentrate around the training data, and hence credible intervals are poorly calibrated.

Original: Post-hoc methods (RUE and
Revised: Post-hoc methods (RUE and

Original: CCP) tend to overestimate conﬁdence in regions with dense training data ( | x | ≤ 4 ) since they rely mainly on the (in-sample) error variance to estimate uncertainty.
Revised: CCP) tend to overestimate conﬁdence in regions with dense training data ( | x | ≤ 4 ) since they rely mainly on the (in-sample) error variance to estimate uncertainty.

Original: DE produces a favorable conﬁdence interval, but with little variation in interval width and with few training points being uncovered.
Revised: DE produces a favorable conﬁdence interval, but with little variation in interval width and with few training points being uncovered.

Original: The DJ conﬁdence intervals hit the right balance between coverage and discrimination.
Revised: The DJ conﬁdence intervals hit the right balance between coverage and discrimination.

Original: The addition of the marginal error (cid:98) Q 0 . 1 , 50 ( R 50 ) in (5) ensures that the training data points and target function are covered everywhere in X .
Revised: The addition of the marginal error (cid:98) Q 0 . 1 , 50 ( R 50 ) in (5) ensures that the training data points and target function are covered everywhere in X .

Original: The variability terms, (cid:98) Q 0 .
Revised: The variability terms, (cid:98) Q 0 .

Original: 1 , 23 ( V +50 ) and (cid:98) Q 0 .
Revised: 1 , 23 ( V +50 ) and (cid:98) Q 0 .

Original: 9 , 27 ( V − 50 ) , add the discrimination components, which enable the interval width to clearly distinguish high-conﬁdence regions ( | x | ≤ 4 ) from low-conﬁdence ones ( | x | > 4 ).
Revised: 9 , 27 ( V − 50 ) , add the discrimination components, which enable the interval width to clearly distinguish high-conﬁdence regions ( | x | ≤ 4 ) from low-conﬁdence ones ( | x | > 4 ).

Original: By contrasting the DJ estimates with m = 1 and m = 2 , we see that adding more HOIF terms gives smoother and tighter conﬁdence intervals.
Revised: By contrasting the DJ estimates with m = 1 and m = 2 , we see that adding more HOIF terms gives smoother and tighter conﬁdence intervals.

Original: [Figure]
Revised: [Figure]

Original: I n t e r v a l w i d t h
Revised: I n t e r v a l w i d t h

Original: [Figure]
Revised: [Figure]

Original: Quantitative assessment.
Revised: Quantitative assessment.

Original: Next, we examine coverage and discrimination performance of all baselines on the following synthetic model.
Revised: Next, we examine coverage and discrimination performance of all baselines on the following synthetic model.

Original: For n = 100 and 1 − α = 0 .
Revised: For n = 100 and 1 − α = 0 .

Original: 9 , we conduct 100 experiments, each for different feature dimensions d = 1 , 11 , 21 , . . .
Revised: 9 , we conduct 100 experiments, each for different feature dimensions d = 1 , 11 , 21 , . . .

Original: , 101 , with i.i.d.
Revised: , 101 , with i.i.d.

Original: samples ( x i , y i ) generated as β ∼ Uniform [ 0 , 1 ] ⊗ d , x i ∼ N (0 , I d ) , and y i | x i ∼ N (( x Ti β ) 3 , 1) .
Revised: samples ( x i , y i ) generated as β ∼ Uniform [ 0 , 1 ] ⊗ d , x i ∼ N (0 , I d ) , and y i | x i ∼ N (( x Ti β ) 3 , 1) .

Original: In each experiment, we ﬁt a 2-layer neural network with 100 hidden units to the training data, and then generate 100 test data points from the same distribution to evaluate the coverage and discrimination performance for different values of d .
Revised: In each experiment, we ﬁt a 2-layer neural network with 100 hidden units to the training data, and then generate 100 test data points from the same distribution to evaluate the coverage and discrimination performance for different values of d .

Original: (Discrimination is evaluated by setting the error threshold E to be the 95 % percentile of the trained model error on the test samples.)
Revised: (Discrimination is evaluated by setting the error threshold E to be the 95 % percentile of the trained model error on the test samples.)

Original: As we can see in Figure 4, DJ is the only method that always manages to achieve the 90 % coverage rate for any number of features d (as predicted by Theorem 2).
Revised: As we can see in Figure 4, DJ is the only method that always manages to achieve the 90 % coverage rate for any number of features d (as predicted by Theorem 2).

Original: It does so while also achieving the most competitive discrimination (AUC-ROC) performance, together with the RUE and PBP baselines.
Revised: It does so while also achieving the most competitive discrimination (AUC-ROC) performance, together with the RUE and PBP baselines.

Original: We also found that, for all values of d , the DJ interval width was strictly tighter than that of the jackknife-minmax (JM) procedure (Barber et al.
Revised: We also found that, for all values of d , the DJ interval width was strictly tighter than that of the jackknife-minmax (JM) procedure (Barber et al.

Original: (2019b)), which is the only other procedure that guarantees the 1 − α target coverage.
Revised: (2019b)), which is the only other procedure that guarantees the 1 − α target coverage.

Original: In this experiment, we found that m = 2 HOIFs are sufﬁcient for achieving the best possible performance by DJ.
Revised: In this experiment, we found that m = 2 HOIFs are sufﬁcient for achieving the best possible performance by DJ.

Original: More HOIFs would generally be needed for more complex networks, with more layers or hidden units (see further results in the supplementary appendix).
Revised: More HOIFs would generally be needed for more complex networks, with more layers or hidden units (see further results in the supplementary appendix).

Original: Because of its post-hoc nature, the computational complexity of DJ was signiﬁcantly less than built-in method (average runtime of
Revised: Because of its post-hoc nature, the computational complexity of DJ was signiﬁcantly less than built-in method (average runtime of

Original: DJ was 50 times faster than DE, 30 times faster than BNN-SGLD and 10 times faster than PBP).
Revised: DJ was 50 times faster than DE, 30 times faster than BNN-SGLD and 10 times faster than PBP).

Original: 5. R EAL D ATA : A UDITING M ODEL R ELIABILITY
Revised: 5. R EAL D ATA : A UDITING M ODEL R ELIABILITY

Original: We conducted a series of experiments on real-world datasets in order to evaluate the accuracy of uncertainty estimates issued by the DJ procedure in various contexts.
Revised: We conducted a series of experiments on real-world datasets in order to evaluate the accuracy of uncertainty estimates issued by the DJ procedure in various contexts.

Original: In this Section, we show how uncertainty estimates can be used to audit the reliability of a predictive model.
Revised: In this Section, we show how uncertainty estimates can be used to audit the reliability of a predictive model.

Original: Further experiments on active learning and out-of-distribution detection are deferred to the supplementary appendix.
Revised: Further experiments on active learning and out-of-distribution detection are deferred to the supplementary appendix.

