Epistemic Uncertainty Estimation in Regression Ensemble Models with Pairwise Epistemic Estimators
Authors: Lucas Berry
David Meger
Venue: NeurIPS 2025 poster ()
Keywords: Uncertainty Quantification
Dynamics Modeling
Ensembles
Primary area: general_machine_learning
TL;DR: This work introduces a fast and efficient method for epistemic uncertainty in ensemble models for regression. Our method significantly improves baseline active learning methods on high-dimensional tasks.

Abstract: This work introduces a novel approach, Pairwise Epistemic Estimators (PairEpEsts), for epistemic uncertainty estimation in ensemble models for regression tasks using pairwise-distance estimators (PaiDEs). By utilizing the pairwise distances between model components, PaiDEs establish bounds on entropy. We leverage this capability to enhance the performance of Bayesian Active Learning by Disagreement (BALD). Notably, unlike sample-based Monte Carlo estimators, PairEpEsts can estimate epistemic uncertainty up to 100 times faster and demonstrate superior performance in higher dimensions. To validate our approach, we conducted a varied series of regression experiments on commonly used benchmarks: 1D sinusoidal data, *Pendulum*, *Hopper*, *Ant*, and *Humanoid*, demonstrating PairEpEsts’ advantage over baselines in high-dimensional regression active learning.

Epistemic Uncertainty Estimation in Regression Ensemble Models with Pairwise Epistemic Estimators
Lucas Berry, David Meger Department of Computer Science
McGill University
lucas. lucas. ber.ymmaillmllll. http://mmmmmmmmll....m mwgill.gll...
Abstract
This work introduces novel approach, Pairwise Epistemic Estimators (PairEpEsts), for epistemic uncertainty estimation in ensemble models for regression tasks using marwise-distanee estimators (PaiDEs). By utilizing the pairwise distances between model components, PaiDEs establish bounds on entropy. We leverage this capa- bility to enhance the performance of Bayesian Active Learning by Disagreement (BALD). Notably, unlike sample-based Monte Carlo estimators, PairFipEsts can estimate epistemic uncertainty up 100 times faster and demonstrate superior performance in higher dimensions. To validate our approach, we conducted a varied series of regression experiments on commonly used benchmarks: ID sinu- soidal data, Pendulum, Hopper, Ant, and Humanoid, demonstrating PairEpEsts' advantage over baselines in high-dimensional regression active learning.
Introduction
In this paper, we propose Pairwise Epistemic Estimators (PairEEpEsts) non-sample based method for estimating epistemic uncertainty in deep ensembles with probabilistic outputs for regression tasks. Epistemic uncertainty, often distinguished from alcatoric uncertainty, reflects model's ignorance and can be reduced by increasing the amount of data available [26, 16, 29]. The significance of epistemic uncertainty particularly pronounced in safety-critical systems, where single erroneous prediction could lead to catastrophic consequences [43]. Moreover, leveraging epistemic uncertainty proves beneficial as an acquisition criterion for active learning strategies [27].
Previously Monte Carlo (MC) methods have been employed for estimating epistemic uncertainty, in regression tasks, due the absence of closed-form expressions in modeling scenarios [15, 4]. However, as the output dimension increases, these MC methods require large number of samples to get accurate estimates. PairEpEEsts leverage Pairwise-Distance Estimators (PaiDEs), which provide non-sample-base alternative for estimating information-theoretic criteria ensemble regression models with probabilistic outputs [35].
Ensembles can seen as committees, with each component acting as member [50]. PairEpEsts syn- thesize the consensus amongst committee of probabilistic learners by calculating the distributional distance between each pair committee members. Aggregating these distances allows accurate estimation of the mutual information between the model output and weight distribution. When pairwise distances are efficiently computable, PairlipEists offer a fast, sample-free method to estimate epistemic uncertainty. Figure shows committee members in agreement with low uncertainty (small distance) and disagreement with high uncertainty (large distance).
This study demonstrates the use of PairEpEsts to estimate epistemic uncertainty in regression ensembles with probabilistic outputs. Unlike classification, regression poses unique challenges because entropy is harder to estimate for mixtures of continuous distributions than for categorical
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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ones. To address this, we focus on Normalizing Flows (NFs), which can capture heteroscedastic and multimodal aleatoric uncertainty [49, 33]. This capability is particularly relevant for robotic locomotion, where nonlinear stochastic dynamics make data acquisition costly and active learning essential. The proposed framework extends epistemic uncertainty estimation to high-dimensional regression tasks that have been largely underserved in the literature. Our contributions are threefold:
We introduce the framework PairEpEssts, which applies PaiDEs estimate epistemic uncer- tainty in deep ensembles with probabilistic outputs (Section 4).
We extend previous epistemic uncertainty estimation methods to settings with higher- dimensional output [4], and demonstrate how PairEpEsts outperform MC and other active learning baselines in these settings with rigorous statistical testing (Section 5).
We provide an analysis of the time-saving advantages offered by PairEpEsts compared to MC estimators for epistemic uncertainty estimation (Section 5.4).
2 Problem Statement
Following a supervised learning framework for regression, let D { ys } denote dataset, where 2; E _ and y; E R and our objective is to approximate a complex multi-modal con- ditional probability plylz)) Let So (r) denote our approximation to the conditional probabil- ity density, where / is set of parameters to be learned and is distributed as p(0).
Leveraging learned model, fo (y;; x), we wish to estimate uncertainty which is typically viewed from a probabilistic perspective [14, 29]. When capturing uncertainty in supervised learning, one common measure is that of conditional dif- ferential entropy,
Figure 1: Epistemic uncertainty in an ensemble of probabilistic learners is high when the mixture components disagree (left) and low when there is agreement (right).
(y) - -- / p(y]x) In plylx)dy
Utilizing conditional differential entropy, we can establish an estimate for epistemic uncertainty as introduced by Houlsby et al. [27], expressed as:
[v, 0 x) (ylæ) - (2læ,
(I)
where I(-) refers mutual information (MI). Equation demonstrates that epistemic uncertainty,
[yy, (r). can be represented by the difference between total uncertainty. H (ylæ),, and aleatoric uncertainty, (y]xx,) 0). MI measures the information gained about one variable by observing another.
To enable our methods to capture epistemic uncertainty, we employ ensembles to create p(0). Ensembles leverage multiple models to obtain the estimated conditional probability by weighting the output distribution from each ensemble component,
M
M
fo(y: x) = Σ (1:
Σ j=1
T
1,
(2)
where 5 π. S. and 0 are the number of model components, the component weights and the parameters for the jthh component, respectively. In order to create an ensemble, one of two ways is typically chosen: randomization [5] or boosting [20]. While boosting has led to widely used machine learning methods [10], randomization has been the preferred method in deep learning [39].
Utilizing our ensemble we can estimate epistemic uncertainty in decision-making scenarios such as active learning. When all components produce the same fo, (y, Tyy 0|τ) is zero, indicating no epistemic uncertainty. Conversely, when the components exhibit varied output distributions, epistemic uncertainty high. In an active learning framework, one chooses, at each iteration, what data points to add to training dataset such that the model's performance improves much possible [41, 52].

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In our context, one chooses the z's that maximize Equation I and adds those data points to the training set as in Bayesian Active Learning by Disagreement (BALD) [27]. It's worth noting that the realm of continuous outputs as in regression and ensemble models, Equation often lacks a closed-fore solution, as the entropy of most mixtures of continuous distributions cannot be expressed in closed form [35]. Hence, prior methods have resorted to MC estimators for the estimation of epistemic uncertainty [15, 48J. One such MC method samples K points from the model, yj ~ foly; x), and then estimates the total uncertainty,
HMC(Y(x = K
Σ ==1
folgix)..
The MC approach approximates the intractable integral over the continuous domain requiring only point-wise evaluation of the ensemble and its components. However, as the number dimensions increases, MC methods typically require greater number of samples which creates greater computational burden [51]. Note that the alcatoric uncertainty can be analytically computed as the average of the entropies of each ensemble component distribution.
Pairwise-Distance Estimators
Unlike MC methods, PaiDEs eliminate sampling by using generalized distance functions between ensemble components. They estimate the entropy of mixture distributions when pairwise distances have closed-fome expressions, enabling estimation of (ylær) from Equation 1. We extend PaiDEs, from Kolchinsky and Tracey [35], to supervised learning and epistemic uncertainty estimation.
3.1 Properties of Entropy
One can treat a mixture model as two-step process: first, a component is drawn and, second, sample taken from the corresponding component. Let ply, 0 0|x) denote the joint our output and model components given input x,
ply. f|z) p(0,1= (0)])))p((())))))))))))))))))) k,p(y108,
Using this representation, following principles of information theory [14], we can write its entropy as, Hly 0) ( + (y)2z)
Additionally, one can show the following bounds for Hyl
Hly00,z) - (y]x) < H 0()).
(4)
Intuitively, the lower bound can be justified by the fact that conditioning on more variables can only decrease or keep entropy the same. The upper bound follows from Equation 3, and H (0]or, v) 2 since 0 is modeled as a discrete random variable (ensemble index), unlike æ and y which are continuous.
3.2 PaiDEs Definition
Let (p: Pş) denote generalized distance function between the probability distributions p, and Pj. which for our case represent ps p(y |r, 0,) and Pj = p(y|æ, i 02). respectively. More specifically, D is referred to premetric, Dp pş) 2 and D, || p3) 0 if p. = Pj- The distance function need not be symmetric nor obey the triangle inequality. As such, PaiDEs can be defined as,
M
M
Ho(v)) := H(100,)) - Σ
n. In Σ
T, exp (-Dp(p: p.)).
(5)
PaiDEs have many options for Dp (p: P3) (Kullback-Leible divergence, Wasserstein distance, Bhattacharyya distance, Chernoff a-divergence, Hellinger distance, etc.).

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Theorem 3.1. from Kolchinsky and Tracey [35], using the extreme distance functions,
Dmnn (pi P2) = Vi,
[0 =
Dma (pi Pς)
00,
otherwise,
one can show that PaiDEs lie within bounds for entropy established in Equation
Refer to Appendix A for the proof of Theorem 3.1. This provides general class of estimators but distance function still needs to be chosen.
3.3
Tighter Bounds for PaiDEs
Let the Chernoff a-divergence, where E [0, 1]. be defined [44],
Dc. (pr Pý) = - In (yla,, 0,)p'="(ylx, 0,)
Corollary 3.2. As from Kolchinsky and Tracey [35], when applying Chernoff o-divergence as our 5, we achieve lighter lower bound than H (yy 0, x) from Equation 4,
distance function &unction in Equation
Hw S Hc. (u(x) S (y)x)).
(6)
Refer to Appendix A for the proof of Corollary 3.2. In addition, the tightest lower bound can be shown to be a = 0.5 for certain situations [35]. This is known as the Bhattacharyya distance,
Dn(Pilles) - In / Vplvla, 0.)p( 0,)
In addition to the improved lower bound, there is an improved upper bound as well. Let Kullback- Liebler (KL) divergence be defined as follows,
(pL ps) - p(ulz 0.) In p(y 04) p(y xr 0,) dy.
Note that the KL divergence is not metric but does suffice generalized distance function.
Corollary 3.3. As from Kolchinsky and Tracey [35], when applying fullback-Lieble divergence as our distance function in Equation 5, we achieve a tighter upper bound than H 0|z) from Equation 4,
Hyy s HK((y( H(y 0lr).
Refer to Appendix A for the proof of Corollary 3.3.
4 Estimating Epistemic Uncertainty with Pairwise Epistemic Estimators
Our extension of prior methodologies enables the estimation of epistemic uncertainty, without the need for sampling. We illustrate this capability for NFs in the main paper and for Probabilistic Network Ensembles (PNEs) in Appendix E [39]. Note that our proposed estimators are applicable to any ensemble model whose component output distributions have closed-fore form pairwise distances. Since many models have these properties, we focus on Nflows Base for its expressiveness with multimodal alcatoric uncertainty and on widely used PNEs.
4.1 Pairwise Epistemic Estimators
By applying our definition of PaiDEs to Equation 1, we obtain the following expression:
io(y.0) 0) H,(v)=) Hvl 0) Σ
In
ny exp (-D,(p: p;)).
(9)
1=1

