HybridFlow: Quantification of Aleatoric and Epistemic Uncertainty with a Single Hybrid Model
Authors: Peter Van Katwyk
Karianne Bergen
Venue: Accepted by TMLR ()

Abstract: Uncertainty quantification is critical for ensuring robustness in high-stakes machine learning applications. We introduce HybridFlow, a modular hybrid architecture that unifies the modeling of aleatoric and epistemic uncertainty by combining a Conditional Masked Autoregressive normalizing flow for estimating aleatoric uncertainty with a flexible probabilistic predictor for epistemic uncertainty. The framework supports integration with any probabilistic model class, allowing users to easily adapt HybridFlow to existing architectures without sacrificing predictive performance. HybridFlow improves upon previous uncertainty quantification frameworks across a range of regression tasks, such as depth estimation, a collection of regression benchmarks, and a scientific case study of ice sheet emulation. We also provide empirical results of the quantified uncertainty, showing that the uncertainty quantified by HybridFlow is calibrated and better aligns with model error than existing methods for quantifying aleatoric and epistemic uncertainty. HybridFlow addresses a key challenge in Bayesian deep learning, unifying aleatoric and epistemic uncertainty modeling in a single robust framework.

Published in Transactions on Machine Learning Research (10/2025)
HybridFlow: Quantification of Aleatoric and Epistemic Un- certainty with a Single Hybrid Model
Peter Van Katwyk
peter van van_ katuykkbrouumeee
Deportment of Earth, Environment, and Planetory Sciences, Brown University Duta Science Institute, Brown Universily
Karianne J. Bergen
koriarne bergen&brroun. edu
Deportment Earth, Enviromment, and Planetary Sciences, Brown University Data Science Institute, Brown University Department of Computer Science, Brown University
Reviewed on OpenReview: /ttps:// net/ forum? id= aBsSddyVVj
Abstract
Uncertainty quantification is critical for ensuring robustness in high-stakes machine learn- ing applications. We introduce HybridFlow, modular hybrid architecture that unifies the modeling of aleatoric and epistemic uncertainty by combining Conditional Masked Autoregressive normalizing flow for estimating alcatoric uncertainty with a flexible proba- bilistic predictor for epistemic uncertainty. The framework supports integration with any probabilistic model class, allowing users to easily adapt HybridFlow to existing architectures without sacrificing predictive performance. HybridFlow improves upon previous uncertainty quantification frameworks across a range of regression tasks, such as depth estimation, collection of regression benchmarks, and scientific case study of ice sheet emulation. We also provide empirical results of the quantified uncertainty, showing that the uncertainty quantified by HybridFlow is calibrated and better aligns with model error than existing methods for quantifying aleatoric and epistemic uncertainty. HybridFlow addresses key challenge in Bayesian deep learning, unifying alcatoric and epistemic uncortainty modcling in a single robust framework.
1 Introduction
Uncertainty quantification plays critical role in modern machine learning, particularly in high-stakes appli- cations such as autonomous driving (Grewal al. 2024), medical diagnosis (Jalal et al. 2024), and scientific modeling (Wang et al. 2023). The ability to estimate uncertainty allows models not only to make predictions but also to provide insights into the reliability of those predictions. This capability is especially valuable in scenarios where decision-making depends heavily on understanding the limitations and confidence of a model's outputs (Thuy & Benoit 2024 Marusich et al.) 2024 Amodei et al.) 2016). In recent years, re- searchers have increasingly focused on improving uncertainty estimation methods deep learning achieve more robust and trustworthy systems (Gal & Ghahramani 2016 Kendall & Gal [017 Mcssuti et al.) 2025;
Yoon & Kim 2024: Hwang Shin 2024). Despite significant progress, key challenges remain, particularly in achieving accurate predictions while effectively quantifying different types of uncertainties [Psaros et al. 2023
In the context of machine learning, uncertainty is broadly calegorized into two lypes: aleatoric uncerlainly, which arises from the inherent noise and variability in data, and epistemic uncertainty, which reflects the model's lack of knowlodge, often due to limited or biased training data (Der Kiureghian & Ditlevson 2009 Gruber al.] 2023). While alcatoric uncertainty is irreducible, epistemic uncertainty can, in principle, be reduced by acquiring additional data or improving the model architecture. Techniques such as Monte Carlo

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(MC) Dropout (Gal & Ghahramani 2016) and ensemble methods [Lakshminarayaaan et al. 2017 have been developed to estimate epistemic uncertainty, while approaches like density estimation (Bishop 1994p and heteroscedastic regression (Kendall & Gal 2017) aim to model aleatoric uncertainty. Despite advancements in single-model frameworks, they often prioritize state-of-the-art uncertainty quantification at the expense of modularity and flexibility, which are critical for integrating uncertainty estimation inlo diverse existing architectures. For example, methods such as heteroscedastic regression (Kendall & Gal] 2017) are widely adopted due to their simplicity and ease of integration, even though they often suffer from calibration issues and suboptimal predictive accuracy (Seitzer et al.) 2022). Heteroscedastic regression methods remain popular because they can be applied to diverse predictive models without significant architectural changes.
This paper introduces a novel hybrid flow-based architecture, HybridFlow, which addresses these challenges by quantifying both aleatoric and epistemic uncertainty without compromising predictive performance. This framework integrates a normalizing flow (NF) (Rezende & Mohamed] 2015) to model alcatoric uncertainty with any probabilistic predictor model for epistemic uncertainty estimation and model prediction. Hybrid- Flow enables precise and reliable uncertainty quantification through leveraging both inpul data and the latent space generated by the NF as model inputs. By decoupling the aleatoric uncertainty quantification mecha- nism from the prodictor model itself, the proposed HybridFlow framework is inherently modular, serving as a flexible method that can be adapted to a wide range of applications. While the current implementation demonstrates its efficacy using specific benchmarks and case studies, the design of the framework allows for the substitution of different predictors tailored to other tasks or domains.
We evaluate the HybridFlow model framework against three widely adopted frameworks (Kendall & Gal, 2017 Seitzer et al., 2022; Caprio et al.] 2024) for quantifying both alcatoric and epistemic uncertainty in single model, and demonstrate that our proposed framework provides accurate predictions and quantified uncertainty for range of tasks: depth estimation in computer vision and a series of regression benchmarks. We also include a scientific case study, where we test the HybridFlow architecture as an emulator for future sea level risc. Another key contribution of this work is that we provide a comprehensive evaluation of quantified uncertainly, which is often overshadowed by the analysis of prediction accuracy alone (Wang) et al.) 2025). We employ a diverse set of metries that collectively assess different aspects of uncertainty estimation to compare the quantified uncortainty from various methods. The proposed framework showcases direct improvement in both predictive accuracy and uncertainty quantification over existing methods, providing a compelling step toward single, flexible models that can effectively model both aleatoric and epistemic uncertainties while maintaining high predictive accuracy.
2 Related Work
Uncertainty quantification has received significant attention in Bayesian deep learning due to its importance in improving model reliability and interpretability. Uncertainty quantification techniques can be model- agnostic, meaning that they can be incorporated into a variety of predictive architectures, or model-specific, where specific constraints are placed on the predictive architecture in order to quantify uncertainty. [Kendall & Gal 2017)) laid the foundation for model-agnostic uncertainty quantification by modeling both aleatoric and epistemic uncertainty within a single framework, specifically the context of computer vision, by using a combination of existing Bayesian approaches and a heteroscedastic log likelihood loss for modeling aleatoric uncertainty. Their approach adds prediction layer and variance layer at the end of a predictor model, such that the model predicts both the mean juxx and variance oxx) for a given input x, with the assumption that the uncertainty follows a Gaussian distribution. To find the optimal weights, maximum likelihood estimation is used, which is equivalent to minimizing the negative log-likelihood (NLL), or heleroscedastic loss, of the predictive distribution:
1
C
II mxx;)12+ + log o² (xx:)
N
20
For the predictor model, Kendall & Gal 2017 integrate MC Dropout (Gal & Ghahramani 2016) to es- timate epistemic uncertainty, and demonstrate the effectiveness of heterosccdastic regression in tasks such as semantic segmentation and depth regression. This approach enables fiexibility, allowing for uncertainty quantification with minimal adjustment to existing predictor model architectures (Smith et al. 2024 How-
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ever, this method is prone to calibration issues, particularly under distributional shifts or when noise patterns deviate from Gaussian assumptions. Additionally, reliance a single combined loss function often leads to overestimation of one type of uncertainty at the expense of the other.
Recent research in model-agnostic uncertainty quantification techniques has highlighted the challenges and pitfalls of heteroscedastic uncertainty estimation. Specifically, (Sitzer et al. (2022) critiqued this standard approach to modeling aleatoric uncertainty, emphasizing the tendency of NLL-trained probabilistic neural networks to underestimate uncertainties in the presence of aut-of-distributio data highly skewed noise distributions. They show that the use of the NLL loss is appropriate if the difference between the model prediction and the true value is solely caused by noise, or alcatoric uncertainty. However, in practice, the inability of the model to perfectly approximate the phenomena may also contribute to error, which leads to an overestimation of aleatoric uncertainty when the data is poorly predicted by the model (Seitzer et al. 2022) To address this, they introduce the Beta-NL.I. loss, a refinement of the NLL loss that adjusts the balance between MSE and uncertainty estimates to address sub-optimal fits and training instability. This method represents an effort to mitigale calibration challenges while retaining the simplicity of the heteroscedastic regression framework. While performance may improve in some tasks, the method remains unstable and fails to resolve miscalibration caused by joint loss functions. Others have attempted to address miscalibration and the decrease in predictive accuracy using differing probabilistic predictors or post-hoc methods, but they similarly encounter comparable trade-offs between predictive accuracy and uncertainty quantification capabilities (Valdenegro-Town & Mori 2022
Yang & Li 2023
These limitations have motivated shift toward more expressive generative approaches for modeling uncer- tainty, particularly those that can capture complex, non-Gaussian output distributions. Normalizing flows [Rezende & Mohamed, 2015) have frequently been used to estimate aleatoric uncertainty by directly esti- mating the conditional output distribution (e.g., (Acharya et al. (0225); Zhang et al. (2024a): Ancha et al. (2024h). To model both aleatoric and epistemic uncertainty using NFs, methods such as NFlows Base and NFlows Out (Berry & Meger 2023) and FlowNet (Zhang et al. 2024b) use NFs to capture both alcatoric uncertainty (Irom the learned distribution) and epistemic uncertainty (via ensembling). While these met hods demonstrate the expressive capabilities of NFs for modeling uncertainty, they either focus solely on aleatoric uncertainty, or require the flow itself to serve as the predictive model, making the approach inherently model- specific and limiting integration into existing predictors. In contrast, HybridFlow uses the normalizing flow as density estimator to model aleatoric uncertainty, while allowing prediction and epistemic uncertainty quantification to be estimated using a probabilistic predictor. To our knowledge, HybridFlow is the first to decouple alcatoric and epistemic uncertainty in this way using conditional flows, while retaining archi- tectural modularity and general-purpose applicability. This decoupled design offers greater flexibility and compatibility across applications, particularly where domain-specifie predictors are already better suited for prediction tasks.