Original: In this experiment, we compare how well DJ and the competing baselines are able to detect when a model’s prediction will be far from the true target value.
Revised: In this experiment, we compare how well DJ and the competing baselines are able to detect when a model’s prediction will be far from the true target value.

Original: We conduct our experiments on 8 common
Revised: We conduct our experiments on 8 common

Original: UCI benchmark datasets for regression — those included: Boston housing, concrete compression strength, energy efﬁciency, robot kinematics (Kin8nm), naval propulsion, combined cycle power plant, red wine quality, and yacht hydrodynamics (Dua & Graff (2017)).
Revised: UCI benchmark datasets for regression — those included: Boston housing, concrete compression strength, energy efﬁciency, robot kinematics (Kin8nm), naval propulsion, combined cycle power plant, red wine quality, and yacht hydrodynamics (Dua & Graff (2017)).

Original: [Table]
Revised: [Table]

Original: For each dataset, we sample 20 train-test splits, and use 90 % of the data points for training.
Revised: For each dataset, we sample 20 train-test splits, and use 90 % of the data points for training.

Original: To control for model complexity, we ﬁx a single feed-forward neural network architecture with two hidden layers and 100 hidden units, logistic activation functions, MSE loss, and a single set of learning hyper-parameters (500 epochs with 100 samples per minibatch, and an Adam optimizer with default settings).
Revised: To control for model complexity, we ﬁx a single feed-forward neural network architecture with two hidden layers and 100 hidden units, logistic activation functions, MSE loss, and a single set of learning hyper-parameters (500 epochs with 100 samples per minibatch, and an Adam optimizer with default settings).

Original: We ﬁt a single model for each split of each dataset, and compute the conﬁdence intervals (with 1 − α = 0 . 9 ) for all baselines on the test data.
Revised: We ﬁt a single model for each split of each dataset, and compute the conﬁdence intervals (with 1 − α = 0 . 9 ) for all baselines on the test data.

Original: We implemented two versions of our method: a
Revised: We implemented two versions of our method: a

Original: DJ with a ﬁrst-order von Mises approximation ( m = 1 ), and a DJ with m = 10 HOIFs.
Revised: DJ with a ﬁrst-order von Mises approximation ( m = 1 ), and a DJ with m = 10 HOIFs.

Original: Similar to the previous experiment, we evaluate the empirical coverage rate and discrimination (AUC-ROC) performance assuming the threshold E to be the 95 % percentile of the empirical distribution over test errors.
Revised: Similar to the previous experiment, we evaluate the empirical coverage rate and discrimination (AUC-ROC) performance assuming the threshold E to be the 95 % percentile of the empirical distribution over test errors.

Original: Results are provided in Table 2.
Revised: Results are provided in Table 2.

Original: We observe that, in terms of discriminative performance, DJ outperforms (or is competitive with) the baselines — the AUC-ROC of detecting low-quality predictions by DJ was better than other methods in 6 out of the 8 datasets.
Revised: We observe that, in terms of discriminative performance, DJ outperforms (or is competitive with) the baselines — the AUC-ROC of detecting low-quality predictions by DJ was better than other methods in 6 out of the 8 datasets.

Original: Including HOIFs was important for achieving good performance in most datasets.
Revised: Including HOIFs was important for achieving good performance in most datasets.

Original: A ﬁrst-order von Mises approximation was sufﬁcient only in the “Energy” dataset, but under-performed compared to the DJ with m = 10 in all other datasets.
Revised: A ﬁrst-order von Mises approximation was sufﬁcient only in the “Energy” dataset, but under-performed compared to the DJ with m = 10 in all other datasets.

Original: Another key observation is that — as predicted by Theorem 2 — the DJ procedure always achieves the 90 % target coverage rate.
Revised: Another key observation is that — as predicted by Theorem 2 — the DJ procedure always achieves the 90 % target coverage rate.

Original: All other methods tend to undercover the test points, and some methods (such as MCDP and
Revised: All other methods tend to undercover the test points, and some methods (such as MCDP and

Original: PBP) can even achieve close to zero coverage.
Revised: PBP) can even achieve close to zero coverage.

Original: It is worth mentioning that on the dataset where DJ provided the biggest improvement in discriminative performance (Boston housing), it was also the only method that achieved the target coverage.
Revised: It is worth mentioning that on the dataset where DJ provided the biggest improvement in discriminative performance (Boston housing), it was also the only method that achieved the target coverage.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: Uncertainty quantiﬁcation is a crucial requirement in various high-stakes applications where deep learning can inform critical decision-making.
Revised: Uncertainty quantiﬁcation is a crucial requirement in various high-stakes applications where deep learning can inform critical decision-making.

Original: In this paper, we developed a rigorous frequentist procedure for quantifying the uncertainty in predictions issued by deep learning models in a post-hoc fashion.
Revised: In this paper, we developed a rigorous frequentist procedure for quantifying the uncertainty in predictions issued by deep learning models in a post-hoc fashion.

Original: Our procedure, which is inspired by classical jackknife estimators, does not require any modiﬁcations in the underlying deep learning model, and provides theoretical guarantees on its achieved performance.
Revised: Our procedure, which is inspired by classical jackknife estimators, does not require any modiﬁcations in the underlying deep learning model, and provides theoretical guarantees on its achieved performance.

Original: Implementation of our procedure using higher-order inﬂuence functions entails minimal computational costs, and allows our method to scale for big datasets, without the need for repeated re-training as in the classical jackknife procedure.
Revised: Implementation of our procedure using higher-order inﬂuence functions entails minimal computational costs, and allows our method to scale for big datasets, without the need for repeated re-training as in the classical jackknife procedure.

Original: While this paper focuses on uncertainty estimation in deep learning, the methods therein can be applied to any machine learning model as long as we are granted access to its loss gradients and Hessian.
Revised: While this paper focuses on uncertainty estimation in deep learning, the methods therein can be applied to any machine learning model as long as we are granted access to its loss gradients and Hessian.

Original: A I NFLUENCE F UNCTIONS : B ACKGROUND & K EY C ONCEPTS A. F ORMAL D EFINITION
Revised: A I NFLUENCE F UNCTIONS : B ACKGROUND & K EY C ONCEPTS A. F ORMAL D EFINITION

Original: Robust statistics is the branch of statistics concerned with the detection of outlying observations.
Revised: Robust statistics is the branch of statistics concerned with the detection of outlying observations.

Original: An estimator is deemed robust if it produces similar results as the majority of observations indicates, regardless of how a minority of other observations is perturbed (Huber & Ronchetti (1981)).
Revised: An estimator is deemed robust if it produces similar results as the majority of observations indicates, regardless of how a minority of other observations is perturbed (Huber & Ronchetti (1981)).

Original: The inﬂuence function measures these effects in statistical functionals by analyzing the behavior of a functional not only at the distribution of interest, but also in an entire neighborhood of distributions around it.
Revised: The inﬂuence function measures these effects in statistical functionals by analyzing the behavior of a functional not only at the distribution of interest, but also in an entire neighborhood of distributions around it.