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PaiDEs estimate epistemic uncertainty using only pairwise component distances, removing reliance on sampling. We propose estimators:
lb(v.0)
Σ
Ty exp exp (-De(pi p,)),
0).(v.0)= Σ m: In exp (- Dir. (ps ps)),
where Dp(p, pý) and (PL(pi |I Pÿ) defined for Gaussians in Appendix B. We refer to [y 0) as PairEpEst-Bhan Bhatt and I (y, 8) as PairEpEst-KL.
4.2 Nflows Base
In this study, we utilize an ensemble technique known as Nflows Base, which has previously demonstrated robust performance in estimating both aleatoric and epistemic uncertainty simulated robotic datasets by leveraging NFs to create ensembles [4].
NFs have traditionally been applied unsupervised tasks [54, 53]. However, NFs have also been adapted for supervised learning tasks, particularly for regression [59, IJ. Using the structure described in Winkler et al. [59], supervised NF is defined as:
Pylx((/x) Pb) x)) (e x))))
(og(p)1=()1x)) log(pb)z.0 (g (y. x))) log(( det(J (g,"(, (),)))))),
where Pale is the output distribution, Pb|2,9 is the base distribution with parameters 0, J refers the Jacobian, and 96 x æ + is the bijective mapping with parameters ø. For complete review of NFs, refer Papamakarios al. [47]. Nflows Base creates an ensemble the base distribution,
Pulz, (2)20, 02) Po|2,8 (g " x:) (
(v,
where Polz.0, (b/r, 03) N N(uo (z)) Ea, (I)). pa, (x) and Ea, (1) denote the mean and covariance for input x and conditioned on 0j. To encourage diversity in the ensemble, each member is initialized with different random weights, trained dropout mask sampled at the start of training (with 0.5), similar to [1]. This mask remains constant throughout training and inference. By constructing the ensemble within the base distribution,
bootstrapped subset of the data, and assigned fixed
one can make use of closed-form pairwise-disse dirrtnced formulae as Berry and Meger [4] showed that
estimating epistemic uncertainty in the base distribution is equivalent to estimating it in the output distribution.
Exploiting these closed-form pairwise distance formulae, Berry and Meger [4] showed that Nifows Base outperforms naive NF ensemble methods when estimating epistemic uncertainty. The aleatoric uncertainty from Equation 1 can be estimated in the base distribution space, and therefore can be computed analytically. However, this approach does not extend to the total uncertainty in Equation 1, which is why MC techniques have traditionally been used to estimate epistemic uncertainty.
4.3 Integrating Pairwise Epistemic Estimators with Nilows Base Ensembles
By combining Nflows Base and PairEpEsts, we construct an expressive non non-ponnnmee con--armettric model capable of capturing intricate aleatoric uncertainty in the output distribution while efficiently estimating epistemic uncertainty in the base distribution. Equation then becomes.
M
M
((, - M n.
Σ 1=1
n exp (-D,(Pola.e, Pbta..)))
(10)
Unlike previously proposed methods, we are able to estimate epistemic uncertainty without taking single sample. Figure in the Appendix shows an example of the distributional pairs that need to be considered in order to estimate epistemic uncertainty for an Nflows Base model. By applying PairEpEstts in the base distribution space we are able to capture epistemic uncertainty in the output space [4]. This approach enables the use of established formulae for computing distributional distances.
5

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PairEpEsts can in principle be extended to skewed or heavy-tailed base distributions, provided that the pairwise premetric admits a flosed-form expression. Recent work on tail-adaptive normalizing flows [40, 25] offers promising direction for incorporating heavy-tailed behavior into normalizing flows. Integrating these approaches with PairEpEsts represents an important avenue for future research, particularly in domains where tail risk is critical.
4.4 Integrating Pairwise Epistemic Estimators with Probabilistic Network Ensembles
In addition to Nflows Base ensembles, we extend PairEpEsts to PNEs, which are commonly used for uncertainty quantification in regression [12, 38, 39]. PNEs comprise multiple components that produce Gaussian predictive distributions with learned means and variances.
Applying PairEpEsts to PNEs leverages the closed-form pairwise distances between these Gaussian outputs to estimate epistemic uncertainty without sampling. Formally, the estimator is:
M
M
(y,,0)= = Σ
n.
In
Ty exp(ppppp (Sp(-DD(pyla.) Pola.o,)),
where Pyla.0; denotes the i-th output distribution. Our experiments in Appendix E show that PairEpEsts with PNEs achieve comparable or better epistemic uncertainty estimates than base- lines, especially in nigh-dimensional settings. Although PNEs are less expressive than normalizing flow ensembles, their combination with PairEpEists offers practical and efficient framework for scalable uncertainty quantification across diverse ensemble models.
Experimental Results
To evaluate our method, we tested each of our PairEpEsts (KL and Bhatt) on two ID envi- ronments, as has been previously proposed in the literature [15]. Additionally, we present 4 multi-dimensional environments. In contrast to previous papers [48, 2], we evaluate each method on high dimensional output space (up to 257) to demonstrate the utility of PairEpEsts in this setting. Note that, for all experiments, the model components assumed to be uniform,
Uncertainty
(a) hetero
Uncoratin
Ty A independent of æ. All model hyper- parameters contained in Appendix B and the code can be found in the supplementary mate-
40
rial. The experimental design follow previous literature [4].
(b) bimodal
5.1 Data
Bhatt
KL.
groundtrath
model
We evaluated our estimators on two 1D bench- marks, hetero and bimodal. The ground-truth data for hetero and bimodal can be seen in Fig- ure 2 on the right graphs with the cyan dots. For hetero, there are two regions with low density (2 and -2) as can be seen by the green bar chart in the left graph which corresponds to the density of the ground-truth data. In these regions, one would expect model to have high epistemic uncertainty. For bimodal, the number of data points drops off as x increases, thus would expect model to have epistemic uncertainty grow does. All details for data generation are contained in Appendix C.
Figure 2: In the right graphs, the brown dots are sampled from Nflows Base and the cyan dots are the ground-truth data. The left graphs depict the epistemic uncertainty estimates corresponding to our two proposed estimators and the density of the ground-trutt data in the green histogram.
In addition to the ID environments, we tested our methods on four multi-dimenssonal environments: Pendulum, Hopper, Ant, and Humanoid [55]. Replay buffers were collected and dynamics for each environment modeled, fe(s,, az) S1-1. The choice of multi- diltii-dieensionll environments is

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Table 1: Mean RMSE on the test set for the last (100th) Acquisition Batch for Nilows Base. Experi- ments were across ten different seeds and the results expressed mean + standard deviation. The best means are in bold and results that are statistically significant are highlighted.
hetero
bimodal
Pendulum
Hopper
Ant
Humanoid
Outpul Dim.
3
11
32
257
Random
1,56 0.14
0.62
0.15
0.97 + 0.2
1.05 + 0.1
6,59 = 1.54
BatchBALD
0.16
6.42 1: 0.65
1.05
0.87 + 0.27
0.94 0.03
5.33 + 1.2
BADGE
1.44 0.12
6.01 = 0.04
0.34
1.11 0.27
0.91 + 0.04
7.31 = 3.11
BAIT
=.11
6,26 0.33
0.17
1.06
0.94
11.01 0.23
MC (BALD)
1.54 0.17
6.01 +: 0.04
0.06 - 0.01
0.32 + 0.02
1.01 0.05
10.71 0.43
KL (ours)
1.47 =
6.01 + 0.04
0.05 0.04
0.3 + 0.03
0.07
3.4 - 0.48
Bhatt (ours)
1.55 0.31
6.0 + 0.04
0.05 + 0.01
0.29 0.03
0.93 + 0.08
3.37 0.44
p < 0.05
p < 0.01
p < 0.001
motivated by their common use as benchmarks and their higher-dimensional output space, providing robust validation of our methods. Note that, for Ant and Humanoid, the dimensions representing their contact forces were eliminated due to a bug in Mujoco-v2.
In the Mujoco environment, the input consists of the current state and action, which are used to predict the next state based on the dynamics of the environment. the hetero- and bimodal environments, we simplify the task to take single input variable x, and predict the output variable y.
5.2 1D Experiments
Our ID environments provide empirical proof that PairEpEsts can accurately measure epistemic uncertainty. Figure 2 depicts that both PairEpEsts are proficient at estimating the epistemic uncertainty as each method shows an increase in epistemic uncertainty around 2 and -2 on the hetero setting. This can be seen from the blue and orange lines with both estimators performing similarly.
A similar pattern can be seen for the bimodal setting in Figure 2, which shows that both PairEpEsts can accurately capture epistemic uncertainty. Each estimator shows the pattern of increasing epistemic uncertainty where the data is more scarce. Both examples show accurate epistemic uncertainty estimation with no loss in aleatoric uncertainty representation, as demonstrated in the right graphs in Figure 2: the brown dots closely match the cyan dots. Note that for visual clarity, the epistemic uncertainty has been scaled to 0-1, while ensuring that the relative properties of the estimators are preserved. Further discussion on over- and underestimation is provided in Section 6. While simple buut- heuristics may suffice in toy ID settings like Figure 2, they fail to scale to higher-dimensional spaces where data sparse and frequency statistics are uninformative. Moreover, they cannot exploit learned representations of the model, which are essential for structured, high- dimensional control tasks such as Hopper, Ant, and Humanoid. For these reasons, principled estimators such as PairEpESst are required in realistic settings.
5.3
Active Learning
While the ID experiments provide evidence of our estimators' effectiveness for estimating epistemic uncertainty, the active learning experiments extend this evaluation to nigher-dimension data and for decision-making task. In order to benchmark our method, we compare against four state-of-the-art active learning frameworks: BatchBALD which maximizes information gain over batches using MC approximations [34], BADGE which uses gradient- based selection to identify the most informative data points for model improvement [3] and BAIT selects data based on uncertainty estimation through Bayesian active learning [2]. Additionally, a random baseline is included. The training set is initialized with 100 or 200 data points for ID and multi-dimensional environments, respectively. In each acquisition batch, 10 points are added. For the MC estimator, 10s candidate inputs were sampled to construct each acquisition batch, with their epistemic uncertainties estimated using K 5000.
7

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hatero
humodal
Pendalim--1
RMSE
1.0
Hopperss2
Antto2
Humonotdl-y?
2.0
2.0
1.5
13
RMSE
155
0.5
1.0
0.5
Training Set Size
Training Set Size
Training Set Size
Bhalt (Ours)
Random
BetchBALD
BAIT
(lus
MC (BALD)
RADGE
Full Dataset
Figure 3: Mean RMSE on the test set as additional data incorporated into training set for Nilows Base. Both proposed methods perform comparably or significantly better than baselines, particularly in nigh-dimensional settings.
In the Humanoid environment, only 100 candidates were used due to computational constraints. In contrast, PairEpEsts sampled 10s candidate inputs and estimated their epistemic uncertainties for each environment including Humanoid. This underscores one advantage of our estimators over MC estimators, they estimate epistemic uncertainty over larger regions cost than their MC counterparts. Note that the other benchmarks each sampled 10 inputs as well and their respective acquisition functions applied. The RMSE the test set was calculated each acquisition batch.
lower computational
Table shows the performance of each frame- work on 100th acquisition batch. Welch's t-test, with Holm-Bonneeroot correction (see Appendix G), compares our estimators to base- lines in each environment. PairEpEsts achieve lower or comparable RMSES, demonstrating their efficacy in estimating epistemic uncertainty. In higher dimensions, our estimators signifi- cantly outperform other methods, as shown in Humanoid. Learning curves are presented in Figure 3. Note that in the bimodal setting, all methods converge to similar performance and are indistinguishable in the plot. BatchBALD and BADGE, originally designed for classifica- tion, required adaptation for regression, facing challenges due to differences in uncertainty mea- sure computation. Similarly, BAIT, designed for 1D regression, did not perform well in high- dimensional settings. The dashed line indicates the performance of our model when trained on the full dataset, serving as an upper bound for comparison.
0,7
0.6
RMSE
0.3
10'
10
10'
10"
Number of Samples
PaiDEs Performance
MC
Figure 4: RMSE the test set at the 100 acqui- sition batch of the MC estimator on the Hopper environment for Nflows Base the number of samples, K, per input, x, increases. Experiment were across 10 seeds and the mean and stdey being reported.