Other model-specific approaches demonstrate similar tradeoffs between flexibility and performance. For example, Credal Bayesian Deep Learning (CBDL) (Caprio et al. 2024) uses credal set of Bayesian neural networks (BNNs), with prescribed set of priors to estimate aleatoric and epistemic uncertainty. CBDL often achieves strong calibration by averaging over the credal set of network, but this approach entails significant computational overhead and requires specialized inference protocols that can be difficult to integrate into existing pipelines. Similarly, Evidential deep learning methods (Sensoy et al.] (018)) learn the paramcters of higher-order distributions directly from the data, but require specialized architectures and loss functions that restrict their usability. Chan et al. (2024) introduce HyperDM,, which employs Bayesian hyper- network to generate an ensemble of model weights and combines it with a conditional diffusion model to estimate predictive distributions. These methods achieve crate-of-the-art uncertainty quantification but require specific model architectures, limiting their practicality for users aiming to enhance existing models or create models for specific tasks and domains where the predictor proposed in these studies may be unfit.
Unlike the aforementioned methods, which integrate uncertainty quantification deeply into the custom ar- chitectures, methods such as our proposed HybridFlow framework and heteroscedastic regression (Kendall & Gal 2017 Seitzer et al. 2022 are flexible by design, allowing integration of uncertainty quantification into any type of probabilistic predictive model. The importance of this capability is evident in the wide-spread
adoption of heteroscedastic regression methods (Roohani et al. 2024 Huang et al. 2023 Yelleni et al. 2024).
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custom models with complex, nonflexible model architectures. In this work, we introduce HybridFlow; hybrid model architecture that surpasses prior flexible heteroscedastic regression frameworks by providing accurate, calibrated prediction and uncertainty estimates while avoiding combined loss function training and generalizing beyond Gaussian assumptions. This enables users to incorporate uncertainty modeling without being tied lo specific predictor architeclures or objectives. This approach aligus with the practical needs of many users who primarily aim to enhance existing predictors with uncertainty quantification, rather than adopting entirely new architectures that are not specific to applications or domains. HybridFlow is able to provide balance between advanced uncertainty quantification and compatibility with diverse predictive frameworks.
HybridFlow remains modular and flexible by using a hybrid model architecture, as introduced in Nalis- nick et al. 2019). Nalisnick et al. (2019) integrate a generative model with a predictive model, enabling exact computation of the joint distribution p(r, v). Their work demonstrates theoretically principled in- tegration of tasks but does not explicitly disentangle alcatoric and epistemic uncertainty, and relics on a non-probabilistic generalized linear model for prediction, which limits the expressiveness of the predictive component. HybridFlow builds on Nalisnick et al.] (2019) by explicitly using the generative component to model alcatoric uncertainty and by introducing flexibility in the choice of predictive architectures.
3 Method
HybridFlow builds upon recent advancements in uncertainty quantification and generative modeling to ad- dress existing challenges in modeling both aleatoric and epistemic uncertainty in single model. It leverages NEs (Rezende & Mohamed 2015; Papamakarios et al., 2017) for modeling aleatoric uncertainty while inte- grating probabilistic prediction model into a hybrid architecture (Nalismick et al. 2019) for both accurate predictions and epistemic uncertainty quantification. This section describes the components of HybridFlow and their integration into a unified framework for uncertainty quantification.
3.1 Conditional Normalizing Flows for Aleatoric Uncertainty
Conditional Normalizing Flows are class of generative models that transform a simple base distribution p(z (() typically a conditional Gaussian [Bishop] 2006), into more complex target distribution p(y|z) z) through a series of invertible transformations f (Papamakarios et al. 2017 2021 In the case of conditional NEs, the transformation is parameterized based on the conditioning variable x:
y = /<(lx) ~~ pr>(()))
(2)
where f is designed to be invertible, ensuring that the probability density function of y given x can be computed via the chango-of-variale variables formula:
p(y[=)= (x((-"(vlx))) det of [))
(3)
Oy
where (y(*( 8y represents the Jacobian of the transformation f-1 conditioned on The Jacobian matrix captures how the volume of the transformed space changes under the inverse mapping f f-1 The absolute value of its determinant accounts for this local volume change and ensures the resulting density p(y)x) x)) remains normalized.
In HybridFlow, Conditional Masked Autoregressive Flow (CMAF) (Papamakarios et al. 2017) is used. The CMAF decomposes the conditional joint distribution p(y(a) into product of autoregressive conditionals:
p(y x)) II pl 1)).,)))
(4)
where d is the dimensionality of y, and each conditional p(y. (okl x)) parameterized by a neural network. This conditional structure allows the flow to explicitly learn the distribution of y given æ, capturing both the inherent noise in y and its dependence sc. Learning ply ax) allows for the calculation of aleatoric
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and the invertibility of the flow enables us to produce latent representation of the target conditioned on the inputs, 2 = -y'((] which serves as an additional feature for the predictor model (Section 3.2p. During training, for a given ground-truth pair (a, y), is deterministically computed using the inverse of the flow. At inference time, however, the ground-truth y is unknown, so z is calculated by taking the expectation over the learned conditional base distribution p(2|z) from the flow. We use a CMAF, as opposed to other NF architectures, due to their strong performance on conditional density estimation tasks, efficient likelihood computation, stable training dynamiis,, and wide adoption (Papamakarios et al,, 2021). However, other flow types (e.g., c-GLOW (Lu & Huang 2020), RealNVP Dinh et al. (2017)) can also be used in HybridFlow when parameterized conditionally. The framework is agnostic to the specific flow architecture, and different flows may be preferable depending on task complexity, output dimensionality, or computational budget.
uncertainty (Section 3.3
While normalizing flows are highly effective generative models for capturing complex conditional distribu- tions, they are not typically used as predictors (Kobyzov et al., 2020). Their primary objective is to model the full data distribution p(y|ær) rather than to optimize predictive accuracy for specific target variable. In contrast, predictive models are explicitly trained to minimize prediction error and can incorporate mecha- nisms for estimating epistemic uncertainty. By introducing dedicated predictor, HybridFlow leverages the strengths of both components: the NF captures the uncertainty inherent in the data, while the predictor focuses on producing accurate point estimates and quantifying model-based uncertainty.
3.2 Hybrid Architecture
To achieve both predictive accuracy and robust uncertainty quantification, the latent space representation z generated by the conditional NF is passed, along with the original inputs I, into a probabilistic predictor, as seen in Figure [ In practice, x and z are concatenated at the feature level; for tabular datasets we directly concatenate the raw input vector with 2, and for digh-dimensional inputs (such as images), feature extractor (e-g., CNN with average pooling) is used to learn a compressed feature representation ((x)) that is concatenated to before being inputted to the predictor (for more details, see Appendix A]
The probabilistic predictor models the distribution p(y x, where y represents the prediction target. By combining > (the latent encoding capturing data-specific uncertainty) and x (the original inputs), the predictor jointly learns the relationship between the inputs and the outputs while retaining uncortainty information from the NF. The design HybridFlow flexible, such that any probabilistic predictor can be used. For example, methods such as MC Dropout (Gal & Ghahramani (0066), Bayesian neural networks, deep ensembles (Lakshminarayanan et al.) 2017). and other Bayesian methods are all viable predictor models that can be implemented within the hy brid architecture.
In the HybridFlow framework, the predictive mean p(z, 2) EEy | æ represents the model's prediction, while the epistemic uncertainty (z, = Varfyl [, x] quantifics the uncortainty due to the model's limited knowledge. The choice of the probabilistic predictor can be tailored based on the application's requirements for computational efficiency, model complexity, or interpretability.
3.3 Aleatoric Uncertainty Estimation
The aleatoric uncertainty is captured directly from the learned data distribution modeled by the conditional NF. By sampling from the posterior distribution p(y x) Icarned by the CMAF, we define the sampled predictions as y ~ p(y 2)). The aleatoric uncertainty is then computed as the variance of these samples, (z) Var(ğ).
This variance can be considered alcatoric because it arises from the distribution P(y]æ) that the flow explicitly learns from the data. The NF is trained to model the full conditional distribution of possible outcomes given each input, including any inherent noise or ambiguity in the data. When we sample multiple outputs for the same input, the spread of those outputs reflocts the learned variability within the data itsclf. Thus, the variance of these samples naturally corresponds to alcatoric uncertainty, as it captures the irreducible variability in outcomes.
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Prediction Maa, Ry s.
Probabilistic Predictor
(learns p 22, 2))
Epistemic
Depth Estimator
Var [
variance
Input data (E))
variance of samples
Aleatoric
dita Var(y)
Sample 1 ~ p(y x) from the learned density
Conditional Masked Autoregressive Flow (learns ( + 2)
Figure 1: HybridFlow architecture, with depth estimation with the NYU Depth v2 dataset as the example task. Conditional Masked Autoregressive Flow (CMAF) learns the density of input data (x) through a latent representation (z). The latent representation and the input data are concatenated and used as inputs to probabilistic predictor model. From the predictor model, the prediction and the epistemic uncertainty can be estimated. The aleatoric uncertainty is estimated by calculating the variance of samples generated from the learned data distribution by the CMAF.
3.4 Training Protocol and Implementation Considerations
A summary of the HybridFlow workflow is follows:
1. Data Transformation:
The CMAF-based NE, conditioned on inputs is used to transform the
inputs into latent representation z.
2. Prediction and Epistemic Uncertainty:
The latent = and input x are fed into a probabilistic predictor that outputs the mean prediction paz,, z) and epistemic uncertainty o² (x, 2). Alcatoric Uncortainty: Sampling from ply x), the alcatoric uncortainty (2(()) is estimated
the variance of the samples.
HybridFlow is trained in two-stage process that reflects its hybrid architecture. First, the normalizing flow (NF) component is trained independently to learn the conditional distribution p(yl æ)) which is used to quantify aleatoric uncertainty. The NF is optimized using the NLL loss of the conditional distribution (Equation 3), standard and widely adopted objective for training normalizing flows to model the full data distribution. After training the flow, the latent representation 2 = f (y ()) extracted and used as an additional input to the predictor model, along with the original input æ. Crucially, because the inverse mapping of the flow is conditioned on x, the resulting z encodes structured information about the conditional distribution, rather than being independent noise. This process ensures z is meaningful representation, distinguishing it from the generative use of the flow (Step 3), where y is sampled from the learned posterior distribution p(y(æ) to estimate alcatoric uncertainty (sce Appendix B During the second stage, the predictor model is trained using a tapk--ppe loss function, such as mean squared error (MSE), and optionally paired with a Bayesian approximation method to estimate epistemic uncertainty.
The NF can be optionally finc-tuned during the training of the prodictor model to allow improved joint representations, though in most applications the flow converges independently and does not require additional updates. Similarly, the predictor model can be fine-tuned after integration with the NF outputs. HybridFlow

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supports any probabilistic predictor architecture for epistemic uncertainty estimation; in experiments, we use MC Dropout for simplicity and compatibility with prior work, but the framework can accommodate deep ensembles, variational inference, or other Bayesian or Bayeesan-approximatton techniques.