Original: Lack of model robustness is a clear indicator of model uncertainty, and hence inﬂuence functions arise naturally in our method as a (pointwise) surrogate measure of model uncertainty.
Revised: Lack of model robustness is a clear indicator of model uncertainty, and hence inﬂuence functions arise naturally in our method as a (pointwise) surrogate measure of model uncertainty.

Original: In this section we formally deﬁne inﬂuence functions and discuss its properties.
Revised: In this section we formally deﬁne inﬂuence functions and discuss its properties.

Original: The pioneering works in (Hampel et al.
Revised: The pioneering works in (Hampel et al.

Original: and (Huber & Ronchetti (1981)) coined the notion of inﬂuence functions to assess the robustness of statistical functionals to perturbations in the underlying distributions.
Revised: and (Huber & Ronchetti (1981)) coined the notion of inﬂuence functions to assess the robustness of statistical functionals to perturbations in the underlying distributions.

Original: Consider a statistical functional T : P → R , deﬁned on a probability space P , and a probability distribution P ∈ P .
Revised: Consider a statistical functional T : P → R , deﬁned on a probability space P , and a probability distribution P ∈ P .

Original: Consider distributions of the form P ε,z
Revised: Consider distributions of the form P ε,z

Original: = (1 − ε ) P + ε ∆ z where ∆ z denotes the Dirac distribution in the point
Revised: = (1 − ε ) P + ε ∆ z where ∆ z denotes the Dirac distribution in the point

Original: z =
Revised: z =

Original: ( x , y ) , representing the contaminated part of the data.
Revised: ( x , y ) , representing the contaminated part of the data.

Original: For the functional T to be considered robust, T ( P ε,z ) should not be too far away from T ( P ) for any possible z and any small ε .
Revised: For the functional T to be considered robust, T ( P ε,z ) should not be too far away from T ( P ) for any possible z and any small ε .

Original: The limiting case of ε → 0 deﬁnes the inﬂuence function.
Revised: The limiting case of ε → 0 deﬁnes the inﬂuence function.

Original: That is, Then the inﬂuence function of T at P in the point z is deﬁned as
Revised: That is, Then the inﬂuence function of T at P in the point z is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: The inﬂuence function measures the robustness of T by quantifying the effect on the estimator T when adding an inﬁnitesimally small amount of contamination at the point z .
Revised: The inﬂuence function measures the robustness of T by quantifying the effect on the estimator T when adding an inﬁnitesimally small amount of contamination at the point z .

Original: If the supremum of I ( . ) over z is bounded, then an inﬁnitesimally small amount of perturbation cannot cause arbitrary large changes in the estimate.
Revised: If the supremum of I ( . ) over z is bounded, then an inﬁnitesimally small amount of perturbation cannot cause arbitrary large changes in the estimate.

Original: Then small amounts of perturbation cannot completely change the estimate which ensures the robustness of the estimator.
Revised: Then small amounts of perturbation cannot completely change the estimate which ensures the robustness of the estimator.

Original: A. T HE VON M ISES E XPANSION
Revised: A. T HE VON M ISES E XPANSION

Original: The von Mises expansion is a distributional analog of the Taylor expansion applied for a functional instead of a function.
Revised: The von Mises expansion is a distributional analog of the Taylor expansion applied for a functional instead of a function.

Original: For two distributions P and Q , the Von Mises expansion is (Fernholz (2012)):
Revised: For two distributions P and Q , the Von Mises expansion is (Fernholz (2012)):

Original: [Equation]
Revised: [Equation]

Original: where I ( k ) ( z ; P ) is the k th order inﬂuence function.
Revised: where I ( k ) ( z ; P ) is the k th order inﬂuence function.

Original: By setting Q to be a perturbed version of P , i.e., Q = P ε , the von Mises expansion at point z reduces to:
Revised: By setting Q to be a perturbed version of P , i.e., Q = P ε , the von Mises expansion at point z reduces to:

Original: [Equation]
Revised: [Equation]

Original: and so the k th order inﬂuence function is operationalized through the derivative
Revised: and so the k th order inﬂuence function is operationalized through the derivative

Original: [Equation]
Revised: [Equation]

Original: A. I NFLUENCE F UNCTION OF M ODEL L OSS
Revised: A. I NFLUENCE F UNCTION OF M ODEL L OSS

Original: Now we apply the mathematical deﬁnitions in Sections A.1 and A.2 to our learning setup.
Revised: Now we apply the mathematical deﬁnitions in Sections A.1 and A.2 to our learning setup.

Original: In our setting, the functional T ( . ) corresponds to the (trained) model parameters ˆ θ and the distribution P .
Revised: In our setting, the functional T ( . ) corresponds to the (trained) model parameters ˆ θ and the distribution P .

Original: In this case, inﬂuence functions of ˆ θ computes how much the model parameters would change if the underlying data distribution was perturbed inﬁnitesimally.
Revised: In this case, inﬂuence functions of ˆ θ computes how much the model parameters would change if the underlying data distribution was perturbed inﬁnitesimally.

Original: [Equation]
Revised: [Equation]

Original: Recall that in the deﬁnition of the inﬂuence function P ε,z
Revised: Recall that in the deﬁnition of the inﬂuence function P ε,z

Original: =
Revised: =

Original: (1 − ε ) P + ε ∆ z where ∆ z denotes the Dirac distribution in the point
Revised: (1 − ε ) P + ε ∆ z where ∆ z denotes the Dirac distribution in the point

Original: z =
Revised: z =

Original: ( x , y ) .
Revised: ( x , y ) .

Original: Thus, the (ﬁrst-order) inﬂuence function in (A.5) corresponds to perturbing a training data point z by an inﬁnitesimally small change ε and evaluating the corresponding change in the learned model parameters ˆ θ .
Revised: Thus, the (ﬁrst-order) inﬂuence function in (A.5) corresponds to perturbing a training data point z by an inﬁnitesimally small change ε and evaluating the corresponding change in the learned model parameters ˆ θ .

Original: More generally, the k th order inﬂuence function of ˆ θ is deﬁned as follows:
Revised: More generally, the k th order inﬂuence function of ˆ θ is deﬁned as follows:

Original: [Equation]
Revised: [Equation]

Original: By applying the von Mises expansion, we can approximate the parameter of a model trained on the training dataset with perturbed data point z as follows:
Revised: By applying the von Mises expansion, we can approximate the parameter of a model trained on the training dataset with perturbed data point z as follows:

Original: [Equation]
Revised: [Equation]

Original: where m is the number of terms included in the truncated expansion.
Revised: where m is the number of terms included in the truncated expansion.

Original: When m = ∞ , the exact parameter ˆ θ ε,z without the need to re-train the model.
Revised: When m = ∞ , the exact parameter ˆ θ ε,z without the need to re-train the model.