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10"
Seconds
Seconds
10
10"
10"
10
10"
10
10
10 Number of Ensembles Bhatt
10
Number Dimensions
MC
Rhart
KL.
KL
Figure 5: The amount of time taken for Nflows Base for each estimator across the different set- tings (1, 3, 11, 27, 257 dimensions). Results averaged over 10 seeds. MC results K = 5000 samples per acquisition candidate across all environments.
Figure The amount of time taken for PaiDEs for Nflows Base the number of ensemble com- ponents increases for Humanoid-v2. Results are averaged over 10 seeds.
For deeper analysis of our method, we compared our estimators MC estimators with a varying sample size, K, in the Hopper environment. As expected, MC estimators perform par with PairEpEsts with sufficient number of samples. This trend is illustrated Figure 4. The MC estimator reaches the peak performance of our estimators given enough samples but does not perform better than PsttE, suggesting that our estimators are sufficient to estimate epistemic uncertainty.
5.4 Time Analysis
In addition to benchmarking estimators active learning experiments, we provide analysis of the time gains across our experiments. Figure 5 depicts the speed increase that can be gained using PairEpEsts over an MC approach. PairEpEsts provide 1-2 order of magnitude runtime improvement over Monte Carlo approaches. The estimates are obtained from the active learning experiments.
Limitations
PairEpEsts have quadratic computational cost as the number of components grows. For asymmetric distances like KL-divergence, .M2 MM pairwise distances are computed, while symmetric distances
like Bhattacharyya require only M 2 Figure shows timing as ensemble size increases. Bhat- tacharyya's is cost could be further reduced using symmetry. Despite this, deep learning ensembles usually have few components (5-10) [45, 12], making the complexity manageable. The limitation is more relevant in large ensembles, such as weather forecasting [57].
PairEpEsts introduce small bias absent in estimators. on the relative ordering of acquisition scores, this bias does not affect performance. As shown in Appendix D, the rankings well preserved (Table 5). The relationships illustrated in Figure with epistemic uncertainty values normalized between 0 and Figure 2.
However, since active learning depends
7 Related Work
Bayesian neural networks with information-based criteria are widely used for active learning in image classification [21, 32, 34], while others use gradient embeddings [3, 2]. Most focus classification, with few methods extending to ID regression [2]. Our work tackles active learning in high-dimensional regression, filling gap relevant to robotic locomotion. Despite limitations of
mutual information classification [58], we demonstrate its value in high- nigh-dimensional regression.
Ensembles have been harnessed for epistemic uncertainty estimation [39, 11, 12]. Specifically related to our work, ensembles have been leveraged to quantify epistemic uncertainty in regression problems and active learning [15, 48, 4J. Depeweg et al. [15] employed Bayesian neural networks to model mixtures of Gaussians and demonstrated their ability to measure uncertainty in low-dimensional environments (1-2D). Building upon this foundation, Postels et al. [48] and Berry and Meger [4]

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extended the research by developing efficient NF ensemble models that capture epistemic uncertainty. Our work advances this line of research by eliminating the need for sampling to estimate epistemic uncertainty, resulting in faster and more effective method, especially in higher dimensions.
In addition to Bayesian neural networks and ensemble methods, the concept of Credal Bayesian Deep Learning has been introduced to enhance uncertainty quantification [7]. Furthermore, distributionally robust statistical verification has been proposed for high-dimensional autonomous systems using Imprecise Neural Networks, which provide uncertainty quantification guarantees [19].
Entropy estimators, which do not rely sampling, is active area of research [30, 31, 28, 35]. Kulak et al. [37] and Kulak and Calinon [36] demonstrated the utility of PaiDEs within Bayesian contexts, employing PaiDEs to estimate conditional predictive posterior entropy. In contrast, our approach provides a more general estimate of epistemic uncertainty, as defined in Equation 1, which can be applied to both ensemble, Bayesian methods and deep learning models.
Several methods have emerged in the literature for estimating epistemic uncertainty without relying on sampling techniques [56, 8J. Both Van Amersfoort et al. [56] and Charpentier et al. [8] focus on classification tasks with ID categorical outputs. Charpentier et al. [9] extends the work Charpentier al. [8] to regression tasks but is limited to modeling outputs members of the exponential family, In contrast, our approach can handle more complex output distributions by directly considering the outputs from NFs and can be applied to larger space of regression models.
8 Conclusions
In this study, we introduced epistemic uncertainty estimators and applied them to Active Learning. We depicted how our method can be used to more efficiently quantify uncertainty by leveraging closed- form formulae instead of sampling. This led to improvements in computational speed and accuracy. As learning becomes pervasive in high- nigh-dimensional tasks in society, our method is well-placed to enable epistemic uncertainty awareness without unnecessary compute.
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A Proofs
The proofs follow from the steps of Kolchinsky and Tracey [35].
Proof of Theorem 3.1
Proof. Applying Dmin(ps (ps || PU) distance in PaiDES,
M
H( -(Y)),)))
Σ
n
ln
Jj exp (-D (p: p,))
i=1
j=1
=H(Y|X,0)
Σ
ln
Jj
=X
Note the last line follows from the fact that the component weights must to one, Σラ T1 1.
Next using Dmaz (p& || Pj)-
M
M Ty exp (- D man (ps |l Pý))
HY = H)
Σ
Tt.
i=1
j=1
M
= =
Σ
Ty. In
T. +
Σ Jj exp (- 1-Dmae || ps)))
i=1
M Σ
O)
m In:;;
HY| θ) + H()
= @|X)
This shows the upper bound with Dmaz (p: || pÿ) and the lower bound with Drain (pe || P5)- Note that in our case XX) = H(O) as the distribution of weights does not depend on our input. O
Proof of Corollary 3.2
Proof. To show the upper bound from Equation 6 we use a derivation from Haussler and Opper [23].
M
(ylx( P((y)z))
H(YIx) H(YIX,))
dy
Pr (
=((IX, 0)
(plp)) In
(.(0))) a Σ, (yps(ylz)1-a dy
Σ, T)P;(1l=)>=a dy
M
2 H(YIX,)) 9)
(ppp(/ In
pr(ulz)a dy
H(1|X,0) Σ
n. In
Pi(ult)a Σ (1)z
1=1
M
M
H(
Σ
m
In
Tt exp (pCapp: ps)).
1=1
The two inequalities follow from Jensen's inequality.
21

---

Proof of Corollary 3.3
Proof. The lower bound can be shown using the definition of entropy for a mixture,
M
H(YI)) =
Σ
T2
F
ln
PP
(v)z))
i=1
j=1
$
ln
J exp (Ep. [In [mnp((ylz))
i=1
j=1
M
M
In
Jj exp
(- (p Pÿ))
1=1
j=1
M
M
M
T H (p:)
Σ 1=1
In
Ty exp (ps p;))
==1
Kj exp (p pj)),
j=1
where Ep; refers to the expectation when distributed as p; and H(p p5) indicates the cross entropy. The incquality in line 2 follows from Hershey and Olsen [24] and Paisley [46].
A.1 PairEpEst-KL vs EPKL
In addition to the previous results from Kolchinsky and Tracey [35], we provide unique proof to our paper demonstrating that the PairEpEst-KK KL estimator provides strictly tighter upper bound on MI than the Expected Pairwise KL (EPKL) introduced by Malinin and Gales [42]. EPKL is another uncertainty measure that captures epistemic uncertainty by averaging the pairwise Kullback-Leibler divergences between predictive distributions of ensemble members. It quantifies how diverse ensemble's is predictions are, reflecting the model's 's uncertainty due to lack of knowledge or data. EPKI. serves as an upper bound on MI. Our PairEpEst-KLL estimator leverages a log-sum-cks formulation, yielding sharper (tighter) upper bound on MI, which leads to more precise uncertainty quantification.
Specifically, consider an ensemble of M predictive distributions p1 (y),.... PM (y). The EPKL is defined as the average pairwise Kullback-Lein divergence between distinct ensemble components:
EPKL
DDL
DKL (P(v)) (v)
M(M
Our PairEpEst-KLL estimator, denoted by B, is given by
1 M
1 M
B = M
Σ
exp (-
M
J=1
The following theorem formalizes the relationship between these two quantities and establishes that
B EPKL,
meaning that PairEpEst-KLL yields a strictly sharper upper bound on MI compared to EPKL. Theorem A.1. Let pı (2) ... PMM (v) be probability distributions, and define
Σ (pL()(())l
= M(M 1) in
and
M
M Σ exp (KL(()(())l))())) j=1
B =
In
M
M
22

---

Then,
B S M
(p <A A.
and in particular,
BAA.
Proof. Define
Sij := DKL (ps S.. O.
By Jensen's 's inequality applied to the concave function In(-), for each fixed i,
1
In
co
8ij-
M
M
j=1
Multiplying both sides by -1 reverses the inequality:
M
M
In
c
1
Σ
Sij.
M
M
j=1
j=1
Substituting back for 8ij. we get
-In
exp (-
DkL. (ka))(()))
M
M ==
j=1
Averaging over i, follows that
M
M
B
In
(p.lps)
Dr.(P:llp))
M
M
M
Since DKL = 0, the double sum over all i, is equal to the sum over i + j:
M
DKL. (p:l(::lp,))
Σ
DKt
i$
Note that
M > M(M 1)
M
M(M-1 - 1)'
so
BS
Σ (KL(Pilee)) <
1
(Ku(pellss) = A,
M
M(M - 1)
i
which establishes the strict inequality
B < A,
completing the proof.
B Compute and Hyper-parameter Details
The Nilows Base model employed one nonlinear transformation, with a single hidden layer containing 20 units, utilizing cubic spline flows as per [17]. The base network consisted of two hidden layers, each comprising 40 units with ReLU activation functions. It is important to note that all base distributions were Gaussian. The PNEs adopted an architecture of three hidden layers each with 50 units and ReLU activation functions. Model hyperparameters remained consistent across all experiments. The base distribution and the bijective mapping are trained simultaneously. Fach base distribution network is mapped through the same bijective mapping, which is shared across ensemble components. Every time base distribution network takes a training step, the bijective mapping is updated accordingly. This approach ensures that the bijective mapping
23

---

Algorithm Active Learning Using PairEpEsts
Input: training dataset Derain, oracle dataset Doraces number of iterations T,, sample size S, and batch size B
for i=1.2.....T do
Randomly initialize each ensemble component 0,. Train our ensemble on Proei using bootstrapped samples. Sample points from Doracl uniformly, Ds.
Calculate the top B values according to Equation 9 from Ds. DB.
Add DB the training dataset while removing from the oracle dataset, Dtrain DB U'Drraia
and D Doracle Doracte D end for train final model on Derain.
Nflows Base
Epistemic Uncertainty
Inpot
Base Distributions
Output Distributions
Base Distribucions
Pairwise Distances
Figure 7: Nilows Base an ensemble of components with one bijective transformation on the left and an example the pairwise comparisons needed to estimate epistemic uncertainty for said model on the right. Note the base distributions, instead of the output distributions, from Nflows Base are used to estimate the epistemic uncertainty which are highlighted by the blue bar.
consistent across all components of the ensemble. This methodology is similar to previous approaches, such those proposed by Osband al. [45]. where shared mappings utilized within ensemble-
models. Training was conducted using 16GB RAM on Intel Gold 6148 Skylake @ 2.4 GHz CPUs and NVidia V V100SXM2 (16G memory) GPUs. For each experimental setting, PNEs and Nflows Base were executed with five ensemble components. The MC estimator sampled 1000 and 5000 points for Nflows Base and PNEs, respectively, for each x conditioned
on. The nflows library [18] was employed with minor modifications. Our code can be found at https://gtt thub. www.cooo/////eeeeeeeee
Note that for the Bhatt estimator, the Bhattacharyya distance between two Gaussians is,
1 In
det. Σ
Dp(Pilly)
8 (Mirs -111)) " - M5]x)) det Enj det 2i
Σ Ejje E,gl
2
24

---

Also note that for the KL estimator, the KL divergence between two Gaussians is,
1
det jia | (151 (4(z) [ S/a mole))
(KL(Pi p3) (r( IT D + In
det Ex
where tr(.) refers the trace of matrix.
To complement the per-step acquisition runtimes reported in Figure 3, also provide end-to-end active learning runtimes in Table 3 and Table 4. These results reflect the total wall-clock time required for full active learning loop in each environment. These tables reinforce the efficiency advantage of PairEpEsts: across both architectures, KL and Bhatt estimators consistently require less total runtime than MC, particularly higher-dimennional environments such Ant and Humanoid. Notably, the runtime gap widens with dimensionality, reflecting the scalability of PairEpEsts relative sampling- based methods.
To justify our choice of five ensemble components, we conducted an additional experiment analyzing the effect of ensemble size active learning performance. This experiment was performed on the Hopper environment, with performance measured as RMSE at the final acquisition batch. Results reported in Table 2. Performance improves markedly when increasing from 3 to 5 members, but remains stable for larger ensembles, suggesting that five components provide a strong balance between efficiency and accuracy.
Table 2: RMSE at the final acquisition batch for different ensemble sizes on Hopper.
3
4
5
7
10
KL
0.51
0.43
0.30
0.31
0.29
Bhatt 0.50
0.45
0.29
0.28
0.30
Table 3: Total active learning runtimes (minutes) for Nflows Base across environments.
Hetero
Bimodal
Pendulum
Hopper
Ant
Humanoid
KL 58.14 14
58.22
76.02
77.06
80.21
96.28
Bhatt
57.72
58.04
76.86
76.94
78.54
95.58
MC
61.87
61.82
94.44
98.66
117.40
155.34
Table 4: Total active learning runtimes (minutes) for PNEs across environments.
Hetero
Bimodal
Pendulum
Hopper
Ant
Humanoid
KI. 22.68
22.50
28.44
28.60
30.58
35.24
Bhatt
22.50
22.36
28.24
28.88
30.14
36.04
MC
24.42
24.62
35.22
46.42
66.96
84.30
C Data
The hetero dataset was generated using two step process. Firstly, categorical distribution with three values was sampled, where p. = = Secondly, x was drawn from one of three different Gaussian distributions (-444, 2). (O, 9 NN, ?)) based on the value of the categorical distribution. The corresponding then generated as follows:
= 7 sin( + 3= cos
On the other hand, the bimodal dataset was created by sampling x from an exponential distribution with parameter and then sampling from a Bernoulli distribution with p = 0.5. Based on the value of n. the value was determined as:
[10sin (x) +
n
y =
10 c0cc((() (r) + = 20 - n
25