A key consideration when implementing HybridFlow is the dimensionality of the target variable y, which constrains the use of normalizing flows. High-dimensional output spaces increase computational and memory requirements due to the need for tractable Jacobian determinants in the NF. In such cases, the NF can be paired with a feature extractor or autocncoder to first compress the output space into a -imensinnni latent representation, ensuring that the method remains tractable for large-scale or high-resolution outputs. In practice, we find that moderate-sized flows (4 6 layers) suffice for the regression tasks we tested, with deeper flows offering diminishing returns (see Appendix C for ablation study). For more complex datasets, additional flow depth or hidden capacity may improve performance, but a computational cost. that scales approximately linearly with the number of flow layers and size of the flow output (Papamakarios et al., 2021 Bhattacharyya et al.
demonstrate in Section 4.1 This preserves the flow's modeling capacity while
2020).
4 Experiments
This section presents the evaluation of the HybridFlow architecture on various distinet datasets, including the NYU Depth v2 dataset for depth estimation and 12 UCT regression benchmarks. The experiments focus assessing both predictive accuracy and uncertainty quantification capabilities. We benchmark HybridFlow against a bascline model based on the framework Kendall & Gal (017)), a similar framework from Seitzer et al. b022) using the Beta-NLL (BNLL) loss, as well as a CBDL model, a of model-specific uncertainty quantification framework (Caprio et al. 2024
4.1 Depth Estimation
We use the NYU Depth v2 dataset Silberman et al. 2012 which is widely used for evaluating models in computer vision tasks, and the primary benchmark in Kendall & Gal (2017). It consists of 400,000 high- dimensional RGB and depth image pairs, covering a variety of envirouments such as offices, homes, and classrooms. For our experiments, we used preprocessed subset comprising 795 training images and 654 test images, following the standard splits used in previous studies. Each RGB image is paired with a depth map representing the ground truth distances of cach pixel from the camera. The diversity and complexity of the scenes, combined with the inherent measurement noise in depth sensors, make this dalaset ideal for evaluating both aleatoric and epistemic uncertainty.
We train four models for quantifying alcatoric and epistemic uncertainty, two baseline heteroscedastic regres- sion models, one with a Gaussian NLL loss (Kendall & Gal 2017). another with a Beta-NLL loss (Seitzer et al. 2022), a deep ensemble of heteroscedastic Gaussian NLL loss models [akshminarayanan et al. (2017). and a hybrid regression model with the proposed HybridFlow architecture. For the heteroscedastic regres- sion models, we include a 2D Convolutional layer at the end of the predictor model to output the mean (prediction) and variance (aleatoric uncertainty) and train with their respective NL.L-based losses. The deep ensemble of heteroscedasttcc regression models includes an ensemble of 5 Gaussian NLL models with the same architecture as the single- model NLL baseline. Each model was initialized and tuned following practices out- lined in Takshminarayanan et al. (207)). For the hybrid regression model, we use a CMAF paired with a UNet-basod autoencoder to quantify the alcatoric uncertainty, sampling p(y x)) with a sample size of 100 and computing the variance (following Section 3.3 While the autoencoder is not a necessary component of HybridFlow, use autoencoder to compress the input images (which are 384 x 512 x 3) into a latent space of size 256, thus demonstrating the proposed approach is computationally manageable when working with high-dimensional data (for more information on the autoencoder, see Appendix Aj. We then use the generated latent. representation from the CMAF and the RGB image as inputs to the predictor model.
For all experiments, we use a predictor model presented in Ganj et al. (2025). which achieves state-of- tho-arr performance on the NYU Depth v2 dataset across key metrics such as Absolute Relative Error (AbsRel) and Root Mean Squared Error (RMSF). Following Kendall & Gal 2017) we implement. MC Dropout (Gal &
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Ghahramani 2016 to make the Ganj et al. (2025) predictor probabilistic for the quantification of epistemic uncertainty. For both Depth Regression and the UCT benchmarks we choose MC Dropout due to the simplicity of incorporating into existing models and to be consistent with the implementations of the baseline model in Kendall & Gal (2017). However, we include a comparison to a deep ensemble (Lakshminarayanan et al. 2017) of heteroscedastic regression models, to showcase that the performance improvements from the hybrid architecture will still be seen, regardless of the specific approach for quantifying epistemic uncertainty. For reference, we have also included a non-probabilistie version of the Ganj et al. (2025) model (without the ability to quantify uncertainty) in Table I in order to compare the effect that quantifying uncertainty has on the predictive performance. We also highlight that our implementation results show improvements over Kendall & Gal (2017), solely because we use a current state-of-the-art predictor model.
Despite its strong performance on a variety of uncertainty quantification tasks (Caprio et al. 2024). CBDL cannot be applicd in this depth estimation task because the underlying UNet-based predictor for the depth estimation task is not Bayesian model. As a result, CBDL's ensemble of variational guides cannot be constructed for a purely deterministic architecture, which limits its flexibility. This incompatibilityy repre- sents significant drawback of model-specific uncertainty frameworks when extending to non- non- non-Bayesian deep learning modules.
Training was conducted for 100 epochs using a batch size of 16, with the Adam optimizer (Kingma & Ba 2015), a learning rate of le-4, and a weight decay of 1e-3. For the NF components, we used a CMAF with flow layers and a conditional diagonal Gaussian base distribution using nflows (Durkan et al.] 2020 We train the CMAF flow with a NLL loss prior to training the prediction model. Then, during the training stage of the predictor, we fine-tune the flow with lower learning rate (1e-6) to integrate the latent representation into the hybrid architecture.
The probabilistic predictor for HybridFlow was configured with MC Dropout at a rate of 0.2 for epistemic uncertainty quantification, and we performed 50 stochastic forward passes at test time to approximate the predictive mean and epistemic variance. The heteroscedastic regression models minimized a Gaussian NLL and Beta-NT.L losses, while the HybridFlow model minimized a combined loss consisting of the flow's negative log- likiiehhoodd and a scale-invariant log loss (Eigen et al., 2014) and multi-scale gradient loss (Ganj et al., 2025) for the prodictor.
All training was conducted on a single Tesla V100 GPU, with the heteroscedasstic models requiring approx- imately 25 hours to converge and HybridFlow requiring 36 hours. The models were evaluated using the standard metrics for depth cstimation, including MSE, AbsRel, and thresholds (8z=z), which indicate the proportion of predictions within 25%, 56%, and 95% of the ground truth values (Eigen et al. 2014).
In this study, we include a robust analysis of the quantified uncertainty from the approaches tested. Following Wang et al. (2025), we evaluate calibration quality using the Expected Calibration Error (ECE), which measures the average discrepancy between the confidence intervals and predicted accuracy, and the Prediction Interval Coverage Probability (PICP) which reports the fraction of true values contained within the computed uncertainty intervals. To assess the sharpness of our uncertainty estimates, we compute the Mean Prediction Interval Width (MPIW), capturing the typical size of the predictive intervals, and the Winkler score, which penalizes both overly wide and under-covering intervals. Additionally, we apply strictly proper scoring rules (Gnciting & Raftery 2007). such as the Negative Log Likelihood score (NLL) and the Continuous Rankod Probability Score (CRPS), which penalize miscalibration in the full predictive dist Total uncertainty, within the scope of this work, is the sum of the aleatoric and epistemic uncertainty, which is common in uncertainty quantification literature (Depeweg et al. 2018 Hüllermeier & Wacgeman 2021 Wimmer et al. 2023). Together, these metrics furnish a concise yet comprehensive characterization of aleatoric and epistemic uncertainty across predictive accuracy, calibration, sharpness, and reliability. Definitions for each uncertainty metric can be found in Appendix D
4.2 UCI Regression Benchmarks
To evaluate the robustness and accuracy of HybridFlow, we compared against the same hetoroscodastic regression methods on the UCI regression datasets, as detailed in Section 4.1 and CBDL model (Caprio et al. ) The UCT datasets widely recognized benchmark suite for regression tasks, offering diverse
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data distributions, varying noise levels, and differing scales that are ideal for evaluating models' predictive accuracy and uncertainty estimation capabilities.
We follow a standard experimental protocol for Bayesian deep learning frameworks (Seitzer et al., 2022; Hernández-Lobato & Adams 2015 Gal & Ghahramani 2016), using the same datasets and splits. Each dataset was evaluated 20 times with random 80/20 splits for training and testing, normalizing input features and target outputs to zero mean and unit variance based on the training split. For more experimental details, see Seitzer et al. 2022
We train a flow using the HybridFlow framework with a CMAF with 5 flow layers. We first train the flow alone using a NLL loss. Then, the latent representations generated by the flow were combined with the original inpuls and fed into the predictor model, which incorporates MC Dropout for modeling epistemic uncertainty. The predictor model used for all three methods is a simple MLP with one hidden layer of 50 ReLU-activated units that is trained using an MSE loss and carly stopping, as in Seitzer et al. (2022).
HybridFlow is compared to heteroscedastie regression models trained with Gaussian NLL loss (Kendall & Gal 2017 and Beta-NLL loss (Seitzer et al. 2022). well trained with an MSE loss. We conduct the experiments using a Tesla V100 GPU with an Adam optimizer and perform grid search of possible learning rates ranging from 1e-3 to 1e-5. To quantify epistemic uncertainty, we applied MC Dropout with a dropout probability of 0.2 and performed 30 stochastic forward passes during evaluation. In HybridFlow, Flow, we sample the learned posterior distribution 100 times to estimate the aleatoric uncertainty learned by the CMAF.
non-probabllistic version of the model
In addition to the heteroscedastic regression baselines, we also benchmark against CBDL (Caprio et al. 2024p. We implement CBDL a credal set of Bayesian models in Pyro (Bingham et al. 2019) that closely resemble the MLP models used for HybridFlow and heteroscedastic regression baselines. We initialize four BNNs, each with one hidden layer of 50 Tanh activated units, by varying the Gaussian prior scale and likelihood over the parameters and train each of them using stochastic variational inference (SVI). Alcatoric and epistemic uncertainties are computed using individual model entropies and variability within the credal set as outlined in Caprio et al. (0044).
We evaluate the predictive accuracy of each model using the mcan RMSE value and the standard deviation of the 20 variations of the model based on the random splits. We then evaluate the uncertainty quantification performance of each model using a robust selection of metrics that collectively capture critical aspects of uncertainty estimation including the NLL, ECE, CRPS, PICP, and MPIW, as described in Section 4.1
5
Results
5.1
Depth Estimation Results
HybridFlow demonstrates superior predictive accuracy on the NYU Depth v2 dataset, outperforming base- line models that use heteroscedastic regression methods. As seen Table I the HybridFlow model achieves improved results across several evaluation metrics, including MSE, AbsRel, and 8 thresholds (ōi-dg). The
Table 1: Comparison of hybrid depth estimation model with the heteroscedasticc depth estimation models (with MC Dropout to quantify uncertainty) and a deep ensemble of NLL models using NYU Depth v2. The Ganj et al. (2025) model does not quantify uncertainty, but is included to compare the predictive accuracy changes when adding uncertainty quantification capabilities.