Original: A. C ONNECTION TO LEAVE - ONE - OUT ESTIMATORS
Revised: A. C ONNECTION TO LEAVE - ONE - OUT ESTIMATORS

Original: Our uncertainty estimator depends on perturbing the model parameters by removing a single training point at a time.
Revised: Our uncertainty estimator depends on perturbing the model parameters by removing a single training point at a time.

Original: Note that removing a point z is the same as perturbing z by ε = − 1 n , hence we obtain an ( m th order) approximation of the parameter change due to removing the point z as follows:
Revised: Note that removing a point z is the same as perturbing z by ε = − 1 n , hence we obtain an ( m th order) approximation of the parameter change due to removing the point z as follows:

Original: [Equation]
Revised: [Equation]

Original: where ˆ θ − z is the model parameter learned by removing the data point z from the training data.
Revised: where ˆ θ − z is the model parameter learned by removing the data point z from the training data.

Original: B P ROOF OF T HEOREM
Revised: B P ROOF OF T HEOREM

Original: Recall that the LOO parameter ˆ θ i,(cid:15) is obtained by solving the optimization problem:
Revised: Recall that the LOO parameter ˆ θ i,(cid:15) is obtained by solving the optimization problem:

Original: [Equation]
Revised: [Equation]

Original: Let us ﬁrst derive the ﬁrst order inﬂuence function I (1) ( x i , y i ) .
Revised: Let us ﬁrst derive the ﬁrst order inﬂuence function I (1) ( x i , y i ) .

Original: Let us ﬁrst deﬁne ∆ i,(cid:15) (cid:44) ˆ θ i,(cid:15) − ˆ θ .
Revised: Let us ﬁrst deﬁne ∆ i,(cid:15) (cid:44) ˆ θ i,(cid:15) − ˆ θ .

Original: The ﬁrst order inﬂuence function is given by:
Revised: The ﬁrst order inﬂuence function is given by:

Original: [Equation]
Revised: [Equation]

Original: Note that, since ˆ θ i,(cid:15) is the minimizer of (B.1), then the perturbed loss has to satisfy the following (ﬁrst order) optimality condition:
Revised: Note that, since ˆ θ i,(cid:15) is the minimizer of (B.1), then the perturbed loss has to satisfy the following (ﬁrst order) optimality condition:

Original: [Equation]
Revised: [Equation]

Original: Since lim (cid:15) → 0 ˆ θ i,(cid:15)
Revised: Since lim (cid:15) → 0 ˆ θ i,(cid:15)

Original: =
Revised: =

Original: ˆ θ , then we can write the following Taylor expansion:
Revised: ˆ θ , then we can write the following Taylor expansion:

Original: [Equation]
Revised: [Equation]

Original: Now by dropping the o ( (cid:107) ∆ i,(cid:15) (cid:107) ) terms, we have:
Revised: Now by dropping the o ( (cid:107) ∆ i,(cid:15) (cid:107) ) terms, we have:

Original: [Equation]
Revised: [Equation]

Original: Since ˆ θ is a indeed a minimizer of the loss function (cid:96) ( . ) , then we have ∇ θ (cid:96) ( . )
Revised: Since ˆ θ is a indeed a minimizer of the loss function (cid:96) ( . ) , then we have ∇ θ (cid:96) ( . )

Original: = 0 .
Revised: = 0 .

Original: Thus, (B.5) reduces to the following condition:
Revised: Thus, (B.5) reduces to the following condition:

Original: [Equation]
Revised: [Equation]

Original: By solving for ∇ θ , we have
Revised: By solving for ∇ θ , we have

Original: [Equation]
Revised: [Equation]

Original: which can be approximated by keeping only the O ( (cid:15) ) terms as follows:
Revised: which can be approximated by keeping only the O ( (cid:15) ) terms as follows:

Original: [Equation]
Revised: [Equation]

Original: Noting that ∇ 2 θ L ( D , ˆ θ ) is the Hessian matrix H ˆ θ , we have:
Revised: Noting that ∇ 2 θ L ( D , ˆ θ ) is the Hessian matrix H ˆ θ , we have:

Original: [Equation]
Revised: [Equation]

Original: By taking the derivative with respect to (cid:15) , we arrive at the expression for ﬁrst order inﬂuence functions:
Revised: By taking the derivative with respect to (cid:15) , we arrive at the expression for ﬁrst order inﬂuence functions:

Original: [Equation]
Revised: [Equation]

Original: Now let us examine the second order inﬂuence functions.
Revised: Now let us examine the second order inﬂuence functions.

Original: In order to obtain I (2) ( x i , y i ) , we need to differentiate (B.6) after omitting the O ( (cid:15) ) once again as follows:
Revised: In order to obtain I (2) ( x i , y i ) , we need to differentiate (B.6) after omitting the O ( (cid:15) ) once again as follows:

Original: [Equation]
Revised: [Equation]

Original: Where we have applied the chain rule to obtain the above.
Revised: Where we have applied the chain rule to obtain the above.

Original: By substituting ∇ 2 θ L ( D , ˆ θ ) =
Revised: By substituting ∇ 2 θ L ( D , ˆ θ ) =

Original: H θ and dividing both sides of (B.11) by (cid:15) 2 , we have
Revised: H θ and dividing both sides of (B.11) by (cid:15) 2 , we have

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: Thus, by re-arranging (B.11), we can obtain I (2) ( x i , y i ) in terms of I (1) ( x i , y i ) as follows:
Revised: Thus, by re-arranging (B.11), we can obtain I (2) ( x i , y i ) in terms of I (1) ( x i , y i ) as follows:

Original: [Equation]
Revised: [Equation]

Original: Similarly, we can obtain the k th order inﬂuence function, for any k > 1 , by repeatedly differentiating equation (B.6) k times, i.e.,
Revised: Similarly, we can obtain the k th order inﬂuence function, for any k > 1 , by repeatedly differentiating equation (B.6) k times, i.e.,

Original: [Equation]
Revised: [Equation]

Original: and solving for ∂ ∆ ki,(cid:15) /∂(cid:15)
Revised: and solving for ∂ ∆ ki,(cid:15) /∂(cid:15)

Original: k .
Revised: k .

Original: By applying the higher-order chain rule to the two added terms in (B.13) (or equivalently, take the derivative of I (2) ( x i , y i ) for k − 2 times), we can represent the formula for
Revised: By applying the higher-order chain rule to the two added terms in (B.13) (or equivalently, take the derivative of I (2) ( x i , y i ) for k − 2 times), we can represent the formula for

Original: I ( k ) ( x i , y i ) using Bell polynomials (Ma (2013)) as follows:
Revised: I ( k ) ( x i , y i ) using Bell polynomials (Ma (2013)) as follows:

Original: [Equation]
Revised: [Equation]

Original: where B m
Revised: where B m

Original: [Equation]
Revised: [Equation]

Original: k is the Bell polynomial
Revised: k is the Bell polynomial

Original: [Equation]
Revised: [Equation]

Original: which is deﬁned as follows:
Revised: which is deﬁned as follows:

Original: [Equation]
Revised: [Equation]

Original: C P ROOF OF T HEOREM
Revised: C P ROOF OF T HEOREM

Original: We start by proving the coverage guarantee in Section C.1 and then prove the discrimination guarantee in Section C.2.
Revised: We start by proving the coverage guarantee in Section C.1 and then prove the discrimination guarantee in Section C.2.

