Nowadays, critical functionalities are increasingly tackled by autonomous decision-making systems, which depend on Artificial Intelligence (e.g. Deep Learning) models. Still, most of these models are designed to maximize the generic performance rather than preventing potential irreversible errors. While robustness and reliability techniques have been developed, in the recent years, to fill this gap, the sources of uncertainty in those decision models are still ambiguous. With a view to standardizing the uncertainty sources, in this paper we present a formal methodology to disentangle those sources from a probabilistic viewpoint for any (regression and classification) supervised learning model. Once we associate a formula to each uncertainty type, we expose the terminology disagreement in the literature and we propose one that is aligned with other previous works. Finally, based on the proposed formulation, we present an integrated visualization method to represent all the uncertainty sources in a single figure to, ultimately, assisting the design of uncertainty-tailored actions.
pervised learning models deal with two diferent kinds of random variables, (, ). The former one, , which corresponds to the available information, and the latter one, , which is assumed to depend on . The supervised goal is to find this probabilistic dependency.
In real-world problems, the joint distribution (, )
is a theoretical construction and we only have access
to the data set, i.e. a certain population of instances,
= {(, )}=1, where each instance is assumed to
be drawn i.i.d. from the aforementioned joint
distribution. By using this existing data set, , the supervised
learning task corresponds to find a (probabilistic)
function, : → , by performing Empirical Risk
Minimization (ERM) [22, 23, 24], which classically takes into
account (
In particular, evaluating the generalization capabilities introduces the problem of detecting when the learnt function will be applied into a lower probability region with respect to (). In other words, it is expected that a new evaluated point, * , could be sampled from a new random variable * ∼ * that hopefully will be similar to . This diference is shown in Figure 2, which illustrates how it changes the “domain” or range of values afecting . In order to detect these changes, we should consider (* | ). At this point, is not considered.
are interested to evaluate the full pair of any new samples,
(* , * ), where the dependent random variable used in
training, , should be also considered apart from .
Therefore, our goal will be to extend the previous
conditional probability to compute (* , * | , ).
Diferently than in the previous (* | ) case, here in order
to forecast the corresponding * value we require the
predictive model, , which we assume it is identified by
its hyper-parameters . Following the representativity
property nº (
strictly a data science or machine learning concern. In fact, their characterization is the main goals of the uncertainty quantification science [ 25] but also a major issue in many fields in natural sciences, engineering and even constitute a philosophical debate [26, 27].
One of the more extended taxonomy is the one that disentangles the types of uncertainty depending on their are reducible or not [28]. Theoretically, we can reduce uncertainty by improving our observational data and experimental techniques (e.g. we can increase the measurement precision, we can collect more high-quality data or we can consider a more proper family of models). However, realistically, there are problems where actual limitations or assumptions (e.g. time or resources constrains) prevent us to perform these improvements. In these scenarios appear irreducible uncertainties. In the literature [29], when the uncertainty is reducible it is called “Epistemic uncertainty” and when is irreducible it is called “Aleatoric uncertainty”.
Compared with Figure 1, this reducible-irreducible
dichotomy do not explicitly refers to the use of a
predictive model. In particular, we can observe that the last
term of Eq. (
paper, we are considering this kind of irreducible uncertainty
negligible. Therefore, it is not present in Eq. (
Given that we are assuming this (non-conditional) in- dure to split the uncertainty sources depending on the
herent noise in the input variable as negligible, in associated probability it has, which are obtained using
consonance with other approaches [30, 31, 32, 33, 19], in the chain rule. Based on this and following [21], in the
this paper we decided to use this irreducible and non- next subsections we will describe how to visualize each
irreducible terms only for the model part. Therefore, in uncertainty type and how to merge them all in a single
our terms, Epistemic uncertainty will correspond to the visualization plot for a regression problem with a single
integral and ( | * , , ) of Eq. (
Finally, we require a name for the (* | ) term of 5.1. The Domain uncertainty
Eq. (
Importantly, the previously exposed DEA disentanglement does not assume any specific supervised task such as regression or classification. Furthermore, the dimensionality of and is not specified. The only crucial point is to compute the Eq 1 integrated term (* , * | , ) and use this information to build reliable DL-based models (e.g. conditioning the human intervention depending on that uncertainty information).