TRAINING METHOD
MSE
ABSREL
81
822
55
NLL KENDALL & GAL
2017
0.129
0.083
0.947
0.987
0.995
BNLL SETTZER ET AL.
2022
0.134
0.087
0.947
0.987
0.994
DEEP ENSEMBLE
0.166
0.100
0.927
0.985
0.994
HYBRIDFLON
0.058
0.041
0.989
0.998
0.999
GANJ ET AL.
2025) (NO UQ)
0.057
0.039
0.989
0.998
0.999
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Table 2: Uncertainty quantification metrics on the NYU Depth v2 datasets. Metrics include NLL (Negative Log- LigeLiikehho)) calibration error (ECE), sharpness, total PICP (Prediction Interval Coverage Probability), CRPS (Continuous Ranked Probability Score), and MPIW (Mean Prediction Interval Width).
METHON
NLL
ECE
WINKLER
MPIW
PICP CRPS
ALE.
EPI.
TOT.
ALE.
EPI.
ToT.
ALE.
EPI.
ToT.
ALE.
EPI.
TOT.
NLL
5.79
10.88
7.39
4.42
2.98
3.07
14.53
11.51
10.16
5.38
0.035
4.04
0.97
2.71
BNLL
2.95
3.37
2.73
2.32
10.96
4.63
6.18
10.52
8.23
3.70
0.78
3.60
0.92
1.66
DEEP ENSEMBLE
6.10
10.22
8.18
3.70
12.25
4.38
10.02
9.78
9.35
5.93
0.061
3.87
0.91
3.17
HYBRIDFLOW
1.96
0.09
0.73
0.68
4.14
0.46
4.82
7.46
4.83
4.70
0.64
4.76
0.99
0.37
Hybrid model achieves an MSE of 0.058 and AbsRel of 0.041, which are improvements over the Gaussian NLL framework (MSE: 0.129, AbsRel: 0.083) and the Beta-NLL approach (MSE: 0.134, AbsRel: 0.087). No- tably, the performance is on par with the Ganj et al. (2025) predictor model when uncertainty quantification is excluded (MSE: 0.057, AbsRel: 0.039), illustrating that the integration of uncertainty estimation ion does not compromise predictive accuracy. We also highlight that the choice of epistemic uncertainty quantification method does not greatly impact the predictive performance, as can be seen by the comparison of NLL re- sults to the Deep Ensemble. We attribute the slightly higher error observed in the Deep Ensemble known training instabilities of heteroscedastic NLL models (Seitzer et al. 2022 Stirn et al. 2023 Wong-Toi et al., 2024), where some ensemble members converge to suboptimal minima, and following the Deep Ensemble protocol (Takshminarayanan et al. 2017), we retain these members rather than restarting or re-tuning them, which increases error relative to the best single model.
HybridFlow also exhibits robust uncertainty quantification capabilities, addressing both alcatoric and epis- temic uncertainties effectively. The aleatoric uncertainty, modeled using a CMAF in the hybrid architecture, captures inherent noise in the dataset, while epistemic uncertainty is quantified through MC Dropout in the predictor model. As shown in Table 2 HybridFlow achieves consistently better alcatoric ECE and PICP with epistemic ECE on par with the other models, indicating that its uncertainty estimates are better cali- brated than those of existing methods. It also shows improved performance across other uncertainty metrics, including lower CRPS and NLL, and more reliable predictive intervals as measured by the Winkler score. While HybridFlow has slightly wider uncertainty intervals on average (as reflected by MPIW), this trade-off supports better coverage and reliability.
Qualitative analyses of depth maps further support the quantitative findings, the HybridFlow model pre- dicts sharper depth edges and maintains consistency across scenes with varied lighting and texture conditions (Appendix E. HybridFlow's uncertainty maps reveal elevated aleatoric uncertainty for areas with abnormal lighting or occlusion boundaries, consistent with regions expected to pose challenges due to alcatoric noise. For epistemic uncertainly, higher uncertainty is more localized to objects or textures that are underrepre- sented in the training dataset. We also see that the aleatoric uncertainty predicted by the heteroscedastic mothods closcly resembles the predicted depth, which may indicate that the model is unable to effectively learn noise distributions at variable depths.
5.2
UCI Regression Benchmarks Results
Table 3 shows that the HybridFlow models consistently are more accurate across the suite of UCI regression benchmarks. Based on RMSE, out of the 12 UCI regression benchnarks we tested, the HybridFlow models outperform the Gaussian NLL, BNLL, and CBDL models on 9 of the datasets. This highlights HybridFlow's ability to excel in diverse conditions, outperforming the other methods in both high-dimensional sottings and tasks characterized by challenging noise distributions.
The NF' not only allows the model to quantify aleatoric uncertainty, but in of the benchmarks the Hy- bridFlow model achieves even greater accuracy than the non-probabilistic models. Of the remaining 6 benchmarks, 5 show RMSE values within 5% of the probabbbaiiiittic MLP baseline. This suggests that the latent feature representations learned by the NF contribute meaningful additional structure to the data,
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Table 3: RMSE values for each the UCI regression datasets from a standard MLP model compared to heteroscodastic training methods (NLL, BNLL), a deep ensemble baseline, and a pon-probaii predictor. N, d, and represent the dataset length, imput dimensions, and output dimensions respectively.
DATASET
k
NLL
BNLL
CBDL
DEEP ENSEMBLE
HYBRIFFLOW
MLP (NO UQ)
HOUSING
3.58
1.07
1.04
3.40
1.12
3.12
L.0
CARRON
10721
3
0.0068
+ 0.003
0.0068
= 0.003
0.0076 + 0.002
0.0072
+ 0.003
0.0068
= 0.002
0.0068 + 0.003
CONCRETE STRENOTH
1030
1
6.08
0.65
5.61 0.65
5.52 + 0.22
6.32 + 0.71
5.07 = 0.52
1.96 + 0.64
CYBLR POWER PLANT
9568
4.06
0.18
4.04
0.15
4.26 + 0.20
4.18 + 0.19
4.02 + 0.18
4.01 + 0.19
ENERGY EFFICIENCY
768
2
2.25
+ 0.34
1.12 + 0.25
0.97 = 0.08
1,30 + 0.22
1,00 + 0.14
0,92 + 0.11
KiN8M
6192
0.087
0.004
0.081
0.003
0.84 0.002
0.091 4: 0.004
0.078 1 0.003
0,081 4: 0.003
NAVAL PROPULSION
11934
0.0021
0.0006
0.0004
0.0001
0.026
0.0043
0.0024
+ 0.0007
0.0004
+ 0.0001
0.0004 + 0.0001
PROTHIN STRECTURE
457.00
1
1.49 + 0.11
1.28 + 0.02
4.91 = 0.11
1.62 - 0.08
4:26 + 0.05
4.28 + 0.07
SUPERCONDUTTIVITY
21263
14.98 2.67
0.45
WINE QUALITY (REd)
1599
1
0.636
0.038
0.635
0.037
0.665 + 0.040
0.652 + 0.041
0.639 + 0.036
0.633 + 0.036
WINE QUALITY ( (Warrs)
4898
11
1
0.691
4 0.032
0.685 + 0.035
0,737 # 0.037
0,702 4 0.033
0.678 # 0.030
0,684 + 0.038
YACIT HYDRODYNARRSS
1.22
1.73 1.00
2.30 1.94
0.61
0.25
Table 4:
Training run-time (in minutes wall-time) for each of the UCI regression datasets from standard MLP model compared to heteroscodastic training methods (NLL, BNLL), a deep ensemble of heteroscedastic regression models, and a non-probabilistic predictor.
DATASRT
NLL
BNLL
CBDL
DREP EXSEMBLE
HYRRMFLoW
ML.P (NO [Q]
BOSTON HOUSING
0.19
0.21
16.91
0.95
1.11
0.37
CARBON
4.49
3.38
24.50
22.45
16.62
2,95
CONCRETE STRENGTH
0.22
0.30
12.27
1.10
2.04
0.80
CYCLE POWER PLANT
5.97
4.60
43.84
29.85
21.95
4,30
ENERGY EFFICIENCY
0.75
0.89
17.15
3.75
2.63
1.19
KIN8M
2.94
3.28
29.51
14.70
18.43
2.99
NAVAL PROPULSION
6.60
6.30
24.79
33.00
16.47
4.93
PROTEIN STRUCTURE
6.05
6.10
44.83
30.25
28.59
4.75
SUPPROONDICCTIIEYY
4.70
4.39
37.28
23.50
43.35
4.95
WINE QUALITY (RED)
0.30
0.30
7.12
1.50
2.73
0.32
WINE QUALITY (WHITR)
0.83
0.82
24.03
4.15
8.61
0.66
YACIEF HYDRODYNAMICS
0.25
0.28
14.48
1.25
1.17
improving predictive performance beyond what achievable with MSE optimization alone (for a detailed ablation study on HybridFlow inputs, see Appendix B Table 7 While the model's performance surpasses others in most scenarios, few cases where it matches the performance of baseline methods suggest the in- fluence of dataset-specific complexities results. Table showcases the training run-times of cach of the 4 models tested. Our results show HybridFlow does add to total training time, but less so than the equivalent Bayesian method, CBDL. However, we consider that a small trade-off when comparing with HybridFlow accuracy and argue that the extra training time provides enough practical benefit to justify its use.
Across the UCI suite HybridFlow showcases superior performance on nearly all uncertainty quantficaation metrics. attains the lowest total NLL on 11 of 12 datasets, the lowest Winkler score on 11 of 12, and the smallest total ECE on 9 of 12 (Table 5 see Appendix F for full results). Despite these sharper and better- calibrated distributions, it still produces the narrowest intervals (lowest MPIW on 8 of 12 datasets) while preserving reliability (PICP remains near 0.97 every benchmark). These results confirm that decoupling alcatoric and epistemic components with HybridFlow yiclds predictive uncertainties that are simultancously sharp, calibrated, and (rustworthy across nearly all of the UCI benchmarks. Additionally, we demonst rate that the uncertainties quantified by HybridFlow are disentangled, as can be seen in Appendix G
6 Case Study: Hybrid Ice Sheet Emulator
To test the efficacy of HybridFlow in real-world secnario and demonstrate how it could be implemented with an existing model structure, we use the HybridFlow framework to create an emulator for projecting ice sheet dynamics and their contributions to sea level rise. Ice sheet evolution governed by complex, nonlincar processes, such as melting, snow accumulation, and shelf instability. These processes involve large amounts of uncertainty, which originate both from the physical modeling of the processes and from the chaotic nature of the processes themselves. Ice sheet. emulators (Van Katwyk et al. 2023 Edwards et al.