Original: C. F REQUENTIST C OVERAGE
Revised: C. F REQUENTIST C OVERAGE

Original: In what follows, we analyze the coverage probability of the exact DJ conﬁdence intervals.
Revised: In what follows, we analyze the coverage probability of the exact DJ conﬁdence intervals.

Original: The proof parallels that of (Barber et al.
Revised: The proof parallels that of (Barber et al.

Original: (2019b)).
Revised: (2019b)).

Original: Recall that the empirical (1 − α ) quantile is deﬁned by: (cid:98) Q α,n ( { X 1 , . . .
Revised: Recall that the empirical (1 − α ) quantile is deﬁned by: (cid:98) Q α,n ( { X 1 , . . .

Original: X n } )
Revised: X n } )

Original: := the (cid:100) (1 − α ) ( n + 1) (cid:101) -th smallest value of { X 1 , . . .
Revised: := the (cid:100) (1 − α ) ( n + 1) (cid:101) -th smallest value of { X 1 , . . .

Original: X n } .
Revised: X n } .

Original: Let the test point be ( x n +1 , y n +1 ) and suppose we have oracle access to the true test label y n +1 .
Revised: Let the test point be ( x n +1 , y n +1 ) and suppose we have oracle access to the true test label y n +1 .

Original: For any two distinct indices i, j ∈ { 1 , . . . , n + 1 } with i (cid:54) =
Revised: For any two distinct indices i, j ∈ { 1 , . . . , n + 1 } with i (cid:54) =

Original: j , let ˜ f − ( i,j ) be the model ﬁtted on the augmented dataset { ( x i , y i ) } ni
Revised: j , let ˜ f − ( i,j ) be the model ﬁtted on the augmented dataset { ( x i , y i ) } ni

Original: =1 ∪ { ( x n +1 , y n +1 ) } , i.e., training and test data, with points i and j removed.
Revised: =1 ∪ { ( x n +1 , y n +1 ) } , i.e., training and test data, with points i and j removed.

Original: Note that ˜ f − ( i,j ) =
Revised: Note that ˜ f − ( i,j ) =

Original: ˜ f − ( j,i ) for any i (cid:54) =
Revised: ˜ f − ( j,i ) for any i (cid:54) =

Original: j , and that ˜ f − ( i,n +1) = (cid:98)
Revised: j , and that ˜ f − ( i,n +1) = (cid:98)

Original: f − i for any i = 1 , . . . , n .
Revised: f − i for any i = 1 , . . . , n .

Original: Next, we deﬁne a matrix K ∈ R ( n +1) × ( n +1) with entries K =
Revised: Next, we deﬁne a matrix K ∈ R ( n +1) × ( n +1) with entries K =

Original: [ K ij ] i,j deﬁned as follows:
Revised: [ K ij ] i,j deﬁned as follows:

Original: [Equation]
Revised: [Equation]

Original: where the off-diagonal entries represent the residual for the i th point when both i and j are left out of model training, in addition to the difference between the model predictions when trained on the entire training dataset and when point i is left out.
Revised: where the off-diagonal entries represent the residual for the i th point when both i and j are left out of model training, in addition to the difference between the model predictions when trained on the entire training dataset and when point i is left out.

Original: Let the set Q be deﬁned as:
Revised: Let the set Q be deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where K αi = (cid:98) Q α,n ( { K ij } n +1 j =1 ) .
Revised: where K αi = (cid:98) Q α,n ( { K ij } n +1 j =1 ) .

Original: Before proceeding, we ﬁrst prove the following Lemma.
Revised: Before proceeding, we ﬁrst prove the following Lemma.

Original: Lemma 1.
Revised: Lemma 1.

Original: Let (cid:98) C n,α ( x n +1 ) be the DJ conﬁdence interval.
Revised: Let (cid:98) C n,α ( x n +1 ) be the DJ conﬁdence interval.

Original: Then coverage condition is:
Revised: Then coverage condition is:

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Suppose that (cid:98) C n,α ( x n +1 ) does not cover the true test label y n +1 , i.e., y n +1 / ∈ (cid:98) C n,α ( x n +1 ) , then this implies that either
Revised: Suppose that (cid:98) C n,α ( x n +1 ) does not cover the true test label y n +1 , i.e., y n +1 / ∈ (cid:98) C n,α ( x n +1 ) , then this implies that either

Original: [Equation]
Revised: [Equation]

Original: , which means that y n +1 > ˆ f ( x n +1 ) + (cid:98) Q + α,n ( { r i } ni =1 ) + v j for at least (1 − α )
Revised: , which means that y n +1 > ˆ f ( x n +1 ) + (cid:98) Q + α,n ( { r i } ni =1 ) + v j for at least (1 − α )

Original: ( n + 1) many indices j ∈ { 1 , . .
Revised: ( n + 1) many indices j ∈ { 1 , . .

Original: . , n } , or y n +

Revised: . , n } , or y n +

Original: <
Revised: <

Original: ˆ f
Revised: ˆ f

Original: [Equation]
Revised: [Equation]

Original: x n +1 ) − (cid:98) Q + α,n ( { r i } ni =1 ) − (cid:98) Q + α,n ( { v i ( x n +1 ) } ni =1 ) , which means that y n +1 < ˆ f ( x n +1 ) − (cid:98) Q + α,n ( { r i } ni =1 ) − v j for at least (1 − α )
Revised: x n +1 ) − (cid:98) Q + α,n ( { r i } ni =1 ) − (cid:98) Q + α,n ( { v i ( x n +1 ) } ni =1 ) , which means that y n +1 < ˆ f ( x n +1 ) − (cid:98) Q + α,n ( { r i } ni =1 ) − v j for at least (1 − α )

Original: ( n + 1) many indices j ∈ { 1 , . .
Revised: ( n + 1) many indices j ∈ { 1 , . .

Original: . , n } .
Revised: . , n } .

Original: In either case, then, we have
Revised: In either case, then, we have

Original: [Equation]
Revised: [Equation]

Original: By nothing that
Revised: By nothing that

Original: [Equation]
Revised: [Equation]

Original: and (cid:98) Q + α,n ( { r i } ni =1 ) + v j = K αn
Revised: and (cid:98) Q + α,n ( { r i } ni =1 ) + v j = K αn

Original: +1 , we have
Revised: +1 , we have

Original: [Equation]
Revised: [Equation]

Original: which implies that n + 1 ∈ Q ( K ) , and hence
Revised: which implies that n + 1 ∈ Q ( K ) , and hence

Original: P
Revised: P

Original: [Equation]
Revised: [Equation]

Original: which concludes the proof of the Lemma.
Revised: which concludes the proof of the Lemma.