---

Note that for both bimodal and hetero data 2 ~ (0,, 1).
Regarding the nulti-dimensionn environments, namely Pendulum, Hopper, Ant, and Humanoid, the training sets and test sets were collected using different approaches. The training sets were obtained by applying random policy, while the test sets were generated using expert policy. This methodology was employed to ensure diversity between the training and test datasets. Distribution shift is inherent challenge in robotic dynamics, where data generated from different policies can vary significantly. These shifts, such as those observed in sim2real and imitation learning contexts. Notably, the OpenAI Gym library was utilized, with minor modifications [6].
D Additional Results
In addition to Table I, present Figure 3 show- ing the full active learning curves and Table detailing additional acquisition batches. The trend Psts outperforming or perform- ing comparably to baselines is consistent across environments and acquisition batches. We also evaluated log-lika logliielihood, proper scoring rule [22], with results shown in Figure 9. Using 10 seeds, report means and standard deviations. The PairEpEst-KL. KL. and PairEpEst Bhatt estima- tors perform similarly better than the MC estimator on all environments.
Uncertaint F.
MC
Blutt
Figure 8: The bias introduced by PairEpEsts for Nflows Base compared to the MC method on the ID environments.
For the high-dimensional Humanoid setting, however, log-likelihood did not improve as data was added. This limitation arises because puting ensemble log- ligliikl requires evaluating each component's 's likelihood and summing them, process prone numerical underflow values round zero. As in Table 1, selected acquisition batches are reported in Table 8. We further compare the MC estimator and PairEpEsts on Hopper the number of drawn samples increases (Figure 10). Additionally, of active learning rankings by computing Spearman's 's rank correlation between the MC estimator across all models and datasets, indi-
assessed the consistency
and PairEpEstts. As shown in Table 5, correlations are close cating strong preservation of relative ordering despite small biases. All correlations are statistically significant with extremely low p-values (maximum p 1.11 1 10 10-83).
As summarized in Table 6, Monte Carlo methods achieve comparable runtime to PairEpEsts with roughly 10 samples in Hopper, and with even fewer samples in higher-dimennsinaal environments such as Ant and Humanoid. This suggests that small-sample MC can, in principle, be competitive in terms of computational cost. However, PairEpEsts provide robust and consistent uncertainty estimates without the need for sample tuning, making them particularly advantageous in high-dimensional tasks where both runtime efficiency and estimator stability critical. Moreover, the accuracy of MC estimates deteriorates as the number of samples decreases, whereas PairEpEstt maintain reliability at low computational cost.
Table 5: Spearman's 's rank correlation coefficients comparing active learning rankings between MC and PatrEpEsts, demonstrating that our estimators preserve relative ranking of points.
Model
Dataset
KL Corr.
Bhatt Corr.
NFlows Base
Bimodal
0.9958
0.9958
NFlows Base
Hetero
0.9976
0.9986
PNEs
Bimodal
0.9893
0.9893
PNEs
Hetero
0.9943
0.9972
26

---

Table Number of MC samples that matched the runtime of PairEpEsts across environments.
Environment
Dimensionality
MC Samples (Runtime Matched)
Hetero
500
Bimodal
500
Pendulum
100
Hopper
11
10
Ant
27
Humanoid
257
Table 7: Mean RMSE on the test set for certain Acquisition Batches for Nflows Base. Experiments were across ten different seeds and the results are expressed as mean plus minus one standard deviation.
Random
BatchBALD
BADGE
BAIT
MC (BALD)
KL (ours)
Bhatt (ours)
Env
Acquisition Batch
10
1.76-0.17
1.86 0.33
1.84 0.33
1.22
1.48 1.2
1.66 0.2
1.56 1
hetero
25
0.45
1.66
1.740025
1.71
1.44
1.122
1.42
50
1.64 0.22
1.54
1.55
1.6=0.22
1.45+0012
1.49 0.29
1.42
10
8443338
1.37
611=0.11
1.1
0.02
6.01 0:00
6.01
bimodal
25
6.14+0088
6.744-0.66
6.02
6,85
6.04 0.044
6.02 0.005
6.01 + 0.04
50
0.72
6.42
6.01 1.04
6.61 1.97
6.01 0.004
6.01 1.044
6.0 1.04
6.62
6.42
6.01 0.04
0.33
6.01
6.01 0.04
6.0 0.04
0.28
0,32
0.32=0027
0.22
0.12
0.00
Penducian
25
0.22-00.8
0.17+:0.66
0.31 0.19
0.19 0.07
0.09 + 0.03
0.09 0.03
0.09 + 0.02
100
0.05
0.03
0.34 0.15
0.17-100.0
0.04 1.00
0.05 0.04
0.05 1 0.01
0.22
0.19
0.09
0.1
Hopper
25
1.24 0.26
0.33
1.37 0.23
1.18 0.22
0.36 0.06
0.38 0.05
0.39 0.06
50
1.14
1.06
1.26 0.23
1,05 0.22
0.31
0.33 0.03
0,34 1 0.04
0.2
0.87
1.12
0.06
0.02
0.3
0.29 + 0.03
10
1.33 0.1
1.11+0.11
1,21 + 0.14
1,25 + 0.08
1.24++0.13
L,09 +.1.1
1.13 + 0.09
Ans
50
L05=0.05
10000
1.02 0.06
1.981000
1.05
0.93 0.05
0.94 1 0.07
1.1
0.04
0.04
0.00
0.07
0.93
10
8.79 1.17
8.26
8.6+
10.97
1.05 10.69
6.82 1.34
7.5 1.63
Humanoid
25
1.48
1.81
6.78 3.224
11.01 1.0.7
10.65
1.088
5.11 1.34
5.3 1,51
50
7.04
6.52
7.874-2.92
11.07 0.25
10.94
4.27 0.94
4.08 0.8
6.59
5.331112
7.31
10.23
10.71
0.05
p < 0.01
p 0.001
27

---

Returo
bunodol
Favdehmesdi
Logg Lilelihood
12
-10
-148
Happprnss2
Ant-22
Plumsnoodd x2
Log: Libelhoood
-70
200
12.0
Training Set Sioe
Training Set Size
Triining Set Stvre
Ihatt
Random
BetchHALD RADGE
BATT
KL.
MC
Figure 9: Mean Log Likelihood on test set as data was added to the training sets for Nilows Base.
26
Likelibood
24
Logg
22
200
10
Number of Samples
MC
PaiDE Performance
Figure 10: Likellool on the test set at the 100 acquisition batch of the MC estimator on the Hopper environment as the number of samples increases, K, for Nflows Base. Experiment run across 10 seeds and the mean is being reported.
28

---

Table 8: Log Likelihood of held out test during training at different acquisition batches for Nflows Base. Experiments were across ten different seeds and the results are expressed as mean plus minus standard deviation.
Ang. Ratch
Random
BatchRALDD
(RAL))
(curs)
(murs)
10
0.54 1 0.27
4067 2.48
0.1++0.12
0.33 0.114
0.12
0.33
30
0.11
092-0:25
0.04
0.16
0.14
100
0.86+0.11 0.14
1.11-
0.7+0.7
1.1+0.114
+.9++01
1.12+00.1
1.14 0.13
10
30.57 12.61
7.99
-5877 1.87
17.2411.26
1.8
1.11
1.11 = 0.1
bunodal
50
25
1.901
-1085
-22.17
10.63
18.86
13,04
0.11
1033
32.88 13.61
-144 L4.8
1.22
1.21
1.11
12200.1
1.27 1 0.09
1.21 0.11
=13.44
-5.53
-47.66 31.67
1,86
Pendulum
25
50
-0.7%
1.8 5.19 2:455
3.77
+-5.22
-12.34 16.57
-2.51 5.06
16.65 -28.0
-7.43 1.7.2
-22.15
$..6.11.91
N:72 4.72+005
0.336
N:45 0.55
9.95 0.5
9.59 1.277
8.3++0.7
100
3,04 2.388
-2.29 2.29
5.31
10.16
10,49 1.291.00.5
9.58 0.8
10
-136.85
53.59
-120.43 77.9 77.99
201221
13,94 5:54
4.332
Hopper
30
1. 6.82
19.33 25.02
38.11 25
1172
26.43
25.72
25.0 1.6
100
8.67 3.39
5.33
6.16666.1
1,06 3.66
26.62 + 1.14
26.96 +: 1.11
26,88 0.91
10
.0.5 16.68 31,08
.0.2 46.22
-4.85 20.71
-T2.13+337.99
25.41 5.008
21 99.99= 9.0
25
17,06 13.16
21.22 10.61
23,12 11.37
39,77 1:3.4
40.62 2.85
37,58 =: 2.37
10
200.6]
20.02
243.1 6.03
-18..66 99.5
230.6] 1.9.2
229. 0.69
221..9
10.68
-201 28 22.78
Humonold
30
25
223
224
54
1
13.72
1.52
-22..22
-36.24 5.887
1 19422
-173.12
-19.007
777755 77.55
96
-28.41 + 0.93
-245 56 1.8
-44.84
247, 16 2.11
1.74
-246.29
-249.39
= 0.43
1.92
-249.57 0.02
-247 .6 1.04
100
247.52
-24.55
-16.778
124.9
249.71
-46.47
-29.76
249.97 1.006
< 0.05
p < 0.01
p < 0.001
29

---

18
04
400
300
Humanold-v22
1.2
200
RM55
200
601
800
1900
12.0
200
-20
000
100
1000
1330
200
400
460
300
2.00
12.0
Training Set Size
Training Set Size
Training Set Size
Bisttt
Raadon
BuichBALD
BAIT
Figure 12: Mean RMSE test set as data was added to the training sets for PNEs.
E Probabilistic Network Ensembles
Uncertainty Domityy
In addition to ensembles of normalizing flows, we also explored the use of probabilistic net- work ensembles (PNEs) as a common approach for capturing uncertainty [12, 38, 39]. The PNEs were constructed by employing fixed dropout masks, where each ensemble compo- nent modeled a Gaussian distribution. The models were trained using negative log like- lihood, with weights randomly initialized and bootstrapped samples from the training set to create diversity amongst the componenents. Our findings paralleled those of Nflows Base, in that PairEpEsts performed similarly or better than baselines and statistically sigmificantly in higher dimensions. These results are presented in Fig- ure and Figure 13, as well as Table and Table 10. Note that PNEs did not perform as well as Nilows Base they are not as expres- sive this can be seen for the bimodal setting in Figure 11. Furthermore, we have included the ID graphs illustrating the performance of PNEs hetero and bimodal in Figure 11. Similarly to before. we also provide comparison of the MC estimator as the number of samples increased in Figure 14. While could have implemented ensembles with different output distributions to better fit the data, decided to stick with Gaus- sians as they seem to be the most frequent choice [12, 38, 39]. Moreover, in order to pick the best distributional fit would require hyper-parameter search or apriori knowledge whereas NFs will learn the best distributional fit.
(a) hetero
Uncertainty Demsity
(b) bimodal
Ehhat
KL.
groundtrath
model
Figure 11: In the right graphs, the cyan dots are the ground-truth data and the brown dots sampled from PNEs. The 2 lines depict epistemic uncer- tainty corresponding different estimators. The left graphs depicts the ground-truth data as the blue dots and its corresponding density the orange histogram. Note the legend refers to the lines in the right graphs.
30