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Table 5: Total uncertainty quantification metrics across models the UCI datasets. Metrics include total NLL (Negative Log- Li-eeiilihood) total calibration error (ECE), total Winkler score, total MPIW (Mean Pre- diction Interval Width), PICP (Prediction Interval Coverage Probability), and CRPS (Continnous Ranked Probability Score). An expanded version this table, which includes metrics for aleatoric, epistemic, and total uncertainty included in Table 10 in Appendix
DATASET
METHOD
NLL
ECE
WINKLER
MPIW
PICP
CRPS
BOSTON HOUSING
NI.I
2,88
2.40
86.35
86.32
1.00
1.98
BNLL
1.88
43
32.94
1.00
DEEP ENSEMBLE
2.20
2.05
48.50
45.00
0.99
1.80
CBDL
5.97
200.15
709.99
662.91
0.94
1.83
HYBRIDFLOW
1.63
1.90
16.63
1.76
CARBON
-2.92
21.06
0.20
0.19
BNLL
-2.79
17.81
0.23
0.23
1,00
0.02
DEEP ENSEMBLE
+2.85
18.90
0.22
0.22
1.00
0.02
CBDL
345.93
4.39
36.61
0.98
0.89
0.03
HYBRDFFOO
52.15
23.90
0.26
0.21
0.99
0.02
CONCRETE STRENGTH
NLL
3.65
4.51
177.13
177.13
1.00
3.58
137.57
137.57
3.29
DEEP ENSEMBLE
3.55
4.30
150.00
145.00
1,00
3.40
CRDL.
6.97
588.65
257. 257.75
348. 11
0.98
3.22
HYBRIDFLOW
2.40
4.44
40.83
39.51
3.76
3.23
BNLL
3.07
3.22
86 03
86.03
1,00
2.64
DEEP ENSEMBLE
3.12
3.18
88
87.00
1.00
2.65
CBDL
55.74
770.32
251.01
436.43
0.72
2.91
HYBRIFFLOW
2.02
3.10
24.61
22.97
0.99
2.66
ENERGY EFFICINCY
NLL
2.90
2.26
83.42
83.42 42
1.00
1.86
BNLL
1.61
36
36. 60
1.00
1.16
DEEP ENSEMBLE
2.20
1,80
40.00
38.00
1,00
1.42
CBDL
6.69
189.01
156. 14
521.30
0.65
1.48
HYBRDFLOW
1.48
1.18
19.28
19.28
1.00
1.16
KIN8M
BNLL
-1.37
15.47
0.69
0.32
0.81
0.07
DEEP ENSEMBLE
-1.33
13.00
0.75
0.33
0.85
0.07
CRDL.
363.54
146.19
10.64
0.33
0.84
0.07
HYBRIFFLOW
-1.86
8.42
0.51
0.50
0.99
0.06
NAVAL PROPUISION
NI.I
-2.89
0.29
0.59
0.26
0.98
0.02
BNLL
-2.33
0.41
0.28
0.97
0.03
DEEP ENSEMBLE
-2.80
0.30
0.58
0.25
0.98
0.02
CBDL
155.82
261.39
0.51
0.40
0.88
0.01
HYSRIFFLOW
-3.32
0.19
0.48
0.22
0.99
0.02
PROTEIN STrUCTURE
4.40
115.13
115.13
4.11
BNLL
3.02
3.66
88.05
87.96
1,00
3.32
DEEP ENSEMBLE
3.20
3.80
95.00
90.00
0.99
3.40
CBDL
4.90
48.17
487.74
487.74
1,00
3,00
HYRRDFLOW
1.95
3.30
22.74
20.04
0.97
3.20
SUPERCCONDECTTIVITY
NLL
6.06
14.55
2033.2 21
2033. 23
1.00
12.49
8.12
647.08
640.82
DEcP ENSUMBLE
4.80
9.00
700.00 00
680,00
0.99
6.80
CBDL
9.47
157.99
1770.72
5461.87
1.00
7.60
HYSRDFFLOW
2.74
7.77
63.02
58.80
0.99
6.40
WINE QUALITY (RED)
NLL
1.38
2.19
0.16
BNLL
0.25
1.95
3.70
1.86
0.86
0.43
DEEP ENSEMBLE
0.30
1.50
3.50
2.00
0.90
0.15
CBDL
47.26
2.65
137.57
2.22
0.76
0.43
HYRRDFFLOW
0.05
0.95
3.27
2.85
0.97
0.44
WINE QUALITY (WHTTE)
NLL
0.16
1.20
3..54
2.42
0.92
0.48
BNLL
2.37
DIOP ENSHMBLE:
0.17
1.10
3.52
2.40
0.93
0.48
CBDL
25.18
3.78
147.42
3.01
0.6-1
0.19
HysuDFLoww
0.13
0.75
3.48
3.14
0.97
0.48
YACHT
BNLL
1.25
3.11
5.52
8.16
1,00
0.25
DEuP ENSEMBLE
3.50
8.80
0.99
0.26
CBDL
30.11
15.50
183.31
40.83
0.71
0.30
HYBRIFFLOR FLow
0.62
2.19
3.17
3.47
0.98
0.22
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Table 6: Comparison of performance metrics for three ice sheet emulators: HybridFlow, a Gaussian Process,
and a non-probabili predictor (Deep Ensemble of LSTMs without uncertainty quantification), evaluated on the Antarctic Ice Sheet (AIS) and Greenland Ice Sheet (GrIS).
EMULATOR
MSE
ECE
PICP
AIS
HYBRIDFLOW
1.20
0.02
0.95
GAUSSIAN PROCESS
3.05
0.08
0.90
LSTM (NO UQ)
1.09
GRIS
HYBRIDFLOW
1.02
0.01
0.96
GAUSSIAN PROCESS
9.87
0.12
0.91
LSTM (NO UQ)
0.87
provide efficient and reliable approximation of the complex physics- based models, which allows ice sheet scientists to perform sensitivity testing to understand the effect that climate variables have on sea level projections, and enables more experimentation to und
2021
Recent work has demonstrated the effectiveness of LSSTM-based ice sheet emulators (Van Katwyk et al. 2023) in improving predictive accuracy over Gaussian process-basedd approaches (Edwards et al.) 2021) the ISMIP6 ice sheet model projection dataset (Nowicki et al.) 2016). Building on these advancements, we use a CMAF and a Deep Ensemble [Lakshminarayanan et al. 2017) of LSTM models to effectively model the temporal structure of ice sheet projections from 2015 to 2100 (sec Appendix for comparison to MC Dropout). We compare the HybridFlow emulator to a Gaussian Process emulator based on the individual projection performance as well as the ability to approximate the full range of ensemble predictions. Consistent with the approach in Section we train non-probabilistic emulator using the LSTM model based on Van Katwyk et al. 2022).
Table G shows that the HybridFlow framework achieves superior performance in both test accuracy and the ability to reproduce the full range of ensemble projections compared to the Gaussian Process-based ice sheet emulator. HybridFlow performs similarly to the non-probabilistic predictor in test accuracy (MSE), highlighting the framework's ability to deliver reliable and actionable projections in the context. of real data. Furthermore, the framework's ability generate calibrated (PICP) and accurate (ECE) uncertainty estimates ensures that it can provide confidence intervals suitable for policy and planning decisions.
7 Conclusion
In this work we introduce HybridFlow, a novel hybrid architocture that combines the strength of NEs and probabilistic prediction models to quantify both aleatoric and epistemic uncertainty within a single unified model. By leveraging the flexibility of NFs to model data-specific aleatoric uncertainty and incorporating probabilistic models for epistemic uncortainty estimation, HybridFlow achieves robust and calibrated pre- dictions without compromising predictive accuracy. HybridFlow's design addresses two gaps in uncertainty quantification methods: the decrease in model predictive accuracy to quantify uncertainty and the miscal- ibration of current state-of-the-art uncertainty quantification methods (Seitzer et al. 2022). HybridFlow achieves this by decoupling the loss functions used for aleatoric and epistemic uncertainty estimation, avoid- ing the pitfalls of joint NL.--ased training methods. Decoupling the loss functions allows for task-specific loss functions for predictors, which results in predictive accuracy levels comparable to non-probabilistic mod- els while maintaining the ability to quantify and separate sources of uncertainty. We demonstrate the ability to consistently achieve better metrics for accuracy (RMSE) and uncertainty quantification (ECE) on ricty of regression tasks, including depth estimation and a scries of standard benchmarks. Furthermore, we present a case study that demonstrates the ability of the HybridFlow framework to be effective in real-world applications, offering accurate emulation of complex systems such as continental-scale ice sheets.
HybridFlow offers a particularly practical solution for domain scientists and rescarchers who seek to in- corporate robust uncertainty quantification into existing modeling workflows without changing their entire system. In many applied settings, such as Earth system modeling or ice sheet forecasting (Section G). pre-
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dictive frameworks already well established and finely tuned (Irgang et al. 2021 ; replacing them with task-specific uncertainty models is often infeasible. HybridFlow avoids this disruption by providing modular framework that can be integrated with existing probabilistic predictors, enabling calibrated and interpretable estimates of both alcatoric and epistemic uncertainty with minimal architectural changes. This makes Hy- bridFlow especially valuable for scientific applications where improving the transparency and trustworthiness of predictions is critical, but where flexibility and compatibility with legacy models remain essential.
HybridFlow's modular design ensures adaptability to wide range of tasks, allowing for future advancements in robust and trustworthy Al systems. However, key drawback of this method is the added computation required for implementation. In the test cases presented in this study, the HybridFlow model for depth estimation took 44% longer to train than the NLL-based models. Similarly, Table 4 shows that HybridFlow was also slower to train for the UCI datasets, as the NF' must be trained before the training of the predictor models. However, HybridFlow requires less computational overhead than the CBDL. method (as can be seen in Table D which took one or two orders of magnitude longer than both the NLL methods and HybridFlow to train, but stiil remains potential drawback of HybridFlow. The NLL and BNLL methods, on the other hand, are simple to implement and the added ability to quantify aleatoric uncertainty requires negligible increase in computation to add to any model. Therefore, future work may focus both on expanding the framework's scalability and exploring alternative generative modeling techniques, including Bayesian NFs, to further enhance its versatility, precision, and reliability in high-dimensional or complex data environments.
While HybridFlow provides an approach to separately estimate aleatoric and epistemic uncertainty, it is im- portant to acknowledge that disentangling these two sources of uncertainty is inherently difficult in practice. Despite conceptual distinctions, they are often entangled in complex models and real-world data, and there is currently no definitive or universally accepted solution to completely separate them (Smith et al. 2024 de Jong et al. 2024 Valdenegro- Toro & Mori 2022). Attempts to decompose uncertainty are still valuable, as demonstrated in our ice sheet case study, where understanding the source of uncertainty informs scientific and policy decisions. Even imperfect separation can provide useful insight into model limitations and the nature of data variability.
Future work should also include the investigation of epistemic uncertainty within the NF itself, as it is not explicitly modeled in the current implementation. Although this practice aligns with conventional blowd modeling approaches, incorporating methods such as flow ensembles (Berry & Meger 2023) could enhance the robustness of aleatoric uncertainty estimates by capturing model-level epistemic uncertainty, even though due to the size of the NF model, the epistemic uncertainty is likely small. Future work may also include the rigorous testing of this framework on a variety of classification tasks, with a comparison to Sale et al. 20024, including detailed analysis of quantified uncertainty.