Moreover, the disentanglement allows the possibility to characterize the tackled problem, by analysing which of the DEA uncertainties dominate over the others, as proposed in [21]. Generically speaking, a problem where the detected Aleatoric uncertainty, (* | * , , , ), lower confidence will be bluish. prevails over the other ones indicates that the input in- Analysing Figure 5 we can see that zones where there formation omits relevant variables for the forecasting are less data points have a whitish colour (e.g. between task, consequently, a point-wise approximation would 0.2 and 0.4 or between 0.8 and 1). Furthermore, we neglect critical information. In contrast, if the predicted should highlight that the (* | ) value is independent uncertainty is mostly the Epistemic one, then it can indi- of the conditional variability (* | * , , ), as we can cates that the considered family of models could be too observe, for instance, between the 0.4 and the 0.6 points. restrictive or there exist a notable model bias. Finally, if 3As a position paper, the implementation details are omitted here for the Domain uncertainty monopolize the integrated value, an extended version of the article to focus now in the combination procedure of the DEA uncertainties. Importantly, this DEA combi2Here the marginalization of M is implicitly considered. nation procedure does not depend on the or dimensionality.
First of all, it is important to highlight that the Domain uncertainty does not consider the variable to be predicted, , nor the model to perform such prediction, characterized by . Therefore, we should be careful when we represent this uncertainty in a standard regression plot where the horizontal axis is some input variable and the vertical axis is the predicted values, due to (* | ) not depend on . Consequently, one way to represent this uncertainty can be using the background colour as represented in Figure 5. In that case, each horizontal value has a diferent background colour where purpler zones correspond to high confidence values while zones with 5.2. The Epistemic uncertainty the response variable and the input one, such as in the horizontal interval from 0.4 to 0.6, does not imply to having an Epistemic discrepancy if the approximated statistic is clearly defined. Therefore, we should need to model Aleatoric uncertainty to detect this extra new source of uncertainty.
Epistemic uncertainty, ( | * , , ), which corresponds to the uncertainty related to selecting a certain family of models M (see Figure 3) is a similar case than the Domain uncertainty: The new response random vari- Figure 7: Conditional distribution approximated using the able * is not involved in this uncertainty. However, here UMAL model [32]. Aleatoric uncertainty is captured as the the prediction of each model, - characterized by , - approximated likelihood. is usually approximating some statistic of given 4.
This last detail produces several ways of visualizing this Our Aleatoric uncertainty is focused on modelling the
uncertainty depending on what is approximating each conditional variability of the response variable (* |
model but we should be careful to distinguish between * , , , ) of Eq. (
At the end, considering a certain family M of such mod- Figure 5 and 6 because now it depends on the vertical els is to consider an ensemble with a finite - or not - num- axis values. Additionally, this uncertainty is irreducible, ber of components or models. Their discrepancy refers therefore, our goal will be to show the distributional to the Epistemic uncertainty we are capturing. There- shape to design shape-tailored techniques. This is why fore, one way to visualize its discrepancy is to plot each it is important to avoid strong assumptions regarding prediction separately, as it is shown in Figure 6. to the conditional distribution, such as symmetry or
Comparing Figure 6 with Figure 5, we can see clearly unimodality, if we do not have clear evidences that they the diference between capturing the Domain or Epis- cannot harm the forecasting or visualization procedure, temic uncertainty: For instance, in the horizontal interval as it is deeply discussed in [21]. from 0.2 to 0.4, the behaviour of both uncertainties are To provide a richer estimation of the likelihood becompletely diferent. This is because the density of () yond the standard aleatoric conditional symmetric and is small in such interval but, diferently, the approximated unimodal approaches [30], in Figure 7 we can see a UMAL conditional medians of the ensemble are producing a sim- forecast5 [32] of the previously presented data set, where ilar forecast given the previous and posterior shape of the blueish areas are the ones that has higher likelihood. data is clearly defined (and the consequence behaviour Overall, we can observe that lower likelihood points that use to perform NN models). This fact could tend to are those where the conditional variability is higher. change when is high dimensional but, if the ensemble Therefore, between [0., 0.2], [0.4, 0.6] and [0.8, 1]. This is naively approximated, we do not have any guarantee behaviour contrast with the presented in previous Figthat the discrepancy will be always higher in zones where ures 5 and 6 as we will discuss in the next subsection () is lower using such NN models. when an integrated approach will be designed.