Original: [Equation]
Revised: [Equation]

Original: Given the result of Lemma 1, we use the exchangeability of the training and test points to further bound the probability P (cid:110)
Revised: Given the result of Lemma 1, we use the exchangeability of the training and test points to further bound the probability P (cid:110)

Original: y n +1 / ∈ (cid:98) C n,α ( x n +1 ) (cid:111) as follows.
Revised: y n +1 / ∈ (cid:98) C n,α ( x n +1 ) (cid:111) as follows.

Original: Since the training and testing points are exchangeable, we have K d = Π K Π T for any ( n + 1) × ( n + 1) permutation matrix Π , where d = denotes equality in distribution.
Revised: Since the training and testing points are exchangeable, we have K d = Π K Π T for any ( n + 1) × ( n + 1) permutation matrix Π , where d = denotes equality in distribution.

Original: Thus, we have
Revised: Thus, we have

Original: [Equation]
Revised: [Equation]

Original: for all j ∈ { 1 , . . . , n } .
Revised: for all j ∈ { 1 , . . . , n } .

Original: Based on the above, we can then evaluate the coverage probability using the equivalent condition in Lemma 1 as follows:
Revised: Based on the above, we can then evaluate the coverage probability using the equivalent condition in Lemma 1 as follows:

Original: [Equation]
Revised: [Equation]

Original: Finally, we compute the expected size of the set Q ( K ) as follows.
Revised: Finally, we compute the expected size of the set Q ( K ) as follows.

Original: By combining the original coverage condition in (C.2) together with the equivalence result in Lemma 1, we have
Revised: By combining the original coverage condition in (C.2) together with the equivalence result in Lemma 1, we have

Original: [Equation]
Revised: [Equation]

Original: |
Revised: |

Original: [Equation]
Revised: [Equation]

Original: which implies that |Q ( K ) | ≤ α ( n + 1) .
Revised: which implies that |Q ( K ) | ≤ α ( n + 1) .

Original: By combining this with (C.3), we arrive at the following inequality
Revised: By combining this with (C.3), we arrive at the following inequality

Original: [Equation]
Revised: [Equation]

Original: The result of the Theorem follows from the fact that the approximate DJ conﬁdence interval (cid:98) C ( m ) n,α ( x n +1 ) converges in probability to the exact interval as m → ∞ .
Revised: The result of the Theorem follows from the fact that the approximate DJ conﬁdence interval (cid:98) C ( m ) n,α ( x n +1 ) converges in probability to the exact interval as m → ∞ .

Original: C. D ISCRIMINATION
Revised: C. D ISCRIMINATION

Original: Recall that the exact DJ interval width is given by:
Revised: Recall that the exact DJ interval width is given by:

Original: [Equation]
Revised: [Equation]

Original: Since the term (cid:98) Q α,n ( R n ) is constant for any x , discrimination boils down to the following condition:
Revised: Since the term (cid:98) Q α,n ( R n ) is constant for any x , discrimination boils down to the following condition:

Original: [Equation]
Revised: [Equation]

Original: Note that to prove the above, it sufﬁces to prove the following:
Revised: Note that to prove the above, it sufﬁces to prove the following:

Original: [Equation]
Revised: [Equation]

Original: If the model is stable (based on the deﬁnition in (Bousquet & Elisseeff (2002))), then a classical result by (Devroye & Wagner (1979)) states that:
Revised: If the model is stable (based on the deﬁnition in (Bousquet & Elisseeff (2002))), then a classical result by (Devroye & Wagner (1979)) states that:

Original: [Equation]
Revised: [Equation]

Original: as n → ∞ , where (cid:96) n ( . ) is the empirical risk on the training sample, and the expectation above is taken over y | x .
Revised: as n → ∞ , where (cid:96) n ( . ) is the empirical risk on the training sample, and the expectation above is taken over y | x .

Original: From (C.7), we can see that an increase in the LOO risk (cid:96) ( y, f ( x ; ˆ θ − i )) implies an increase in the empirical risk (cid:96) n ( y, f ( x ; ˆ θ )) , and vice versa.
Revised: From (C.7), we can see that an increase in the LOO risk (cid:96) ( y, f ( x ; ˆ θ − i )) implies an increase in the empirical risk (cid:96) n ( y, f ( x ; ˆ θ )) , and vice versa.

Original: Thus, for any two feature points x and x (cid:48) , if v ( x ) is greater than v ( x (cid:48) ) , then on average, the empirical risk at x is greater than that at x (cid:48) .
Revised: Thus, for any two feature points x and x (cid:48) , if v ( x ) is greater than v ( x (cid:48) ) , then on average, the empirical risk at x is greater than that at x (cid:48) .

Original: D E XPERIMENTAL D ETAILS
Revised: D E XPERIMENTAL D ETAILS

Original: In this Section, we provide more details for the experimental setups in Section 5, in addition to further experimental results in the context of active learning and out-of-distribution detection.
Revised: In this Section, we provide more details for the experimental setups in Section 5, in addition to further experimental results in the context of active learning and out-of-distribution detection.

Original: D. C ODE SNIPPETS FOR THE DJ PROCEDURE
Revised: D. C ODE SNIPPETS FOR THE DJ PROCEDURE

Original: The DJ procedure is very simple to implement.
Revised: The DJ procedure is very simple to implement.

Original: The key component of its implementation is the computation of HOIFs.
Revised: The key component of its implementation is the computation of HOIFs.

Original: Let model be an instance of the nn module of Pytorch , with a set of parameters model.parameters() that correspond to ˆ θ in our formulation.
Revised: Let model be an instance of the nn module of Pytorch , with a set of parameters model.parameters() that correspond to ˆ θ in our formulation.

Original: Below is a Python code snippet for computing the ﬁrst-order inﬂuence function of model .
Revised: Below is a Python code snippet for computing the ﬁrst-order inﬂuence function of model .

Original: def influence_function(model, training_data_indexes): .......
Revised: def influence_function(model, training_data_indexes): .......

Original: IF ....
Revised: IF ....

Original: =
Revised: =

Original: [] ....... H ...... =
Revised: [] ....... H ...... =

Original: hessian(model.loss, model.parameters(), create_graph=True) .......
Revised: hessian(model.loss, model.parameters(), create_graph=True) .......

Original: Lamb = 1e-
Revised: Lamb = 1e-

Original: Hinv = torch.inverse(H + Lamb * torch.eye(H.shape[0])) .......
Revised: Hinv = torch.inverse(H + Lamb * torch.eye(H.shape[0])) .......