Broader Impact Statement
HybridFlow provides a practical and accessible framework for uncertainty quantification, enabling users to enhance the reliability of their models without significant architectural changes or specialized expertise. By prioritizing case of integration, HybridFlow lowers the barrier to adopting robust uncertainty estimation, making it especially valuable for applications where trust and interpretability are critical.
Rather than focusing solely novelty, this work emphasizes usability and support broader adoption across disciplines, including scientific. modeling, environmental forecasting, and healthcare. While users should remain mindful of data limitations, HybridFlow equips them with a straight- forward tool for improving model transparency and decision decision-making confidence, contributing to more trust- worthy Al systems.
that
Acknowledgments
This effort was funded by the Office of Naval Research (ONR), under Grant Number N00014-23-1-27 as well as the National Science Foundation Graduate Research Fellowship Program under Grant 2040433. Computational resources and services required for this study were provided by the Center for Computation and Visualization at Brown University. We would also like to thank the three anonymous reviewers for their invaluable feedback on improving this work.
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Appendix
A Autoencoder for Depth Estimation
For the depth estimation experiments dctailed Section 4.1 we utilized an autoencoder to compress the dimensionality of RGB images from the NYU Depth v2 dataset. The autoencoder employs UNet-inspired architecture with encoder-decoder structure enhanced by skip connections. The encoder gradually re- duces the spatial dimensions of the input images through six convolutional layers, each followed by batch normalization and ReLU activation. The bottleneck compresses the high- oigh-dimensiona input (384 x 512 x 3) into a compact 256-dimensional latent representation. The decoder reconstructs the images using trans- posed convolutions, with skip connections from the encoder aiding in preserving spatial details. A sigmoid activation in the final layer ensures that pixel values are mapped to the range [0, 1].
The autoencoder was trained using a composite loss function designed ensure high-quality reconstruction and edge prescrvation. This loss combines Mean Squared Error (MSE) with the Structural Similarity Index Measure (SSIM) to focus perceptual similarily, alongside edge preservation term that penalizes dif- ferences in gradients between the input and reconstructed images. Training was conducted with the Adam optimizer at learning rate of 1 x 10-4, decayed by factor of 0.5 based on validation loss, for a maximum of 50 epochs. Early stopping with a patience of 5 epochs was applied to prevent overfitting. Data augmentation, including random cropping, flipping, and brightness adjustments, was employed to improve generalization.
Extensive testing was conducted with latent dimensions of 512, 1024, 128, and 64, but it was observed that performance improvements plateaued beyond 256 dimensions, space to balance computational efficiency and reconstruction quality. On the NYU Depth v2 validation set., the autoencoder achieved an MSE of 0.012, SSIM of 0.943, and an edge loss of 0.005.
selected the 256-dimensional latent
The latent representations generated by the autoencoder were used as inputs to the Conditional Masked Au- toregressive Flow (CMAF) within the HybridFlow framework. This approach allowed for accurate modeling of alcatoric uncertainty while maintaining computational efficicncy.
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Original
Reconstructed
Figure 2: Reconstruction results from the autoencoder (rained on the NYU Depth v2 dataset. Each column shows the true depth image (left) and the corresponding reconstructed depth map (right). The autoencoder effectively compresses high-dinensional input data into 256-dimensional latent representation, enabling efficient dimensionality reduction while preserving fine- ine-grained spatial details.
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B Evaluating the Contribution of Latent Variable 2 in HybridFlow
Table 7: Ablation study for inputs the HybridFlow model, showing the RMSE values for each of the UCT regression datasets comparing the full HybridFlow model (predictor with both x and 2), an ablation with only x as input to the predictor, and non-probabilistie MLP baseline.
DATASET
AND
ONLY
NON-PRORABILISICC, ONLY
BOSTON HOUSING
3.12 + 0.85
3.26
1.32
4.01
1.12
3.24 1.08
CARBON
0.0068 1 0.0018
0.0068
0.0024
0.0081
0.0026
0.0068 1 0.0028
CONCRETE STRENGTH
5.07 + 0.52
0.77 5.04
5.61 + 0.83
4.96 = 0.64
CYCLE POWER PLANT
4.01
1.59 0.21
4.01 0.19
ENERGY EFFICIENCY
1.00 = 0.14
1.15 + 0.21
1.69 + 0.23
0.92 = U.11
KINSM
0.094 0.005
0.081
NAVAL PROPULSION
0.0004 = 0.0001
0.0011
0.0001
0.0012
0.0009
0,0001 0.0001
PROTEIN STRUCTURE
4.26 + 0.05
4.33 0.05
5,03 1 0.07
4.28 0.07
SUPFRCONNUCTTIIIY
11.87 0.45
13.49 0.53
14.88 0.62
12.48 0.40
WINE QUALITY (Ruu)
0. 0.639 + 0.036
0.637 + 0.040
0.680 + 0.045
0 636 = 0.036
WINE QUALITY (WaTTE)
0.678
0.738 0.038
0.681
YACHT HYDRODYNAMICS
1.55 + 0.79
1.47 1 0.58
2.38 + 1.04
0.78 0.25
To better understand the contribution of the latent representation from the normalizing flow to predictive performance, we conducted an ablation study across the UCT regression benchmarks. Table compares four model inputs: the full HybridFlow model inputs (inputs x and latent representation 2), the HybridFlow architecture but only using the raw inputs x, the HybridFlow architecture but only using the computed 2, and an MLP baseline (like that which was used for the HybridFlow predictor) but without uncertainty quantification capabilitics. All other models use MC Dropout for uncortainty quantification, explained in section B
This ablation helps isolate the impact of the NF-derived latent variable on performance, as well as disen- tangle improvements due to the hybrid architecture versus uncertainty quantification.
Table shows that most datasets, the model using the full HybridFlow inputs (both æ and 2) achieves the lowest, or near-lowest RMSE. The combination appears especially beneficial for larger or more complex datasets, such the Superconductivity,, Protein Structure, and Naval Propulsion datascts, where the NF may extract structure not directly visible in the input space.
However, in a few datasets, the model trained only on raw inputs æ slightly outperforms the combined input model, indicating that the added latent structure may not always provide a benefit, particularly in smaller datasets. Models trained with & alone consistently under-perform, confirming that the NF alone does not encode all information necessary for accurate prediction, further supportive the interpretation of z as a complementary, uncertainty-aware feature rather than a standalone representation of the data.
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C Computational Cost and Flow Depth Ablation
To better understand the computational tradcoffs associated with deeper flow-based models, we conducted an ablation study examining how the number of flow layers affects both training time and predictive per- formance. Table reports the training time (in minutes) required for HybridFlow models using 2 to 12 flow layers across three representative UCI regression benchmarks: Housing, Concrete, and Energy. Table g reports the corresponding RMSE accuracy for each configuration.
As expected, training time increases with the depth of the flow model. This trend is particularly pronounced for higher--iimensional datasets like Energy and Concrete, where each additional layer adds nontrivial com- putational overhead. However, we observe diminishing returns (and even degradation) in predictive accuracy beyond a moderate number of flow layers. For all three datasets, RMSE is minimized between and flow layers. Deeper models (10+ layers) yield no consistent gains and can introduce training instability, especially for smaller datasets.
These results highlight key practical consideration: while deeper normalizing flows can model more complex conditional distributions, they may not be necessary (or beneficial) for moderato-sized tabular datascts. For many weal-world settings, 4 6 layer flow offers a favorable balance between expressivily and elficiency.
Table 8: Training time (in minutes) for HybridFlow models with varying numbers of flow layers across three UCI datasets.
DATASET 2 LAYERS
4 LAYERS
6 LAYERS
8 LAYERS
10
LAYERS
12 LAYERS
HOUSING
1.01
1.49
1.92
2.69
2.71
3.11
CONCRETE
2.24
3.12
3.70
5.21
6.03
6.65
ENERGY
2.79
3.83
4.92
5.83
6.91
7.44
Table 9: RMSE accuracy for HybridFlow models with varying numbers of flow layers across three UCT datasets.
DATASET 2 LAYERS
4 LAYERS
6 LAYERS
8 LAYERS
10 LAYERS
12 LAYERS
HOUSING
3.11
2.56
2.71
2.93
2.87
3.34
CONCRETE
5.52
5.10
5.28
5.41
5.34
5.48
ENERGY
1.52
1.01
1.41
1.54
1.27
4.92
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D Uncertainty Quantification Metrics
This appendix provides precise mathematical descriptions for the uncertainty quantification and evaluation metrics used in this paper.
Expected Calibration Error (ECE)
The Expected Calibration Error (ECE) measures the difference between a model's predicted confidence and its empirical accuracy. For a model that outputs a prodictive distribution, the prediction space is partitioned into M bins, each corresponding a range of probability values. The ECE then computed as weighted average of the absolute difference between the mean predicted probability and the fraction of positive outcomes in each bin:
M
Bml N
ECE
|ace(Bm) Bm)l
(5)
m=1
where N is the total number of samples, Bm is the set of predictions in bin m, acc(Bm) is the accuracy of the predictions in bin and (onf(Bm) is the average confidence of the predictions in bin A perfectly calibrated model has an ECE of O, with lower values indicating better alignment between predicted confidence and empirical accuracy.
Prediction Interval Metrics
For given confidence level (1 - o), a prediction interval is defined by a lower bound L(x) and an upper bound Uxx). The following metrics evaluate the quality of these intervals:
Prediction Interval Coverage Probability (PICP)
PICP is the proportion of true target values yi that fall within their predicted interval [L(x;), Ux:)). Ideally, for (1 a) prediction interval, the PICP should be close to (1 (1-a).
N T(y E [L(x))
PICP =
(6)
N
where T(.) is the indicator function.
Mean Prediction Interval Width (MPIW)
MPIW is the average width of the prediction intervals. Smaller MPIW values reflect narrower prediction intervals and correspond to sharper, more informative uncertainty estimates, assuming coverage adequate.
N MPIW
Σ
N
where (xx)) and L(x;) represent the upper and lower bounds of the predicted confidence interval for the i-th sample.
Winkler Score
The Winkler score is a proper scoring rule that combines PICP and MPIW into single metric. It penalizes wide intervals and intervals that do not contain the true value.
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y y [,, (U L) (L v)
(U 1 L)
(L,
(8)
(U - L) -v y
The Winkler score rewards narrow intervals that contain the true value and penalizes both over-wide intervals and those that miss the target. Lower scores indicate better-calibrated and sharper predictive intervals.
Probabilistic Forecasting Metrics
Negative Log-Likelihood (NLL)
NLL, also known the logarithmic score, evaluates the quality of model's predicted probability distri- bution. It measures the likelihood of observed outcomes under predicted distribution. For model that outputs probability density function p(y(x), the NLL is:
NLL
Σ log ((g((());
(9)
N
Lower NLL values indicate higher likelihood assigned to the observed outcomes under the predicted distri- bution, reflecting better-calibrated and more confident predictions.
Continuous Ranked Probability Score (CRPS)
The CRPS is a proper scoring rule that generalizes the Mean Absolute Error to probabilistic forecasts. measures the difference between the predicted cumulative distribution function (CDF), F, and the empirical CDF of the obs.rvation.