---

himodal
Poncuuum
1.5
Log Liketihond
0.5
-0.5
-13
Hopper-y2
Ant v2
#hmonoudd-12
L.keelhood
-35
-50
-.00
-230
-20
-300
Training Set
Training Size
Trnining Set
Bhatt
Random
BaichBALD RADGE
BAIT
KI.
MC
Figure 13: Mean Log Likelihood on the set as data was added to the training sets for PNEs.
20.0
12
Log Likelitood
RMSE
12.5
0.6
10.0
7.5
10'
10a
10
Number Samples
Number Samples
PaiDEs Performance
Performance
Figure 14: RMSE and Log-Likelihood on the test set at the 100 acquisition batch of the MC estimator on the Hopper environment the number of samples increases, K, for PNEs. Experiment run across 10 seeds and the mean is being reported.
31

---

Table 9: Mean RMSE the test set for certain Acquisition Batches for PNEs. Experiments across ten different seeds and the results are expressed as mean plus minus one standard deviation. Note that nothing bolded for bimodal many methods performed similarly.
Env
Acq. Batch
Random
BatchBALD
BADGE
BAIT
MC (BALD)
KL (ours)
Bhatt (ours)
10
3.9 1.77
69- 1.15
6.91 1.77
69 1.58
1.94 0.34
2.11 0.334
2.36 0.75
4.42
4.43
4.06
2.92
1.75
1.36
1.96 0.44
hetero
3,96 1.71
4.06112.66
5.12
4.22:22.95
1.78
1.68 0.33
1.67 1: 0.27
4.28 1.45
5.12
4.54
6.01 2.82
1.66
1.83
1.98 0.81
0.82
6.05
6.19:0227
6.11
6.01 0.044
6.04
bimodal
6.18
6.26++0.75
6.31 +.03
6.3 0.88
6.01 +.004
6.01 +00.4
6.01 0.04
0.13
0.07
6.03
0.044
6.11 0.004
0.04
6.01
100
6.06++0.00
6.01 + 0.03
6.03 0.03
6.01 0.04
6.01 +.0.4
6.01 0.044
6.01 0.04
0.31
0.15
0.09
0.2++
0.27 00099
0,11 1 0.03
0.12 0.03
0.11 1 0.03
0.16
0.12
0.13 0.04
Pendulan
0.09
100
0.06
0,07 + 0.03
0,07 0.02
0.08 0.02
0.16
0.09 0.00
0.08 0.05
185 0.14
1.44 0.31
1.22
1.16 0.37
0.94
1.19
1.14
Hopper
1.9 0.14
1: 0.13
0.66 0.19
0.55 0.12
0.73 0.13
0.72.10.00 0.08
0,71 1 0.1
183
0.54 0.15
0.55 0.15
0.12
0.57 0.06
0.57 1: 0.06
0.57 1: 0.05
100
1,72 0.144
0,4 0.05
0.49 0.133
0.43 0.05
0.5 0.04
0.44 1. 0.04
0.49 1. 0.05
0.13
2.24
0.12
Ant
0.14
2.15+0211
2.14+00118
0.12
1.67
0.12
1,39 + 0.12
1.5
0.13
2.16
0.11
0.22
1.29 : 0.06
1.3 0.04
0..33
11.63
01.22:00.9
10.15
10.41
9,95
10.65
Hamanoid
2.47
50
10.29
0.19
11.37
10.98 0.22
0.29
0.881
4,91 0.24
6.33 2.24
100
1..17
1.13
10.99 1: 0.26
11.05
10.57 0.41
4.84 :: 0.24
6.25 + 2.32
p < 0.05
P < 0.01
p < 0.001
12
Uncerainnt
cD
MC
Bhatt
KL.
Figure 15: The bias introduced by PairEpEsts for PNEs compared to the MC method on the 1D envi- ronments.
32

---

Table 10: Log Likelihood the test during training at different acquisition batches for PNEs. Experiments were across ten different seeds and the results are expressed as mean plus minus standard deviation. Note that values were highlighted for Hopper despite the results being statistically significant. In this case, the results are statistically significantly worse and we found misleading highlight.
Em
Acq.
Butch
Random
BatcBBALD
BADGE
BAIT
(CALD)
KL. (ours)
Bhalt (ours)
Neturo
50
25
0.23 0.57
0.03
0.76=0.76
0.78=0071
2:14
0.38 0.76
0
0.13+0022
=
1,2 L012=0111 0.11
0.5 0.014
1110
1.09 2: 0.11
0.24
100
0:23+00222
0.23+0.71 0.78
0.78 0.85
0.03++0.74
1.1
1.11 1.11+
1,08 0.2
10.01
0.32+0.33
0.21
4.136.44.14
0.6+0.1
0.59
bimosia
50
-20.78
-12.3 8.55
0.66 0.09
0.59 0.16
0.61 0.118
0.69
0.02
= 0.03
0.69
100
-5.79 7.19
0.66 0.1
0.64::0.11
0.68 0.08
0,7 0.02
0.7 1:
0.7 0.02
10
2.35
-10.66 + 7.78
8:88
-9.76: 9.09
4.94 0,48
5.51 0.77
5.21 1-6.1
Pendulam
25
4.19-+0..0
0.0.
100
4.69
9.53 = 0.96
7.07 5.49
3.8 1:23
6.09 0.42
0.64
7.77 0.68
10
-31.44
28.24
26.65 + 15.57
12.96
10.81 + 4,25
9.36 1.79
8.55 1.34
8,55 1.88
Hopper
25
-5.74 + 2.66
27.28
12,62 4.32
23.49 2.97
13.88 +.27
28.08
17.29 3.45
13.31
17.6+
13.9++1.
16.93 1,43
13.93 + 1.6
11.83 2:29
4,91 12.37
6.29
5
13.8
23.13
23.32 2.73
Ant
25
17.78
7.68
3.64
11.97 5:41
18.69
26.96
27.66 1788
50
22.21 2:11
12.24
14.91
36.16.132
20.51 8.88
29.84 1.09
29.62 0.7
100
23.4 1.85
15.26 6,5
16.87 3,75
19.6 2,07
20,03 9.87
31.24 1.09
31,63 0.9)
19.87
Humanoid
25
-245.45
2.24
37.12 -224.95
-245
-247.3 1.26
-247.83 1.19
2247.39
232.33
12.02
50
-246.91
1,28
-247. 38 3.28
-249 78 0.52
-249.2 0.52
-24% 06 1.82
-249 0.7.5
-223.49 77.83
100
-247.74 0.6
-248.99 +11.1
-24944
-249.66 0...66-0.53
-247.25 3.36
-249.91 0.12
-223.29 79.8
< 0.05
>< 0.01
p < 0.001
33

---

Introduction to Normalizing Flows
NFs are powerful non-parametric models that have demonstrated the ability to fit flexible multi- modal distributions [54, 53]. These models achieve this by transforming simple base continuous distribution, such as Gaussian or Beta, into a more complex one using the change of variable formula. By enabling scoring and sampling from the fitted distribution, NFs find application across various problem domains. Let B represent with (b) as its density function, and let Y gBB), where is invertible function with an existing inverse 9 g-,, and both y and g are differentiable. Leveraging the change of variable formula, we can express the distribution of Y as follows:
base distribution, D-dimensional continuous random vector
(v) pB (w))) det(J(
(11)
where (-) denotes the Jacobian, and det signifies the determinant. The first term on the right-hand side of Equation 11 governs the shape of the distribution, while de (v))) normalizes it, ensuring the distribution integrates to onc. Complex distributions can be effectively modeled by
making gb) learnable function with tunable parameters 0. denoted go)b). However, it is essential to select carefully to guarantee its invertibility and differentiability. For examples of suitable choices, please refer to [47].
G Hypothesis Testing Details
We conducted Welch's t-tests to compare means (14 H4) between different estimators, as this test relaxes the assumption of equal variances compared to other hypothesis tests [13]. The means for both KL and Bhatt were compared to each of the baseline methods: BatchBALD, BADGE, BAIT, MC and random. To control the family-wiss error rate (FWER), we performed a Folm-Bonferron correction across each setting, environment, and acquisition batch. This follows best practices to ensure our results are statistically significant and not occur just by random chance.
34