Original: ....... for u in
Revised: ....... for u in

Original: (training_
Revised: (training_

Original: _indexes)): .............. u]], .............
Revised: _indexes)): .............. u]], .............

Original: IF
Revised: IF

Original: ....... return IF .......
Revised: ....... return IF .......

Original: Computing the HOIFs can be easily implemented by recursively evaluating the gradients and lower order inﬂuence function as stated in Theorem 1.
Revised: Computing the HOIFs can be easily implemented by recursively evaluating the gradients and lower order inﬂuence function as stated in Theorem 1.

Original: Once all the ingredients of the HOIFs are evaluated, then the perturbed parameters can be evaluated as explained in the paper, and a modiﬁed LOO model is created for every sample in the training data.
Revised: Once all the ingredients of the HOIFs are evaluated, then the perturbed parameters can be evaluated as explained in the paper, and a modiﬁed LOO model is created for every sample in the training data.

Original: D. I MPLEMENTATION OF B ASELINES
Revised: D. I MPLEMENTATION OF B ASELINES

Original: In what follows, we provide details for the implementation and hyper-parameter settings for all baseline methods involved in Section 5.
Revised: In what follows, we provide details for the implementation and hyper-parameter settings for all baseline methods involved in Section 5.

Original: Probabilistic backpropagation (PBP).
Revised: Probabilistic backpropagation (PBP).

Original: We implemented the PBP method proposed in (HernándezLobato & Adams (2015)) with inference via expectation propagation using the theano code provided by the authors in ( github.com/HIPS/Probabilistic-Backpropagation ).
Revised: We implemented the PBP method proposed in (HernándezLobato & Adams (2015)) with inference via expectation propagation using the theano code provided by the authors in ( github.com/HIPS/Probabilistic-Backpropagation ).

Original: Training was conducted via 40 epochs.
Revised: Training was conducted via 40 epochs.

Original: Monte Carlo Dropout (MCDP).
Revised: Monte Carlo Dropout (MCDP).

Original: We implemented a Pytorch version of the MCDP method proposed in (Gal & Ghahramani (2016)).
Revised: We implemented a Pytorch version of the MCDP method proposed in (Gal & Ghahramani (2016)).

Original: In all experiments, we tuned the dropout probability using
Revised: In all experiments, we tuned the dropout probability using

Original: Bayesian optimization to optimize the AUC-ROC performance on the training sample.
Revised: Bayesian optimization to optimize the AUC-ROC performance on the training sample.

Original: We usedsamples at inference time to compute the mean and variance of the predictions.
Revised: We usedsamples at inference time to compute the mean and variance of the predictions.

Original: The credible intervals were constructed as the (1 − α ) quantile function of a posterior Gaussian distribution deﬁned by the predicted mean and variance estimated through the Monte Carlo outputs.
Revised: The credible intervals were constructed as the (1 − α ) quantile function of a posterior Gaussian distribution deﬁned by the predicted mean and variance estimated through the Monte Carlo outputs.

Original: Similar to the other baselines, we conducted training via 40 epochs for the SGD algorithm.
Revised: Similar to the other baselines, we conducted training via 40 epochs for the SGD algorithm.

Original: Bayesian neural networks (BNN).
Revised: Bayesian neural networks (BNN).

Original: We implemented BNNs with inference via stochastic gradient
Revised: We implemented BNNs with inference via stochastic gradient

Original: Langevin dynamics (SGLD)
Revised: Langevin dynamics (SGLD)

Original: (Welling & Teh (2011)).
Revised: (Welling & Teh (2011)).

Original: We initialized the prior weights and biases through a uniform distribution over [ − 0 .
Revised: We initialized the prior weights and biases through a uniform distribution over [ − 0 .

Original: We run 40 epochs of the SGLD inference procedure and collect the posterior distributions to construct the credible intervals.
Revised: We run 40 epochs of the SGLD inference procedure and collect the posterior distributions to construct the credible intervals.

Original: Deep ensembles (DE).
Revised: Deep ensembles (DE).

Original: We implemented a Pytorch version of the DE metho (without adversarial training)d proposed in (Lakshminarayanan et al.
Revised: We implemented a Pytorch version of the DE metho (without adversarial training)d proposed in (Lakshminarayanan et al.

Original: We used the number of ensemble members
Revised: We used the number of ensemble members

Original: M = 5 as recommended in the recent study in (Ovadia et al.
Revised: M = 5 as recommended in the recent study in (Ovadia et al.

Original: Predictions of the different ensembles were averaged and the conﬁdence interval was estimated as 1.645 multiplied by the empirical variance for a target coverage of 90 % .
Revised: Predictions of the different ensembles were averaged and the conﬁdence interval was estimated as 1.645 multiplied by the empirical variance for a target coverage of 90 % .

Original: We trained the model through 40 epochs.
Revised: We trained the model through 40 epochs.

Original: Resampling uncertainty estimation (RUE).
Revised: Resampling uncertainty estimation (RUE).

Original: We implemented RUE (Schulam & Saria (2019)) in
Revised: We implemented RUE (Schulam & Saria (2019)) in

Original: Pytorch using the same Hessian computation routines involved in implementing our method.
Revised: Pytorch using the same Hessian computation routines involved in implementing our method.

Original: We tuned the dampening factor λ in the stabilized Hessian H
Revised: We tuned the dampening factor λ in the stabilized Hessian H

Original: + λ
Revised: + λ

Original: I used for matrix inversion.
Revised: I used for matrix inversion.

Original: We used 20 bootstrapped sub-samples of the training data to compute the prediction variance.
Revised: We used 20 bootstrapped sub-samples of the training data to compute the prediction variance.

Original: D. F URTHER R ESULTS ON S YNTHETIC D ATA
Revised: D. F URTHER R ESULTS ON S YNTHETIC D ATA

Original: The minimum number of HOIFs m ∗ required to achieve the best possible AUC-ROC for the quantitative experiment ( d = 20 ) in Section 5.1 is shown in Figure 5.
Revised: The minimum number of HOIFs m ∗ required to achieve the best possible AUC-ROC for the quantitative experiment ( d = 20 ) in Section 5.1 is shown in Figure 5.

Original: As expected, more complex models will generally require more high order terms of the von Mises expansion in order to recover the parameters of the LOO re-trained model.
Revised: As expected, more complex models will generally require more high order terms of the von Mises expansion in order to recover the parameters of the LOO re-trained model.

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: We performed another series of experiments to evaluate the accuracy of the estimates of the uncertainty estimates produced by different methods.
Revised: We performed another series of experiments to evaluate the accuracy of the estimates of the uncertainty estimates produced by different methods.

Original: We study two particular applications of uncertainty quantiﬁcation: active learning and out-of-distribution detection (OOD).
Revised: We study two particular applications of uncertainty quantiﬁcation: active learning and out-of-distribution detection (OOD).

Original: Active Learning.
Revised: Active Learning.