CRPS(F, v) - [ (F(z) (F(2) 11 S 2))
(10)
where F is the predicted CDF and is the observed value. Lower CRPS values correspond to predicted distributions that more closely match the observed outcome, terms both sharpness and calibration.
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E
Depth Estimation Visualizations
Thun
Thun Doyeh
Prodectin
Esteiami: Unvartenty
Aisttark Unzeremry
HybridFlow
BNLL
NLL
HybridFlow
BNLL
Tre Imaze
Trac Death
Iedidiion
Buskomi Uncortaiity
Alcatox x Uncoraiitty
HybridFlow
BNLL
NLL
Figure 3: Visualization of depth estimation results using the HybridFlow framework compared to the BNLL (Seitzer et al. 2022) and NLL (Kendall & Gal 2017] the NYU Depth v2 dataset. Each row showcases sample scene, with the columns representing (from left to right): the input RGB image, ground truth depth map, predicted depth map, model error, alcatoric uncertainty map, and epistemic uncertainty map. The predicted depth maps closely align with the ground truth, demonstrating accurate scene reconstruction. The aleatoric uncertainty maps highlight areas with inherent measurement noise, such as changes in image lighting (glare) occlusion boundaries, while the epistemic uncertainty maps identify regions of the scene where the predictive model is uncertain. These visualizations illustrate the HybridFlow model's ability to provide both accurate predictions and reliable uncertainly quantification.
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F Full UCI Uncertainty Results
Table 10: Uncertainty quantification metrics across models on the UCI datascts. Metrics include NLL (Negative Log- -og-Liicchhodd), calibration error (ECE), Winkler score, MPIW (Mean Prediction Interval Width), total PICP (Prediction Interval Coverage Probability), and CRPS (Continuous Ranked Probability Score). a )
Metrics are reported for each quantified aleatoric, epistemic, and total (additive) uncertainty. C
PICP CRPS
1.98
1.65
1.83
1.76
0.01
0:02
0.03
0.02
3.58
3:29
8.22
3.76
2.64
2.64
2.91
2,66
1.86
1.46
1:48
1.16
0.08
0.07
0.06
0.02
0.03
0.01
0.02
4.11
3.32
8.00
3.20
12.19
6.51
7:60
0.40
0.43
0.43
0.44
0.47
0.49
0.48
0.28
11.5
0.30
0.22
1.00
1.00
0.94
0.98
1.00
1.00
0.80
0.99
1.00
1.00
0.98
0.99
1.00
1.00
0.72
0:99
1.00
1.00
0.65
1.00
0.79
0.84
0.09
0.98
0.97
0.88
0.99
1.00
1.00
1.00
0.97
1.00
1.00
1.00
0.99
0.86
0.76
0.97
0.92
0.91
0.64
0.97
1.00
1.00
0.71
0.98
Tor.
80,30
02.32
32 91
662.91
16.63
0.19
0 0.23
0.98
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473
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20
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80
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31 31-78
101 18
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0. 0.00
0.05
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112.45
84.45
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17.17
2013
14
629.24
3730
41.91
1.76
1.35
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2.38
1.02
1.81
0.34
2 2.65
6.30
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20
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Ton.
80 35
33 13
769
999
69.65
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36
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137 57
257.75
40.83
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80 83.03
251.01
24.61
83.42
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156 11
19.28
0.81
10.64
0-51
0.59
0.51
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115.13
88.05
187.71 71
22.74
DG 2033
647.08
1770.7
63.02
3.41
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187.57 ST
3.27
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147 42
3.48
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8.17
WINKLER
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36
28.28
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37
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GU 02
54.34
254
80
78.5
78.59
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72
253.72
51.85
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27
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210
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133
39
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313
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173 173.80
128
11.94
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164
03
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5.57
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5. 80
AGE.
80.82 09.82
29. 96
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18.77
0.61
0.46
0.46
126.25
1567.2
35.74
82.94
40
80
20.53
76.70
31.78
703 48
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2.49
2.32
8.87
0.43
0.39
0.46
0.50
0.31
112.45
84.57
662.00
21.54
2013.1
629.83
402.27
57.08
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165
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1.95
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1.79
1.38
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587.23
23
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261
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157.97
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1.
83
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4.15
588.80
1.10
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3.
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2.25
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225.
21
1.20
27
76.
16.96
67.0
11.61
0.26
0.38
778.68
0.17
4.40
3.00
40.0
3.26
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8.12
580
43
7.65
1.91
3.18
8.63
1.26
1.70
1.88
6.96
0.99
1.07
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363
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33.3
155.82
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3.37
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4.53
9.47
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2.09
0.25
47.26
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0.16
28.18
0.13
1.31
1.25
30.11
0.62
NLL
EPI.
4.11
3.15
6.28
4.42
-2.98
-2.56
317.27
7.68
4.03
3,80
7.07
7.85
5.45
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60 87
6.53
3.49
2.76
7.8T
2.08
0.29
-0.60
400.10
400.
1.76
-0.00
-0.31
132. 27
-0.47
63.53 53
33.52 82
4.88
38.85 88
10.57
6.01
8.45
6.67
57
17.24
106. 97
19.97
20.24
18.62
33 25
22.47 47
3.98
17
5.0 00
ALe.
2.50
2.39
191.23
1.64
NLL 43.71
23-53
370,94
24.63
3.56
3.36
148.05
2 2.29
3.03
3.00
135,34 34
1.93
2.79
1.64
171.68
1.17
20.67
23. 49
2001. 21
-1.94
-3.24
-2 -2.61
201.33 38
-3.79
3.34
2.97
92. 66
1.97
6.04
4 L8
707 73
8.63
1.03
1.57
75.83
0.08
0.49
26 46
0 0.22
1.21
1.35
29. 11
0.63
HODRRDFLOW
HYBRID LOW
HYYRIIFLOON LOW
HYBRIDELON LOW
HYURDF LOW
HYRRIDFLLW
HYBRIDFLOW
HYURIDFLOW
LLW
HYBRIDFLOW
HVURIDFLOW
HYBRIDELLN
MRTHOD
BNLL
CBDL
BNLL
CBDL
BNLL
CIRD.
BNLL
CBDL
BNLL
CBDL
BNLL
CBDL
BNLL
CBDL
BNLL..
CBDL
BNLL
CBDL
BNLL
CBDL
BNLL
CBDL
BNLL
CBDL.
NLL
NLL
NLL
NL.
NLL
NLL
NLL
NLL
NLL
WINE QUALITY (WHITEE
YACHT HYDRODYNAMICS
CONCRETE STRENOTH
ENERGY EFFICIENCY
NAVAL PROPULSION
PROTEIN STRUCTURE
SUPERCUNDECTIVIEY
WINE QUALITY (NED)
BOST HOUSING
CYcLe POWER
DATASET
CANBON
KINSM
26

---

Published in Transactions on Machine Learning Research (10/2025)
G Disentanglement of Aleatoric and Epistemic Uncertainties
To further assess whether HybridFlow effectively separates alcatoric and epistemic uncertainty, we analyze their joint behavior across three regression datasets In Figure 4 we plot aleatoric uncertainty (x-axis) against epistemic uncertainty (y-axis) for all test samples across three methods: HybridFlow, standard Gaussian negative log-likelihood (NLL), and beta-NLL (BNLL). Each point represents a single test prediction.
HybridFlow exhibits dispersed epistemic values across narrow bands of aleatoric uncertainty, suggesting separation of the two uncertainty sources, while NLL displays more tightly coupled uncertainty values, with both types increasing together, especially in the Energy Efficiency dataset where strong negative Spearman correlation is observed. BNLL shows partial disentanglement but still exhibits moderate correlation between the two uncertainties.
To quantify the level of dependency between aleatoric and epistemic uncertainty, we compute both Pearson's correlation coefficient (r) and Spearman's rank correlation coefficient (p) for each method and datasel. Spearman p is emphasized due its robustness to nonlinearity and non-Gaussian behavior.
Table 11: Correlation between Aleatoric and Epistemic Uncertainty (Spearman p Pearson r). Values closer zero indicate greater disentanglement between alcatoric and epistemic uncertainty.
Dataset
Method
Spearman
Pearson r
HybridFlow
0.149
0.168
Energy Efficiency
NLL
-0.334
-0.324
BNLL
0.240
0.182
HybridFlow
0.333
0.373
Boston Housing
NLL
0.456
0.373
BNLL
0.562
0.576
HybridFlow
0.147
0.235
Concrete Strength
NLL
0.265
0.440
BNLL
0.163
0.317
HybridFlow consistently demonstrates the lowest or near-lowest correlation between aleatoric and epistemic uncertainty across datasets. In contrast, NLL and BNLL exhibit stronger coupling, suggesting entanglement in their uncertainty estimates.
Boston Howsing
Efficiency
Concrete Strength
inpodiion Ma.
Sproaee Unoutarr
Anattree Uncertaley
Alvataric Unierr nby
Alsatanc Uria
Figure 4:
Scatter plots of aleatoric (x-axis) vs. epistemic (y-axis) uncertainty across test points for three datascts and three methods.
The smaller alcatoric intervals for HybridFlow in Boston Honsing and Energy Efficiency are supported quan- titatively by lower aleatoric MPIW /Winkler and aleatoric NLL (Table 10 while total PICP remains high. This aligns with the decoupled design: the flow models p(ylz) and avoids absorbing model misspecification, which NLL/BNLL often express inflated aleatoric variance (Seitzer al. 2022).
27

---

Published in Transactions on Machine Learning Research (10/2025)
H Impact of Epistemic Uncertainty Method in HybridFlow Ice Sheet Emulator
HybridFlow is designed to be agnostic to the choice of epistemic uncertainty quantification method. In the main text, we adopt Monte Carlo Dropout (MCD) Gal & Ghahramani (2016) for its computational efficiency and ease of integration with the baseline architectures used in prior work. To verify that our conclusions for the ice sheet emulator case study are not specific to MCD, we conducted an ablation in the Ice Sheet Emulator case study (Sec. 6 where we compare MCD with the five-member Deep Ensemble (DE) lakshminarayanan et al. (217) as the epistemic uncertainty quantification method in HybridFlow.
Table 12 reports mean squared error (MSE), expected calibration error (ECE), and prediction interval coverage probability (PICP) for the Antarctic Ice Sheet (AIS) and Greenland Ice Sheet (GrIS) emulators. As expected, DE achieves slightly lower MSE and improved calibration compared to MCD, consistent with prior findings that ensembles produce stronger epistemic estimates. Crucially, the qualitative conclusions of the case study remain unchanged: HybridFlow yields well-calibrated predictions and high coverage under both epistemic methods, demonstrating that its benefit arises from the hybrid architecture rather than a specific epistemic estimator.
Table 12: Performance comparison between Monte Carlo Dropout (MCD) and Deep Ensembles (DE) in the Ice Sheet Emulator case study. Best values in bold.