Reviews:
[{"review_id": "tFgnvLA5ek", "reviewer": "Reviewer_3iVP", "summary": "The work proposes a novel estimator for epistemic uncertainty estimation in ensemble models for regression tasks, which does not require Monte Carlo sampling. The estimator (Pairwise Epistemic Estimators) is based on Pairwise-Distance Estimators (PaiDEs) to obtain a lower/upper bound for the mutual information (or equivalently, the epistemic uncertainty). The PaiDEs require closed-form expressions for the divergences/distances between distributions, which motivates the authors to use Normalizing Flows (NFs) to model the distribution and reduce the target distributions' premetrics to the base distributions. The algorithm is validated by active learning experiments compared to other uncertainty-based baselines.", "questions": "1. Have you ever tried other active learning benchmark algorithms? (See Weaknesses point 3)\n\n2. How precise is your evaluation of epistemic uncertainty (as a final output rather than an intermediate product)?\n\n3. Typos: line 118 \"wtighter\" should be \"tighter\"?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1. The proposed method is sampling-free, which will boost the computation once we are in a high-dimensional space.\n\n2. The theoretical understanding of the algorithm is solid.\n\n3. The empirical evidence is impressive.\n\n4. The presentation is clear and easy to follow.\n\nWeaknesses:\n\n1. The method only applies when all premetrics are of closed-form expressions. This limits the applicability in many scenarios. Also, the numerical experiments are all conducted in Gaussian bases. It remains unclear how robust it is with skewed or heavy-tailed distributions.\n\n2. The estimator of the epistemic uncertainty is biased (e.g., either a lower bound or an upper bound in the two examples). It shall be fine if the uncertainty is only an intermediate product (e.g., uncertainty-based active learning), but if one wants the epistemic uncertainty estimator as the final output, it becomes a big problem. The paper also lacks an analysis of the potential bias.\n\n3. The active learning algorithm benchmark is restricted. Uncertainty-based query strategy is only one category of active learning algorithms; it remains unclear if they are SOTA in many scenarios. More comparisons (e.g., coverage/representation-based active learning algorithms) should be included. (update: solved)\n\n4. The theoretical contribution of this paper is limited. All the theory parts are from the previous PaiDE paper.", "was_revised": true, "final_justification": "I think this paper gives a good sampling-free way of estimating the epistemic uncertainty (though with many limitations to its applicability). The theoretical part is largely due to previous works (which limits the originality), but the application of the theory is worth noticing. Therefore, I will raise my score to 4.", "rebuttal": "Dear Reviewer,\n\nThank you for your thoughtful review and for recognizing the strengths of our work. We appreciate your acknowledgment that our proposed method is sampling-free, offering a significant computational advantage. We're also glad you found the theoretical foundations of our approach to be solid and the empirical results compelling.\n\n**Legend:**  \n**W** – Reviewer *Weakness*  \n**R** – Author *Response*  \n**Q** – Reviewer *Question*  \n**A** – Author *Answer*\n\n**W1:** The method only applies when all premetrics are of closed-form expressions. This limits the applicability in many scenarios.\n\n**R1:** We agree this is an important consideration and note it explicitly in Section 4. However, PairEpEsts are broadly applicable, including to non-parametric models such as normalizing flows, which capture complex, multimodal, and skewed distributions. This is because PairEpEsts can operate on latent distributions such as the tractable base distributions in normalizing flows where closed-form premetrics are available.\n\nPairEpEsts are also flexible in their choice of distributional distance measures. They support a wide range of options, such as KL divergence, Bhattacharyya distance, Hellinger distance, total variation distance, and Wasserstein distance, provided these measures admit closed-form or efficiently computable expressions. Many commonly used models, including those that produce outputs in the exponential family (e.g., Gaussian, Bernoulli, Poisson, Dirichlet), as well as models with tractable latent distributions, satisfy this condition.\n\nIn summary, while PairEpEsts rely on closed-form premetrics, they can be applied to a broad class of predictive models, making them well-suited for a variety of machine learning applications.\n\n**W2:** The numerical experiments are all conducted in Gaussian bases. It remains unclear how robust it is with skewed or heavy-tailed distributions.\n\n**R2:** Our choice of Gaussian bases aligns with standard practice in robotic control tasks and many other fields, where they’ve proven to be highly effective. This also ensures more comparability to prior work.\n\nThat said, we acknowledge the importance of modeling heavy-tailed and skewed predictive distributions in other domains such as finance and insurance. PairEpEsts are not inherently limited to Gaussian bases. They are compatible with skewed or heavy-tailed base distributions—so long as the pairwise premetric remains tractable. Recent work [1, 2] introduces tail-adaptive normalizing flows specifically designed for such heavy-tailed distributions. We’ve clarified this in the updated manuscript with the following addition:\n\n> PairEpEsts naturally extend to skewed or heavy-tailed base distributions, as long as the pairwise premetric admits a closed-form expression. Tail-adaptive normalizing flows [1, 2] offer a promising direction for incorporating heavy-tailed behavior into normalizing flows. Exploring their integration with PairEpEsts remains an exciting avenue for future work, especially in domains where tail risk is critical.\n\n**W3:** The estimator of the epistemic uncertainty is biased (e.g., either a lower bound or an upper bound in the two examples). The paper also lacks an analysis of the potential bias.\n\n**R3:** Unlike aleatoric uncertainty, there is no universally accepted ground truth for epistemic uncertainty. However, it is agreed that inputs with fewer observations should exhibit higher epistemic uncertainty, as in Figure 2. In this context, biased metrics for epistemic uncertainty estimation have become a common tool [3, 4].\n\nTo further assess whether our methods introduce reordering in the estimated uncertainties, we computed Spearman’s rank correlation coefficient, a non-parametric statistic that measures the strength and direction of the monotonic relationship between two ranked variables. Specifically, we compared the rankings produced by our estimators (KL and Bhatt) against those from MC estimation on the data used in Figure 8. A coefficient of 1 indicates perfect agreement in ranking, 0 indicates no correlation, and -1 indicates complete inverse ranking.\n\nAs shown below, the correlation coefficients are all very close to 1, suggesting that our estimators preserve the relative ordering of uncertainty values nearly perfectly. This implies that any bias introduced by our method has minimal impact. Moreover, all p-values are extremely small (largest p = 1.11e-83), confirming the statistical significance of the correlations. \n\nThis analysis has been added to the paper.\n\n| Model       | Dataset | KL     | Bhatt  |\n|-------------|---------|--------|--------|\n| NFlows Base | Bimodal | 0.9958 | 0.9958 |\n|             | Hetero  | 0.9976 | 0.9986 |\n| PENs        | Bimodal | 0.9893 | 0.9893 |\n|             | Hetero  | 0.9943 | 0.9972 |\n\n**W4:** The active learning algorithm benchmark is restricted.\n\n**R4:** We include BADGE, BAIT, and BatchBALD, three widely used and highly cited active learning methods that incorporate representation and diversity-based strategies:\n\n- **BADGE**: Combines feature space diversity via gradient embeddings.  \n- **BAIT**: Utilizes influence functions to promote diversity-aware selection.  \n- **BatchBALD**: Combines uncertainty with feature space diversity.  \n\nThese methods are not purely uncertainty-based; rather, they are varied approaches that represent state-of-the-art acquisition strategies beyond uncertainty alone. We believe this selection strikes a strong balance between methodological relevance, algorithmic diversity, state-of-the-art, and computational feasibility in the active learning setting.\n\n**W5:** The theoretical contribution of this paper is limited. All the theory parts are from the previous PaiDE paper.\n\n**R5:** We transparently cite prior work [5] for the theoretical foundations underlying PaiDEs. However, our contribution lies in how we operationalize this theory for scalable epistemic uncertainty estimation, especially in high-dimensional regression tasks where existing methods falter.\n\nWe introduce:\n\n- A principled epistemic uncertainty estimator using PaiDEs.  \n- A practical implementation using normalizing flows and PNEs.  \n- Extensive empirical validation across six environments, two model families, and five acquisition methods.\n\nThis work tackles a concrete gap in the literature: how to estimate epistemic uncertainty scalably and meaningfully for continuous, high-dimensional outputs—critical in domains like robot learning, where data collection is expensive and risk-sensitive.\n\n**Q1:** Have you ever tried other active learning benchmark algorithms?\n\n**A1:** We include BADGE, BAIT, and BatchBALD, which incorporate both uncertainty and diversity. See **R4** above.\n\n**Q2:** How precise is your evaluation of epistemic uncertainty (as a final output rather than an intermediate product)?\n\n**A2:** While epistemic uncertainty is used here primarily as an acquisition signal, we also provide qualitative visualizations in 1D settings (Figure 8) to assess estimator behavior across the input space. These plots show that PairEpEsts effectively capture key structural patterns in uncertainty, assigning higher uncertainty to under-sampled regions as expected. See **R3** above.\n\n**Q3:** Typo on line 118 (\"wtighter\")\n\n**A3:** Thank you for spotting this typo. It has been corrected.\n\n**References**\n\n[1] Laszkiewicz, M., Lederer, J., & Fischer, A. (2022). *Marginal tail-adaptive normalizing flows*. ICML.  \n[2] Hickling, T., & Prangle, D. (2024). *Flexible Tails for Normalizing Flows*. arXiv:2406.16971  \n[3] Malinin, A., & Gales, M. (2020). *Uncertainty estimation in autoregressive structured prediction*. arXiv:2002.07650  \n[4] Schweighofer, K. et al. (2023). *An improved information-theoretic measure of predictive uncertainty*. arXiv:2311.08309  \n[5] Kolchinsky, A., & Tracey, B. (2017). *Estimating mixture entropy with pairwise distances*. *Entropy*, 19(7), 361."}, {"review_id": "ENJBv9DThH", "reviewer": "Reviewer_5FYN", "summary": "The paper studies an alternative to sample-based estimation of epistemic uncertainty via pairwise distance estimators. The paper derives estimators based on two measures, Bhattacharyya distance and Kullback-Leibler divergence. Across two 1d synthetic examples and high-dimensional active learning tasks, the paper shows that the proposed sample-free estimators yield improved performance at matched compute budgets.", "questions": "* In Figure 2, it appears that also a count-based method could perform well, at an even lower cost. I think this could be worth mentioning.\n* In line 103, why does $H(\\theta \\mid y, x) \\geq 0$ hold, given that this is a differential entropy?\n* In Figure 5, for which #samples is MC evaluated here? For which #samples would MC have the same compute time as Bhatt/KL?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n* The paper correctly identifies the computational burden of MC in most settings. Even though in many active learning settings, sample efficiency is the main bottleneck, improving compute efficiency can allow for screening more options in very high-dimensional settings.\n* The experimental evaluation is solid, with a range of settings from low-dimensional to high-dimensional.\n\nWeaknesses:\n* The theoretical contribution beyond reference [35] appears limited. Nevertheless, the manuscript could benefit from a clearer outline of the novelty of the analysis.\n* I am missing a comparison of MC and the proposed methods in terms of compute cost (time). For example, in Figure 3, it would be useful to indicate the #samples for MC such that the overall compute cost in terms of time is matched between MC and PaiDEs.\n* While the experimental evaluation is solid, the conclusion around the benefits of sample-free estimation in high-dimensional tasks would be more significant if evaluated also on another (very) high-dimensional task beyond Humanoid. It would also be nice to see an evaluation on a common benchmark for active learning (e.g., [1]).\n\nTypos:\n* line 118: \"wtighter\"\n\n[1]: Navigating the pitfalls of active learning evaluation: A systematic framework for meaningful performance assessment (https://proceedings.neurips.cc/paper_files/paper/2023/file/1ed4723f12853cbd02aecb8160f5e0c9-Paper-Conference.pdf)", "was_revised": true, "final_justification": "In my view this paper is a solid contribution. Though its novelty and originality is somewhat limited, it evaluates scalable and efficient methods for uncertainty estimation on high-dimensional tasks. Since the authors have addressed my remaining concerns, I increased my score to 5. Overall, I would rate this paper between 4 and 5.", "rebuttal": "Dear Reviewer,\n\nThank you for your thoughtful review. We appreciate your recognition that the paper correctly identifies the computational burden of MC methods, particularly in settings like active learning where epistemic uncertainty must be evaluated repeatedly across large candidate pools. While sample efficiency is often the primary focus, improving compute efficiency can dramatically expand the feasible scale of screening, especially in very high-dimensional output spaces. We're also glad you found our experimental evaluation solid. We emphasize that it covers a diverse range of tasks, two model types, and five baselines, highlighting the versatility and practical impact of our method.\n\n**Legend:**  \n**W** – Reviewer *Weakness*  \n**R** – Author *Response*  \n**Q** – Reviewer *Question*  \n**A** – Author *Answer*\n\n**W1:** The theoretical contribution beyond reference [1] appears limited. Nevertheless, the manuscript could benefit from a clearer outline of the novelty of the analysis.\n\n**R1:** We transparently cite prior work [1] for the theoretical foundations underlying PaiDEs. However, our contribution lies in how we operationalize this theory for scalable epistemic uncertainty estimation, especially in high-dimensional regression tasks where existing methods falter.  \nWe introduce:  \n- A principled epistemic uncertainty estimator using PaiDEs.  \n- A practical implementation using normalizing flows and ensembles.  \n- Extensive empirical validation across six environments, two model families, and five acquisition methods.\n\nThis work tackles a concrete gap in the literature: how to estimate epistemic uncertainty scalably and meaningfully for continuous, high-dimensional outputs—critical in domains like robot learning, where data collection is expensive and risk-sensitive.  \nWe have added the following text to clarify this:  \n\n> We introduce PairEpEsts, a novel framework that leverages pairwise distance estimators [1] for epistemic uncertainty in deep ensembles with probabilistic outputs—focusing on high-dimensional regression tasks previously underserved by the literature (see Section 4).