Original: For this setup, we replicate the active learning experiment in (Hernández-Lobato & Adams (2015)).
Revised: For this setup, we replicate the active learning experiment in (Hernández-Lobato & Adams (2015)).

Original: In active learning, it is necessary to produce accurate estimates of uncertainty in order to decide which data samples we should actively collect.
Revised: In active learning, it is necessary to produce accurate estimates of uncertainty in order to decide which data samples we should actively collect.

Original: In this experiment, we used a neural network with a single hidden layer and 50 hidden units.
Revised: In this experiment, we used a neural network with a single hidden layer and 50 hidden units.

Original: We split each data set into training and test sets with 20 and 100 data instances, respectively, and pool sets with all the remaining data.
Revised: We split each data set into training and test sets with 20 and 100 data instances, respectively, and pool sets with all the remaining data.

Original: After this, one data point is collected from the pool set and then moved into the training set.
Revised: After this, one data point is collected from the pool set and then moved into the training set.

Original: The process repeats until 9 of these active additions to the training set have been completed, that is, until we have performed 10 evaluations on the test set.
Revised: The process repeats until 9 of these active additions to the training set have been completed, that is, until we have performed 10 evaluations on the test set.

Original: The entire process, including the random data set splitting, is repeated 40 times.
Revised: The entire process, including the random data set splitting, is repeated 40 times.

Original: The pool data is initially lacking the target variables and these become available only once the data is moved to the training set.
Revised: The pool data is initially lacking the target variables and these become available only once the data is moved to the training set.

Original: We compare a baseline neural network model trained on random selections from the pool data with version of the model trained by data collected actively (PBP-A, MCDP-A, DE-A, BNN-A and DJ-A).
Revised: We compare a baseline neural network model trained on random selections from the pool data with version of the model trained by data collected actively (PBP-A, MCDP-A, DE-A, BNN-A and DJ-A).

Original: Results are provided in Table 3.
Revised: Results are provided in Table 3.

Original: As we can see, DJ-based active collection of data is signiﬁcantly better than the random baseline model on all datasets.
Revised: As we can see, DJ-based active collection of data is signiﬁcantly better than the random baseline model on all datasets.

Original: Moreover, data actively collected using
Revised: Moreover, data actively collected using

Original: DJ-based uncertainty estimates were superior to all other methods on 6 out of the 8 datasets.
Revised: DJ-based uncertainty estimates were superior to all other methods on 6 out of the 8 datasets.

Original: [Table]
Revised: [Table]

Original: Out-of-distribution detection.
Revised: Out-of-distribution detection.

Original: Next
Revised: Next

Original: [Table]
Revised: [Table]

Original: we evaluate the accuracy of the DJ uncertainty estimates in detecting whether a new test example came from the same distribution as the training data
Revised: we evaluate the accuracy of the DJ uncertainty estimates in detecting whether a new test example came from the same distribution as the training data

Original: [Table]
Revised: [Table]

Original: or from another unknown distribution.
Revised: or from another unknown distribution.

Original: This task is known as Out-of-distribution detection (OOD), and is commonly used to benchmark uncertainty estimation methods (Ovadia et al.
Revised: This task is known as Out-of-distribution detection (OOD), and is commonly used to benchmark uncertainty estimation methods (Ovadia et al.

Original: In this experiment, we use data from the Infant Health and Development Program (IHDP) (Alaa & van der
Revised: In this experiment, we use data from the Infant Health and Development Program (IHDP) (Alaa & van der

Original: Schaar (2017)).
Revised: Schaar (2017)).

Original: The IHDP is an interventional program intended to enhance the health of premature infants.
Revised: The IHDP is an interventional program intended to enhance the health of premature infants.

Original: This dataset comprises observational data with treatment assignments from a real-world clinical trial, and introduced selection bias to the data artiﬁcially by removing a subset of the patients.
Revised: This dataset comprises observational data with treatment assignments from a real-world clinical trial, and introduced selection bias to the data artiﬁcially by removing a subset of the patients.

Original: Every sample in the dataset is of the form ( x , y, w ) , where w ∈ { 0 , 1 } is a binary treatment assignment variable which is equal to 1 if the patient was treated, and is 0 otherwise.
Revised: Every sample in the dataset is of the form ( x , y, w ) , where w ∈ { 0 , 1 } is a binary treatment assignment variable which is equal to 1 if the patient was treated, and is 0 otherwise.

Original: Since patients are not treated at random, i.e., the likelihood of a patient getting treated depends on their feature x , the treated and untreated patient populations have different distributions, i.e., we have P ( x | w = 0) (cid:54) =
Revised: Since patients are not treated at random, i.e., the likelihood of a patient getting treated depends on their feature x , the treated and untreated patient populations have different distributions, i.e., we have P ( x | w = 0) (cid:54) =

Original: P ( x | w = 1) .
Revised: P ( x | w = 1) .

Original: The entire dataset comprised 747 subjects, with 25 features for each subject, of which 165 were treated and the rest were untreated.
Revised: The entire dataset comprised 747 subjects, with 25 features for each subject, of which 165 were treated and the rest were untreated.

Original: [Table]
Revised: [Table]

Original: We conducted our experiment as follows.
Revised: We conducted our experiment as follows.

Original: We train the model on 100 samples for treated patients, and leave out 65 samples for testing.
Revised: We train the model on 100 samples for treated patients, and leave out 65 samples for testing.

Original: In testing, we use the interval width to predict whether the test samples comes from the 65 treated patient (whose features resemble the training data), or from the untreated patients who were not involved in training.
Revised: In testing, we use the interval width to predict whether the test samples comes from the 65 treated patient (whose features resemble the training data), or from the untreated patients who were not involved in training.

Original: We repeated the experiment 20 times with random train/test splitting.
Revised: We repeated the experiment 20 times with random train/test splitting.

Original: The model we used is a feed-forward neural network with 2 hidden layers and 100 hidden units.
Revised: The model we used is a feed-forward neural network with 2 hidden layers and 100 hidden units.

Original: AUC-ROC performance is reported in Table 4.
Revised: AUC-ROC performance is reported in Table 4.

Original: We found that DJ outperforms other methods in detecting the OOD samples, with DE and RUE being the most competitive baselines.
Revised: We found that DJ outperforms other methods in detecting the OOD samples, with DE and RUE being the most competitive baselines.

Original: This result is of practical signiﬁcance since it implies that we can accurately determine which patients should be treated based on the recommendations of a machine learning-based decision-support system, and which patients should be examined in a case by case basis through experts.
Revised: This result is of practical signiﬁcance since it implies that we can accurately determine which patients should be treated based on the recommendations of a machine learning-based decision-support system, and which patients should be examined in a case by case basis through experts.

Original: Moreover, by determining which samples in an observational dataset have the most unreliable predictions, and consequently determine what features are required in a prospective design of a clinical trial.
Revised: Moreover, by determining which samples in an observational dataset have the most unreliable predictions, and consequently determine what features are required in a prospective design of a clinical trial.