Importantly, similarly to the Domain case, it is worth to Importantly, isolated aletoric uncertainty fixes a cerhighlight that high conditional variability zones between tain model parametrized by . This can be seen as one
mate the conditional mean, - or median -, which comes from min- 5The UMAL model learns a conditional mixture of an infinite number imizing the mean square error [34], - or mean absolute error, re- of Asymmetric Laplacians using a neural network. Therefore, it spectively [35]-. can learn multi-modalities and asymmetries if they appear. of the components of the ensemble in the previous Epistemic subsection and, based on this idea, we can build an integration procedure to visualize all the presented uncertainty sources as follows.
uncertainty types is useful to synthesize all this complex information in a single plot. Based on the previously introduced visualization types, we can represent the presented Domain, Epistemic and Aleatoric uncertainties in an integrated visualization using Eq. 1, as shown in Figure 8. This representation displays the confidence in all the uncertainty sources using the (* , * | , ) estimated information, which includes modelization of the outlier detection, model uncertainty and conditional irreducible uncertainty.
crease the use of Artificial Intelligence (AI) systems for critical autonomous decision-making. This emphasizes the importance of developing robust and reliable methods, where determining the sources of the uncertainty constitutes an essential pillar.
Still, in the literature, there is not a clear standard to identify the sources of the uncertainty. To support this objective, in this paper we have presented the Domain, Epistemic and Aleatoric (DEA) disentanglement; a formal methodology to divide the uncertainty sources from a probabilistic viewpoint for any (regression or classiifcation) supervised learning model. Furthermore, we presented a comparison of the DEA disentanglement to other literature nomenclatures and approaches.
possibility to combine all the literature uncertainty modelling techniques. Furthermore, it provides an integrated procedure to visualize them together for the sake of recognizing which is the efect of each uncertainty type.
Overall, we hope that the presented framework can help to build a backbone where to connect future research in designing autonomous AI-based systems that requires Safety certifications and to tackle other needs beyond.
ing from the Horizon Europe Programme under the SAFEXPLAIN Project (www.safexplain.eu), grant agreement num. 101069595 and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 772773). Additionally, this work has been partially supported by Grant PID2019-107255GB-C21 funded by MCIN/AEI/ 10.13039/501100011033. [5] H. Tunga, R. Saha, S. Kar, A method of fully au- temic uncertainty in machine learning: An introtonomous driving in self-driving cars based on ma- duction to concepts and methods, Machine Learnchine learning and deep learning, Intelligent Multi- ing 110 (2021) 457–506.
modal Data Processing (2021) 131–156. [20] H. Lin, Bayesian epistemology (2022). [6] R. Culkin, S. R. Das, Machine learning in finance: [21] A. Brando, Aleatoric uncertainty modelling in rethe case of deep learning for option pricing, Journal gression problems using deep learning, Universitat of Investment Management 15 (2017) 92–100. de Barcelona (2022). [7] J. De Spiegeleer, D. B. Madan, S. Reyners, [22] V. Vapnik, Principles of risk minimization for learnW. Schoutens, Machine learning for quantitative ing theory, Advances in neural information proifnance: fast derivative pricing, hedging and fitting, cessing systems 4 (1991).
Quantitative Finance 18 (2018) 1635–1643. [23] J. Berner, P. Grohs, A. Jentzen, Analysis of the [8] A. Diez-Olivan, J. Del Ser, D. Galar, B. Sierra, Data generalization error: Empirical risk minimization fusion and machine learning for industrial progno- over deep artificial neural networks overcomes the sis: Trends and perspectives towards industry 4.0, curse of dimensionality in the numerical approximaInformation Fusion 50 (2019) 92–111. tion of black–scholes partial diferential equations, [9] F. Noé, G. De Fabritiis, C. Clementi, Machine learn- SIAM Journal on Mathematics of Data Science 2 ing for protein folding and dynamics, Current opin- (2020) 631–657.