\n\n**W2:** I am missing a comparison of MC and the proposed methods in terms of compute cost (time). For example, in Figure 3, it would be useful to indicate the #samples for MC such that the overall compute cost in terms of time is matched between MC and PaiDEs.\n\n**R2:** We have clarified this in Figure 3. MC matches the compute time of KL and Bhatt estimators when using approximately 10 samples. This now appears in the figure.\n\n**W3:** While the experimental evaluation is solid, the conclusion around the benefits of sample-free estimation in high-dimensional tasks would be more significant if evaluated also on another (very) high-dimensional task beyond Humanoid. It would also be nice to see an evaluation on a common benchmark for active learning (e.g., [2]).\n\n**R3:** We chose high-dimensional regression environments from the MuJoCo suite (Ant, Humanoid) to explicitly address a gap in the literature: most benchmarks, including [2], focus on classification or 1D regression, which do not reflect the challenges of real-world control tasks. Ant and Humanoid environments test uncertainty estimation in more realistic, complex settings with structured, high-dimensional outputs. While we do include 1D toy examples—similar to prior work—our primary goal was to evaluate performance in more demanding settings. Although our approach builds on [3], its extension to these control domains is not straightforward; it requires adapting to the computational challenges of continuous, high-dimensional spaces. Our work provides this bridge. Applying our method to even higher-dimensional problems like image generation is a promising direction, though it demands additional architecture-specific considerations, which we leave for future work.\n\n**Q1:** In Figure 2, it appears that also a count-based method could perform well, at an even lower cost. I think this could be worth mentioning.\n\n**A1:** We agree that in the 1D toy setting of Figure 2, a simple count-based method could serve as an estimate for epistemic uncertainty and potentially perform well at low computational cost. However, count-based methods generally fail to scale to higher-dimensional input spaces such as those in Hopper, Ant, and Humanoid. In these settings, the input space becomes exponentially larger and sparser, making it difficult to obtain meaningful counts. Moreover, count-based methods lack the ability to leverage the learned representations of the model, which are essential in high-dimensional and complex tasks. For these reasons, more principled estimators, such as those we propose, are necessary to provide reliable uncertainty estimates in realistic settings.  \nWe have added the following text to the paper to make this clear:\n\n> While count-based methods may suffice in low-dimensional settings, they do not generalize well to high-dimensional spaces where data is sparse and frequency-based heuristics fail. Our approach leverages learned representations, enabling uncertainty estimation in complex, high-dimensional scenarios.\n\n**Q2:**  In line 103, why does $H(\\theta|y,x) \\geq 0$ hold, given that this is a differential entropy?\n\n**A2:** $\\theta$ denotes the discrete index of ensemble components (we use 5), making $H(\\theta|x,y)$ a Shannon entropy, not differential entropy. This ensures $H(\\theta|x,y) \\geq 0$. We’ve clarified this in the text:\n\n> The upper bound follows from Equation 3, and $H(\\theta|x,y) \\geq 0$ since $\\theta$ is modeled as a discrete random variable (ensemble index), unlike $x$ and $y$ which are continuous.\n\n**Q3:** In Figure 5, for which #samples is MC evaluated here? For which #samples would MC have the same compute time as Bhatt/KL?\n\n**A3:** The MC results in Figure 5 use 5000 samples per acquisition candidate across all environments, consistent with the setup in Figure 4. We have updated the caption of Figure 5 to explicitly clarify this.\n\nThe number of samples for MC to match compute time with Bhatt/KL varies by environment due to dimensionality. We’ve added a table to the paper summarizing these values:\n\n| Environment | Dimensionality | MC Samples (Runtime Matched) |\n|-----------|------------------------|--------------------------|\n| Hetero    | 1 dim                  | 500                      |\n| Bimodal   | 1 dim                  | 500                      |\n| Pendulum  | 3 dims                 | 100                      |\n| Hopper    | 11 dims                | 10                       |\n| Ant       | 27 dims                | 5                        |\n| Humanoid  | 257 dims               | 5                        |\n\nAs noted in Section 5.3, we use 100 acquisition candidates for MC in the Humanoid environment, compared to 1000 in other tasks, due to computational constraints. This reduced set allows Monte Carlo to match the performance of Bhatt and KL methods.\n\nThis analysis has been added to the paper.\n\n**Typo:** Line 118: \"wtighter\"\n\n**R:** Thank you, this has been corrected.\n\n**References**\n\n[1] Kolchinsky, A., & Tracey, B. D. (2017). Estimating mixture entropy with pairwise distances. *Entropy*, 19(7), 361.  \n[2] Lüth, C., Bungert, T., Klein, L., & Jaeger, P. (2023). Navigating the pitfalls of active learning evaluation: A systematic framework for meaningful performance assessment. *Advances in Neural Information Processing Systems*, 36, 9789-9836.  \n[3] Berry, L., & Meger, D. (2023). Normalizing flow ensembles for rich aleatoric and epistemic uncertainty modeling. arXiv:2302.01312."}, {"review_id": "PRBA3CTgqx", "reviewer": "Reviewer_q5B8", "summary": "The paper introduces Pairwise Epistemic Estimators, a new method for estimating epistemic uncertainty in regression tasks. This approach leverages an ensemble of models derived from Normalizing Flows and uses pairwise distance estimations to quantify uncertainty. The authors provide theoretical grounding, demonstrating that PaiDEs establish bounds on entropy and enhance the performance of the BALD method. Notably, PaiDEs efficiently estimate uncertainty using only a few members in ensemble, making them computationally faster than Monte Carlo-based alternatives. The effectiveness of the method is demonstrated on the active learning task on two 1D and four multidimensional regression datasets.", "questions": "Given that the proposed method only requires receiving a probability distribution from each ensemble member, could it be applied to other domains, such as image classification using Bayesian Neural Networks?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n1. The paper is well-structured with clear clearly written, providing a comprehensive background on normalized flows and probabilistic network.\n2. The paper offers theoretical justification for how PaiDEs establish entropy bounds using various generalized distance functions and leverage these to enhance the BALD method.\n3. The authors validate the proposed epistemic uncertainty estimation method on simple 1D regression tasks, showing expected uncertainty behavior across different regions. More importantly, they demonstrate its effectiveness for active learning in multidimensional regression tasks.\n4. Additionally, the paper highlights the computational efficiency of the method, achieving competitive or superior performance to Monte Carlo-based approaches with a small ensemble.\n\n\n\n**Weaknesses:**\n1. The approach for ensuring diversity of the ensemble (lines 152–153) is not clearly explained. Specifically, how is the fixed dropout mask computed? Is a different mask generated for each ensemble member? Furthermore, it is unclear for what part of the ensemble randomization or bootstrapping is applied. \n2. The computational efficiency of the proposed approach appears to result from a smaller committee size, and even this small ensemble achieves competitive results. However, it remains unclear how varying the committee size impacts active learning performance. While Figure 5 presents computational time across different ensemble sizes, including active learning performance metrics for the same range would significantly strengthen the empirical analysis. \n3. The paper would benefit from comparisons with additional baselines. In particular, methods based on Monte Carlo [1]. Moreover, an adaptation of least confidence method, which commonly used in classification, could be applied, given that the model outputs a distribution. \n4. Minor issue: there is a typo in line 118 “wtighter” should be corrected to “tighter”.\n\n[1] https://arxiv.org/pdf/1806.09856", "was_revised": true, "final_justification": "The rebuttal and follow-up discussions addressed most of my initial concerns: ensemble diversity is now clearly described; prior work and new appendix experiments justify the choice of five ensemble members, showing diminishing returns for larger ensembles; and the authors clarified that entropy-based criteria were evaluated in prior work and found ineffective in this setting.\n\nWhile the addition of some extra baselines could further enrich the comparison, the current set of state-of-the-art baselines and the diversity of tasks provide sufficient empirical support. Overall, the work is technically sound, clearly presented, and addresses an important problem in uncertainty estimation for regression, with both theoretical and practical contributions. I maintain my positive assessment and recommend acceptance.", "rebuttal": "Dear Reviewer,\n\nThank you for your thoughtful and encouraging review. We’re glad you found the paper well-structured and the background on normalizing flows and probabilistic networks helpful. We appreciate your recognition of our theoretical contributions and the empirical results—both in 1D regression and high-dimensional active learning settings.\n\n**Legend:**  \n**W** – Reviewer *Weakness*  \n**R** – Author *Response*  \n**Q** – Reviewer *Question*  \n**A** – Author *Answer*\n\n**W1:** The approach for ensuring diversity of the ensemble (lines 152–153) is not clearly explained. Specifically, how is the fixed dropout mask computed? Is a different mask generated for each ensemble member? Furthermore, it is unclear for what part of the ensemble randomization or bootstrapping is applied.\n\n**R1:** Each ensemble member is initialized with a unique random seed, trained on a bootstrapped subset of the data, and assigned a distinct dropout mask (sampled once at initialization with probability $p = 0.5$). This mask is fixed for the duration of both training and inference. These techniques—random weights, data bootstrapping, and fixed dropout—collectively ensure diversity across the ensemble. For clarity, we have added the following to the paper:\n\n> To encourage diversity in the ensemble, each member is initialized with different random weights, trained on a bootstrapped subset of the data, and assigned a fixed dropout mask sampled at the start of training (with $p = 0.5$), similar to [1]. This mask remains constant throughout training and inference.\n\n**W2:** The computational efficiency of the proposed approach appears to result from a smaller committee size, and even this small ensemble achieves competitive results. However, it remains unclear how varying the committee size impacts active learning performance. While Figure 5 presents computational time across different ensemble sizes, including active learning performance metrics for the same range would significantly strengthen the empirical analysis.\n\n**R2:** We experimented with different ensemble sizes and found that 5 components consistently provided reliable uncertainty estimates across both low- and high-dimensional tasks (see Figures 2 and 4). This is consistent with prior work [2], which also uses 5 components for normalizing flow ensembles. Our choice was thus not a limitation imposed for efficiency, but one grounded in empirical sufficiency and precedent. Additionally, other recent works [3, 4] also find that ensembles of 5 members are effective for epistemic uncertainty estimation.\n\nWe also acknowledge in Figure 6 that our method becomes less effective as the number of ensemble components increases. However, this is less concerning in most deep learning applications, where ensembles are typically kept small due to computational considerations.\n\n**W3:** The paper would benefit from comparisons with additional baselines. In particular, methods based on Monte Carlo. Moreover, an adaptation of least confidence method, which is commonly used in classification, could be applied, given that the model outputs a distribution.\n\n**R3:**  Our experiments include a diverse and widely used set of baselines: BADGE, BAIT, BALD, BatchBALD, and random. These cover both uncertainty-based and diversity-driven acquisition strategies. We also evaluate across two model classes (PNEs and Nflows Base) and six environments, ranging from simple 1D tasks to high-dimensional control (Humanoid), providing a comprehensive assessment of epistemic uncertainty.\n\nMC Dropout is conceptually and practically similar to PNEs. It performs approximate Bayesian inference by applying dropout at test time, inducing a distribution over models. Each stochastic forward pass samples from this implicit posterior, analogous to how PNEs sample from an explicit weight distribution. In our setup, PNEs are instantiated by fixing different dropout masks at initialization to define each ensemble member. This approach mirrors the stochasticity of MC Dropout, while producing deterministic members that enable consistent downstream evaluation and acquisition.\n\nRecent work [2, 4, 5] has shown that deep ensembles often yield more robust and reliable uncertainty estimates than MC Dropout, particularly for regression and under distribution shift—both of which are central to our active learning setting. For example, Ovadia et al. [4] report:\n\n> Deep ensembles seem to perform the best across most metrics and be more robust to dataset shift.\n\nAs for the least confidence criterion, [2] used the entropy of the output distribution as a proxy for least confidence in multi-dimensional regression, but found it performed poorly in practice. This underscores the difficulty of using such heuristics for epistemic uncertainty estimation.\n\n**W4:** Minor issue: there is a typo in line 118 “wtighter” should be corrected to “tighter”.\n\n**R4:** Thank you, this has been corrected.\n\n**Q1:** Given that the proposed method only requires receiving a probability distribution from each ensemble member, could it be applied to other domains, such as image classification using Bayesian Neural Networks?\n\n**A1:** While our method can be applied to classification tasks using ensembles, its advantages are most pronounced in regression with high-dimensional outputs. In classification, predictive entropy $H(y \\mid x)$ can already be directly estimated from the categorical distribution produced by an ensemble, reducing the need for more complex estimators. In contrast, high-dimensional regression presents unique challenges for epistemic uncertainty estimation, where reliable techniques are still limited. Our approach directly addresses this gap, extending uncertainty estimation to a setting that has received far less attention than classification or 1D regression.\n\n**References:**\n\n[1] Durasov, N., Bagautdinov, T., Baque, P., & Fua, P. (2021). Masksembles for uncertainty estimation. *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition*, 13539–13548.  \n[2] Berry, L., & Meger, D. (2023). Normalizing flow ensembles for rich aleatoric and epistemic uncertainty modeling. *arXiv preprint arXiv:2302.01312*.  \n[3] Chua, K., Calandra, R., McAllister, R., & Levine, S. (2018). Deep reinforcement learning in a handful of trials using probabilistic dynamics models. *NeurIPS*.  \n[4] Ovadia, Y., Fertig, E., Ren, J., Nado, Z., Sculley, D., Nowozin, S., ... & Snoek, J. (2019). Can you trust your model's uncertainty? Evaluating predictive uncertainty under dataset shift. *NeurIPS*.  \n[5] Fort, S., Hu, H., & Lakshminarayanan, B. (2019). Deep ensembles: A loss landscape perspective. *arXiv preprint arXiv:1912.02757*."}, {"review_id": "x5GGq50XUo", "reviewer": "Reviewer_yg5h", "summary": "The paper utilizes Paired Distance Estimators to efficiently measure epistemic uncertainty in regression ensemble models, i.e. estimating the differential entropy of the mixture distribution.\nSpecifically, two practical implementations based on the Bhattacharyya distance and KL divergence are introduced.\nThey demonstrate the performance of this approach on two synthetic tasks and for active learning on real-world datasets for an ensemble of NFlow base models.", "questions": "- How does PairEpEst-KL relate to the Expected Pairwise KL-divergence discussed in Malinin et al. 2021 and Schweighofer et al. 2023 as measure of epistemic uncertainty in the classification setting? Is it an extention to the regression setting or is there a fundamental difference between those quantities?\n\n- Given the multimodel output distribution of an ensemble of NFlows base, how was the point prediction for calculating the RMSE obtained?\n\n- Why not reporting the MC estimate in Figure 2, when this is done in Figure 8 in the appendix? \nAlso, why does the estimated uncertainty look different in the two figures?\n\n- Why are many different colors introduced for different significant levels in Table 1, when there are only two entries with p < 0.01 and no entries that are significant for p < 0.05 and p < 0.001?\n\n- Why was a uniform weighting of components, i.e. $\\pi_j = 1/M$ (line 198) chosen? Are there alternative ways to weight ensemble members that could be interesting?\n\n\nMalinin, A., & Gales, M. (2021). Uncertainty estimation in autoregressive structured prediction. \n\nSchweighofer, K., Aichberger, L., Ielanskyi, M., & Hochreiter, S. (2023). Introducing an improved information-theoretic measure of predictive uncertainty.", "limitations": "I don't agree that the stated limitation of introducing bias in the estimation of epistemic uncertainty (line 287) does not affect the relative ordering of points.\nThe bias is input dependent as can also be seen in Figure 8, thus might change the relative ordering of individual inputs for active learning, unless I missed some technical result on the monotonicity of the bias depending on $H(y|x)$?\nNevertheless, it could be that the statement is true in practice, but I would like to see an experiment looking at ranking correlations between PairEpEsts and the MC estimate with a very high number of samples (to rule out MC variance) on the ordering of inputs on one of the tasks to confirm this.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\n- Good motivation for why using PaiDEs is necessary for estimating epistemic uncertainty in regression ensemble models. While this is especially true for their utilized NF predictive model, this may also be a concern for usual PNEs.\n- While the theoretical results are following the steps of Kolchinsky and Tracey and are not novel per se, they are relevant and interesting for the uncertainty estimation community in machine learning.\n\n## Weaknesses\n\n- The experimental results are unfortunately not very encouraging. The synthetic experiments look reasonable, but the active learning results only look convincing on the Humanoid task. Also adding results for the full dataset would put the active learning results better into perspective.\n- Figure 3 only reports the difference between means. Having variances or CI would put the differences between PaiDEs and MC estimation better into perspective.\n- The runtime issue of MC estimation is a bit unclear to me. Is this a general issue or intrinsic to an ensemble of NFlows base models? Because I don't see how sampling would be so much slower when given the output distributions of members of the PNE. Also, I did not find a time comparison for PNEs.", "was_revised": true, "final_justification": "As detailed in my response to the rebuttal, many of my points of critique have been adequately addressed.\nHowever, the presentation in Figure 2 is still not comprehendable to me, I would go with the presentation as in Figure 8.\nFurthermore, my doubts regarding the effectiveness of the method beyond the humanoid task have not been resolved.\nIn principle, also efficiency (i.e. runtime) could have been the major metric, controlling only that performance is comparable.\nHowever, the current presentation in Table 2 focus entirely on performance, where PaiDEs perform significantly better on humanoid, but not on other tasks.\n\nOverall, I found the paper very interesting and think its merits would warrant acceptance, yet the experimental evalution or at least its presentation is on the weaker side and could be improved further.", "rebuttal": "Dear Reviewer,\n\nThank you for your thoughtful review. We're glad you found the motivation for using PaiDEs to estimate epistemic uncertainty in regression ensembles compelling—particularly with Normalizing Flows, but also potentially for standard PNEs. We also appreciate your recognition that, while our theoretical results build on prior work by Kolchinsky and Tracey, they remain relevant and valuable to the uncertainty estimation community.\n\n**Legend:**  \n**W** – Reviewer *Weakness*  \n**R** – Author *Response*  \n**Q** – Reviewer *Question*  \n**A** – Author *Answer*\n\n**W1:** Limited effectiveness beyond Humanoid; need full-dataset performance\n\n**R1:** While the gains from PairEpEsts are most pronounced in the highest-dimensional environment, they consistently match or outperform baselines across all environments, model types, and acquisition batches (see Tables 2 and 4). This includes low-dimensional synthetic tasks and mid-dimensional control tasks such as Hopper and Ant.\n\nCrucially,  PairEpEsts deliver these improvements while being more computationally efficient and robust. We also include statistical significance testing (Table 1), which is often omitted in related work but is key to verifying that improvements are meaningful rather than due to noise.\n\nWe believe the breadth, consistency, and statistical rigor of our evaluation make the results compelling.\n\nRegarding full-dataset performance: we agree this adds valuable perspective. Figure 4 and Figure 12 now include RMSEs using the full dataset. We reproduce the values below:\n\n| Model        | Hetero | Bimodal | Pendulum | Hopper | Ant  | Humanoid |\n|--------------|--------|---------|----------|--------|------|----------|\n| NFlows Base  | 1.4    | 6       | 0.02     | 0.21   | 0.85 | 2.97     |\n| PNEs         | 1.6    | 6       | 0.04     | 0.31   | 1.17 | 4.57     |\n\n**W2:** Figure 3 lacks variances or confidence intervals\n\n**R2:** We have now added mean ± standard deviation values in the table below and the paper to better reflect the variance in performance. These were initially omitted as the standard deviations were consistently small:\n\n| Samples | RMSE ± Std |\n|---------|------------|\n| 5       | 0.60 ± 0.09 |\n| 10      | 0.54 ± 0.05 |\n| 50      | 0.39 ± 0.05 |\n| 100     | 0.36 ± 0.03 |\n| 500     | 0.32 ± 0.03 |\n| 1000    | 0.31 ± 0.02 |\n| 5000    | 0.30 ± 0.02 |\n| 10000   | 0.29 ± 0.03 |\n| 20000   | 0.29 ± 0.02 |\n| 50000   | 0.30 ± 0.02 |\n\n**W3:** Clarify runtime cost of MC estimation and lack of PNE runtime\n\n**R3:** The runtime bottleneck arises primarily in active learning, where epistemic uncertainty must be estimated for every input in a large candidate pool (e.g., 1000 inputs per acquisition step). MC-based methods require drawing K = 5000 per candidate then estimating mutual information, which becomes expensive in high-dimensional output spaces like Humanoid.\n\nThis cost scales with both the number of samples and the output dimensionality. As a result, MC estimation is especially burdensome under these settings.\n\nWe do not include a separate runtime plot for PNEs because they share a similar architecture to the base distribution used in NFlows Base. Both models sample from Gaussians whose parameters are predicted by neural networks. As shown in Appendix A.10 of [1], computing epistemic uncertainty in the base distribution space is equivalent to the output space. Therefore, the runtime of PNEs is effectively reflected in Figures 5 and 6. We’ve clarified this in the manuscript.\n\n**Q1:** Relationship between PairEpEst-KL and EPKL.\n\n**A1:** PairEpEst is a general framework that extends beyond KL-divergence, allowing practitioners to apply a range of premetrics, such as KL divergence, Bhattacharyya distance, Hellinger distance, total variation distance, and Wasserstein distance. This generality is important when KL is undefined between ensemble members.\n\nIn the specific case of KL divergence, we show that PairEpEst-KL provides a strictly tighter upper bound on mutual information than the EPKL estimator used in prior work [2,3]. This theoretical analysis has been added to the paper.\n\nWe formally prove that:\n\n$\\text{PairEpEst-KL} < \\text{EPKL}$\n\n**Theorem:** Let $\\( p_1(y), \\ldots, p_M(y) \\)$ be probability distributions. Define\n\n$A := \\frac{1}{M(M-1)} \\sum_{i \\neq j} D_{\\mathrm{KL}}(p_i(y) \\|\\| p_j(y)),$\n\nand\n \n$B := - \\frac{1}{M} \\sum_{i=1}^M \\ln \\left( \\frac{1}{M} \\sum_{j=1}^M \\exp \\big(- D_{\\mathrm{KL}}(p_i(y) \\|\\| p_j(y)) \\big) \\right).$\n\nThen,\n\n$B \\leq \\frac{1}{M^2} \\sum_{i,j} D_{\\mathrm{KL}}(p_i(y) \\|\\| p_j(y)) < A,$\n\nand in particular,\n\n$B < A.$\n\n**Proof:** Set\n\n$s_{ij} := - D_{\\mathrm{KL}}(p_i(y) \\|\\| p_j(y)) \\leq 0.$\n\nBy Jensen’s inequality applied to the concave function $\\ln(\\cdot)$, for each fixed $i$,\n\n$\\ln \\left( \\frac{1}{M} \\sum_{j=1}^M e^{s_{ij}} \\right) \\geq \\frac{1}{M} \\sum_{j=1}^M s_{ij}.$\n\n\nMultiplying both sides by -1 reverses the inequality:\n\n$- \\ln \\left( \\frac{1}{M} \\sum_{j=1}^M e^{s_{ij}} \\right) \\leq - \\frac{1}{M} \\sum_{j=1}^M s_{ij}.$\n\nSubstituting back for $s_{ij}$,\n\n$- \\ln \\left( \\frac{1}{M} \\sum_{j=1}^M \\exp\\big(- D_{\\mathrm{KL}}(p_i(y) \\|\\| p_j(y)) \\big) \\right) \\leq \\frac{1}{M} \\sum_{j=1}^M D_{\\mathrm{KL}}(p_i(y) \\|\\| p_j(y)).$\n\nAveraging over $i$, we get\n\n$B = \\frac{1}{M} \\sum_{i=1}^M \\left[- \\ln \\left( \\frac{1}{M} \\sum_{j=1}^M e^{-D_{\\mathrm{KL}}(p_i \\|\\| p_j)} \\right) \\right] \\leq \\frac{1}{M^2} \\sum_{i=1}^M \\sum_{j=1}^M D_{\\mathrm{KL}}(p_i \\|\\| p_j).$\n\nSince $D_{\\mathrm{KL}}(p_i \\|\\| p_i) = 0$, the sum over all $\\(i,j\\)$ equals the sum over $i\\neq j$:\n\n$\\sum_{i=1}^M \\sum_{j=1}^M D_{\\mathrm{KL}}(p_i \\|\\| p_j) = \\sum_{i \\neq j} D_{\\mathrm{KL}}(p_i \\|\\| p_j).$\n\nNote that\n\n$M^2 > M(M-1) \\implies \\frac{1}{M^2} < \\frac{1}{M(M-1)},$\n\nso\n\n$B \\leq \\frac{1}{M^2} \\sum_{i \\neq j} D_{\\mathrm{KL}}(p_i \\|\\| p_j) < \\frac{1}{M(M-1)} \\sum_{i \\neq j} D_{\\mathrm{KL}}(p_i \\|\\| p_j) = A.$\n\nHence,\n\n$B < A,$\n\ncompleting the proof.\n\n**Q2:** How point prediction was computed for RMSE for Nflows Base?\n\n**A2:** For each test input, we draw samples from the ensemble’s output distribution and compute the mean prediction. This mean is then compared to the ground-truth value to calculate RMSE.\n\n**Q3:** Why MC estimate is missing in Figure 2 and uncertainty appears different?\n\n**R3:** Figure 2 applies a min-max normalization to each estimator, scaling the values to the [0, 1] range for improved visual clarity. This makes it easier to interpret relative uncertainty and observe trends, such as lower uncertainty for points with fewer observations. For more details, please see our response to Limitations below.\n\n**Q4:** Color coding for significance in Table 1.\n\n**R4:** We included the full range of commonly reported significance levels in statistics to emphasize that none of the results reached the p < 0.05 or p < 0.001 thresholds. This also ensures consistency across the paper, as all colors are used in later tables where more significance levels are present.\n\n**Q5:** Why use uniform weighting of ensemble components?\n\n**R5:** We use uniform weighting because it is standard in ensemble learning and uncertainty estimation. While alternative weightings may be interesting (e.g., to induce certain multimodal properties), we are not aware of prior work exploring this in our context.\n\n**Limitations:** Does estimator bias affect active learning rankings?\n\n**R:** Thank you for this thoughtful point. To directly test whether bias alters active learning rankings, we computed Spearman’s rank correlation between our estimators (KL and Bhatt) and high-sample MC estimates on the data from Figure 8.\n\nSpearman’s coefficient measures monotonic agreement in rankings (1 = perfect correlation). Our results show values close to 1 across all settings, indicating that relative ordering is preserved. All p-values were extremely small (max p = 1.11e-83), confirming statistical significance.\n\nWe have updated the manuscript to reflect this and revised the original statement to clarify that while bias exists, its impact on active learning ranking is negligible in practice. Not that the order is preserved exactly.\n\nThis analysis has been added to the paper.\n\n| Model        | Dataset  | KL Corr | Bhatt Corr |\n|--------------|----------|---------|------------|\n| NFlows Base  | Bimodal  | 0.9958  | 0.9958     |\n| NFlows Base  | Hetero   | 0.9976  | 0.9986     |\n| PNEs         | Bimodal  | 0.9893  | 0.9893     |\n| PNEs         | Hetero   | 0.9943  | 0.9972     |\n\n**Formatting:** Excessive use of negative \\vspace under Figure 4. The figure description and main text nearly overlap each other.\n\n**R:** Thank you for noticing, we have corrected this in the manuscript. \n\n**References**\n\n[1] Berry, L., & Meger, D. (2023). *Normalizing flow ensembles for rich aleatoric and epistemic uncertainty modeling*. arXiv preprint arXiv:2302.01312\n\n[2] Malinin, A., & Gales, M. (2020). *Uncertainty estimation in autoregressive structured prediction*. arXiv preprint arXiv:2002.07650  \n\n[3] Schweighofer, K., Aichberger, L., Ielanskyi, M., & Hochreiter, S. (2023). *Introducing an improved information-theoretic measure of predictive uncertainty*. arXiv preprint arXiv:2311.08309"}]

Author rebuttal:

Meta-review: This paper is about the use of pairwise distance estimators in ensemble models for epistemic uncertainty estimation, which can be computed without performing sampling, providing computational advantages versus other standard methods that require sampling.

Reviewers appreciated the proposed method being sampling-free, good experimental evaluation and theoretical justifications. The negatives are the limited set of baselines and comparison using more complex datasets.

There was a good amount of discussion and all reviewers vote to accept. This paper makes a good contribution of providing a uncertainty estimation method that does not require sampling, being much faster than alternative methods, which is a key weakness of uncertainty estimation methods and improving computational complexity is required for real-world application use.

I recommend that the authors consider the feedback from reviewers in their final version, in particular from reviewers 5FYN about pitfalls in evaluating active learning, and comparisons with standard baselines like MC-Dropout (Reviewer q5B8), clarify some figures (Reviewer yg5h) and more complex datasets, which can provide more information for future research.
Decision: Accept (poster)