EMULATOR
MSE
ECE
PICP
AIS HYBRIDFLOW (DE)
1.20
0.02
0.95
HYBRIDFLOW (MCD)
1.27
0.04
0.92
GRTS
HYBRIDF1LOW (DE)
1.02
0.01
0.96
HYBRIDFLOW (MCD)
1.31
0.03
0.93
28


Reviews:
[{"review_id": "FFGcSK2XGK", "reviewer": "Reviewer_H6s6", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**:\n- The paper is well-written and easy to follow.\n- The core idea of decoupling aleatoric and epistemic uncertainty using separate models is both intuitive and valuable. The authors motivate this effectively with clear examples.\n- The proposed method is simple and accessible, making HybridFlow appealing to both researchers and practitioners.\n\n**Weaknesses**:\n- **Repetitions, conciseness, and representation**:  The paper would benefit significantly from a more concise presentation. There are frequent repetitions, especially regarding the modularity of the approach, which could be removed without loss of clarity. Additionally, the abstract and introduction describe HybridFlow too vaguely. It should be clearly stated which type of normalizing flow is used, and that, in principle, any predictor (e.g., deep ensembles) can be integrated. From a layout perspective, I recommend avoiding sections that consist solely of bullet points. The workflow summary should instead be integrated into the training protocol at the beginning of Section 3.5.\n    \n- **Computational considerations**: Section 3.5 mentions the use of an autoencoder to address the high dimensionality of the output space. However, the computational cost of the full method remains unclear. How does the complexity of the model scale with the dataset's complexity? Is a more complex normalizing flow necessary for more challenging datasets? This point is important for practical applications and should be at least discussed. Ideally, the authors could include additional experiments (e.g., in the appendix) to support this.\n    \n- **Choice of normalizing flow**: As someone not deeply familiar with normalizing flows, I wonder why the Conditional Masked Autoregressive Flow (C-MAF) was chosen. Is it particularly well-suited for HybridFlow, or could other flows be used as well? Some justification for this design choice would be helpful. I would have expected a comparison with at least one alternative normalizing flow to validate this aspect.\n    \n- **Novelty and related work**: The novelty of the proposed method is somewhat difficult to assess without a stronger contextualization in related literature. The current related work section is rather general. A more focused discussion of prior work using normalizing flows in a similar way would help clarify the paper’s contribution. If no such work exists, the authors should clearly state that they are the first to model aleatoric uncertainty with a normalizing flow.\n    \n- **Minor remark**: Ensure that all tables use a consistent font size and span the full `\\linewidth`.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose HybridFlow, an approach that models aleatoric uncertainty using a Conditional Masked Autoregressive Flow and epistemic uncertainty using a predictor such as deep ensembles. They argue that employing a separate normalizing flow to capture aleatoric uncertainty helps preserve generalization performance and does not interfere with the modeling of epistemic uncertainty.\nPredictions are generated by the predictor model using its expected value. Epistemic uncertainty is estimated as the variance across the predictor’s outputs, while aleatoric uncertainty is derived from the sample variance of the predictive distribution produced by the normalizing flow.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "See above.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "pkqUOuIAOX", "reviewer": "Reviewer_Edrs", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- Uncertainty estimation for regression is somewhat understudied compared to classification, it's an interesting and important problem.\n- The paper is well written overall, it contains basically not a single typo or similar issues. Figure 1 illustrates the proposed approach well.\n- The proposed approach, to separately train a conditional flow model just for doing aleatoric uncertainty estimation, is quite interesting and makes sense overall.\n- The proposed approach seems to perform well overall compared to heteroscedastic regression baselines.\n\n\n\n\nWeaknesses:\n- Section 4 is not ideally structured, and some experimental/implementation details are somewhat unclear.\n- The simple baseline of creating a deep ensemble of Gaussian NLL models is not included in any of the experiments.\n- The experimental results (especially Table 5) are somewhat difficult to overview and interpret (a lot of datasets and different metrics).\n\n\n\n\nQuestions/suggestions:\n- Why do you never compare the uncertainty estimation quality to a regular deep ensembles model, I would expect that to outperform MC dropout? Should at least add this baseline to Table 6?\n- It's not entirely clear to me how $x$ and $z$ are combined as input to the model? If $x$ is an image, is a feature vector first extracted, which is concatenated with $z$?\n- Why is $z$ combined with $x$ as input to the model, is it obvious that this should help improve the performance? Have you done a direct ablation, comparing performance with/without $z$? In Section 5.2 you write _\"This suggests that the latent feature representations learned by the NF contribute meaningful additional structure to the data, improving predictive performance beyond what is achievable with MSE optimization alone\"_, but is $z$ really the only difference compared to that non-probabilistic MLP? No? Your model also adds MC dropout, which should affect the RMSE performance as well?\n- In Table 1 and 2, I suppose that NLL and B-NLL also are implemened with MC dropout in order to estimate epistemic uncertainty? This could be made a bit more clear.\n- You sample $\\tilde{y} \\sim p(y|x)$ in order to get an aleatoric uncertainty estimate, how many samples do you use?\n- CMAF with 4 flow layers in Section 4.1, but 5 layers in 4.2? Any particular reason why you don't use the same number of layers?\n- MC dropout with 30 samples in Section 4.1, but 50 samples in 4.2? Any particular reason why you don't use the same number of samples?\n- The uncertainty estimation metrics are described in both Section 4.1 and 4.2? Better to do this just once before 4.1? Also, _\"We evaluate aleatoric, epistemic, and total uncertainty with all of the aforementioned metrics. Total uncertainty, within the scope of this work, is the sum of the aleatoric and epistemic uncertainty, which is common in uncertainty quantification literature (Depeweg et al., 2018; Hüllermeier & Waegeman, 2021; Wimmer et al., 2023)\"_ at the end of 4.2 is relevant for both 4.1 and 4.2, and could thus also be put before 4.1?\n- In Section 4.2, the paragraph _\"In addition to the heteroscedastic regression baselines, we also benchmark against CBDL (Caprio et al.,2024), a model-specific uncertainty framework [...] that is simpler to train and can be used with existing predictive models\"_ does perhaps seem a bit superfluous? Or, could at least make more sense to describe this somewhere else, not in the experimental details for the UCI benchmarks here in Section 4.2?\n- In Table 4, why are the training run-times sometimes so different for NLL and MLP? Should NLL not always be very similar, or a bit slower? Also, how come that HybirdFlow is almost 10x slower than MLP on \"Superconductivity\" but basically the same on \"Yacht Hydrodynamics\"?\n\n\n\n\nMinor things:\n- In Table 2, why not aleatoric, epistemic and total uncertainty also for PICP and CRPS? Just to save space?\n- Start of Section 5.1: _\"outperforming baseline models that use heteroscedastic regression methods and the CBDL model\"_, but CBDL is not included here in 5.1?\n- The Table 2 caption is closer to Table 1 than to Table 2, could perhaps tweak this a bit.\n- Table 5 is too wide.\n- Figure 1 is not referenced anywhere in the text?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose an approach for estimating aleatoric and epistemic uncertainty, called HybridFlow.\n\nHybridFlow entails separately training a conditional flow model to estimate $p(y|x)$, which at test-time is sampled to obtain an estimate of _aleatoric_ uncertainty.\n\nThe latent variable $z$ from the flow model is also fed as additional input to e.g. a deep ensemble or MC dropout model (which is trained separately in a second stage), which outputs a model prediction and an estimate of _epistemic_ uncertainty.\n\nThe approach is evaluated on the NYU Depth v2 dataset for dense depth estimation and on 12 datasets from the UCI regression benchmarks, being compared with heteroscedastic regression models. It's also applied to two ice sheet dynamics datasets.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Overall, I think this is a quite interesting and solid paper, and I definitely think it could be relevant for the TMLR audience.\n\nHowever, I think the current version requires some clarifications and tweaks, see \"Questions/suggestions\" above.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "R0Y9zYDpbq", "reviewer": "Reviewer_ijDQ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n1. The empirical evaluation considers various tasks and metrics to evaluate performance.\n\n## Weaknesses\n\n1. The paper is poorly written. It is overly verbose, yet some key pieces of the method are not clearly explained. In particular, the flow trained in the first step is meant to provide a representation $z$ of $x$, as per Figure 1. However, the flow takes $y$ as an input, so I do not understand how this representation can be computed without access to the ground truth $y$ which the model is trying to predict in the first place.\n\n2. I also found the method to be poorly motivated, and I do not believe it disentangles aleatoric and epistemic uncertainties in the way claimed by the authors. In the second step, a model $p(y|x, z)$ is trained, but $z$ is meant to be a deterministic transformation of $x$. Since $x$ is also given as an input, adding $z$ to the conditioning adds no additional information, and so if the models in the first and second steps are flexible enough, they should learn the exact same distribution. To me this is a fundamental issue with the paper, since the authors claim that the variance of one model measures aleatoric uncertainty, while the variance of the other measures epistemic uncertainty.\n\n3. Although the experiments consider enough tasks and metrics, they do not consider enough baselines. Deep ensembles are cited, but not compared against. I would like to see a comparison against an ensemble of heteroscedastic regression models, where aleatoric uncertainty is estimated as the average (over ensemble members) predicted variance, and where epistemic uncertainty is estimated as the variance (over ensemble members) of the mean prediction.\n\n4. Lastly, a list of more minor points:\n- The related work section should discuss deep ensembles; although this method is cited in the paper, it should not be cited in passing.\n- Some statements should make it clear that the proposed quantities are estimates/measurements of aleatoric and epistemic uncertainty. For example, \"the aleatoric uncertainty $\\sigma_{al}^2(x)$ is calculated ...\" makes it sounds like you are computing the ground truth aleatoric uncertainty.\n- Although plenty of metrics are used, a precise mathematical description of every metric should be included in the appendix. As it stands, it is unclear what some of the entries in the tables represent. For example, what is the difference between the reported aleatoric and epistemic ECEs?\n- It is unclear why deep ensembles are marked as \"No UQ\" in the ice sheet emulator experiments.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes HybridFlow, a method to quantify uncertainty in regression models. HybridFlow is a two-step approach. In the first step, the authors train a conditional normalizing flow to learn the distribution of regression target $y$ given features $x$, $p(y|x)$. The flow has two purposes: first, computing the element-wise variance of its learned distribution provides an estimate of aleatoric uncertainty. Second, the flow produces a representation $z$ of $x$. In the second step, this representation is taken, along with $x$, to train a probabilistic model $p(y|x, z)$. The mean of this model is used for predictions, and its variance as an estimate of epistemic uncertainty.\n\nThe authors provide experiments on a depth estimation task where a depth scalar is predicted for every pixel of an image, UCI datasets, and on a task involving an ice sheet emulator.\n\nOverall, I found the paper to be poorly written and the method to lack conceptual soundness.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "No", "audience_explanation": "", "requested_changes": "Please address the weaknesses mentioned above. In particular, for weakness 2, in addition to an explanation of why you believe the two models' variances are quantifying fundamentally different quantities (or of any misunderstanding I might have had), I would like to see scatterplots plotting your estimates of aleatoric and epistemic uncertainties. I suspect these estimates will tend to exhibit a very large correlation.", "broader_impact_concerns": "I have no broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)