ion in structural biology 60 (2020) 77–84. [24] H. Zhang, M. Cisse, Y. N. Dauphin, D. Lopez-Paz, [10] J. Jumper, R. Evans, A. Pritzel, T. Green, M. Figurnov, mixup: Beyond empirical risk minimization, in: O. Ronneberger, K. Tunyasuvunakool, R. Bates, International Conference on Learning RepresentaA. Žídek, A. Potapenko, et al., Highly accurate tions, 2018. protein structure prediction with alphafold, Nature [25] T. J. Sullivan, Introduction to uncertainty quantifi596 (2021) 583–589. cation, volume 63, Springer, 2015. [11] A. Hevelke, J. Nida-Rümelin, Responsibility for [26] I. Hacking, et al., The emergence of probability: A crashes of autonomous vehicles: an ethical analysis, philosophical study of early ideas about probabilScience and engineering ethics 21 (2015) 619–630. ity, induction and statistical inference, Cambridge [12] P. Lin, Why ethics matters for autonomous cars, in: University Press, 2006.
Autonomous driving, Springer, Berlin, Heidelberg, [27] J. Von Plato, Creating modern probability: Its math2016, pp. 69–85. ematics, physics and philosophy in historical per[13] A. Maxmen, Self-driving car dilemmas reveal that spective, Cambridge University Press, 1994. moral choices are not universal, Nature 562 (2018) [28] A. Der Kiureghian, O. Ditlevsen, Aleatory or epis469–469. temic? does it matter?, Structural safety 31 (2009) [14] M. Abdar, F. Pourpanah, S. Hussain, D. Rezazade- 105–112.
gan, L. Liu, M. Ghavamzadeh, P. Fieguth, X. Cao, [29] H. G. Matthies, Quantifying uncertainty: Modern A. Khosravi, U. R. Acharya, et al., A review of uncer- computational representation of probability and tainty quantification in deep learning: Techniques, applications, in: Extreme man-made and natural applications and challenges, Information Fusion 76 hazards in dynamics of structures, Springer, 2007, (2021) 243–297. pp. 105–135. [15] W. J. Maddox, P. Izmailov, T. Garipov, D. P. Vetrov, [30] A. Kendall, Y. Gal, What uncertainties do we need A. G. Wilson, A simple baseline for bayesian un- in bayesian deep learning for computer vision?, Adcertainty in deep learning, Advances in Neural vances in neural information processing systems Information Processing Systems 32 (2019). 30 (2017). [16] A. Loquercio, M. Segu, D. Scaramuzza, A general [31] N. Tagasovska, D. Lopez-Paz, Single-model unframework for uncertainty estimation in deep learn- certainties for deep learning, Advances in Neural ing, IEEE Robotics and Automation Letters 5 (2020) Information Processing Systems 32 (2019). 3153–3160. [32] A. Brando, J. A. Rodriguez, J. Vitria, A. Ru[17] Y. Gal, Z. Ghahramani, Dropout as a bayesian ap- bio Muñoz, Modelling heterogeneous distributions proximation: Representing model uncertainty in with an uncountable mixture of asymmetric lapladeep learning, in: international conference on ma- cians, Advances in neural information processing chine learning, PMLR, 2016, pp. 1050–1059. systems 32 (2019). [18] A. Ashukha, A. Lyzhov, D. Molchanov, D. Vetrov, [33] G. Li, L. Yang, C.-G. Lee, X. Wang, M. Rong, A Pitfalls of in-domain uncertainty estimation and bayesian deep learning rul framework integrating ensembling in deep learning, arXiv preprint epistemic and aleatoric uncertainties, IEEE TransacarXiv:2002.06470 (2020). tions on Industrial Electronics 68 (2020) 8829–8841. [19] E. Hüllermeier, W. Waegeman, Aleatoric and epis- [34] R. Moore, J. DeNero, L1 and l2 regularization for multiclass hinge loss models, in: Symposium on machine learning in speech and language processing, 2011. [35] M. Schmidt, G. Fung, R. Rosales, Fast optimization methods for l1 regularization: A comparative study and two new approaches, in: European Conference on Machine Learning, Springer, 2007, pp. 286–297.