Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?

The notion of uncertainty has recently drawn increasing attention in machine learning (ML) and artificial intelligence (AI) due to the fields’ burgeoning relevance for practical applications, many of which have safety requirements, such as in medical domains [Lambrou et al., 2010, Senge et al., 2014, Yang et al., 2009] or socio-technical systems [Varshney, 2016, Varshney and Alemzadeh, 2017]. These applications to safety-critical contexts show that a suitable representation and quantification of uncertainty for modern, reliable machine learning systems is imperative.

In general, the literature makes a distinction between aleatoric and epistemic uncertainties (AU and EU, respectively) [Hora, 1996]. While the former is caused by the inherent randomness of the data-generating process, EU results from the learner’s lack of knowledge regarding the true underlying model; it also includes approximation uncertainty. Since EU can be reduced per se with further information (e.g., via data augmentation using semantic preserving transformations), it is also referred to as reducible uncertainty. In contrast, aleatoric uncertainty, as a property of the data-generating process, is irreducible [Hüllermeier and Waegeman, 2021]. The importance of distinguishing between different types of uncertainty is reflected in several areas of recent machine learning research, e.g. in Bayesian deep learning [Depeweg et al., 2018, Kendall and Gal, 2017], in adversarial example detection [Smith and Gal, 2018], or data augmentation in Bayesian classification [Kapoor et al., 2022]. A qualitative representation of total uncertainty, AU, and EU, and of their asymptotic behavior as the number of data points available to the learning agent increases, is given in Figure 1.

Typically, uncertainty in machine learning, artificial intelligence, and related fields is expressed solely in terms of probability theory. That is, given a measurable space $(\Omega,\mathcal{A})$, uncertainty is entirely represented by defining one single probability measure $P$ on $(\Omega,\mathcal{A})$. However, representing uncertainty in machine learning is not restricted to classical probability theory; various aspects of uncertainty representation and quantification in ML are discussed by Hüllermeier and Waegeman [2021]. Credal sets, i.e., (convex) sets of probability measures, are considered to be very popular models of uncertainty representation, especially in the field of imprecise probabilities (IP) [Augustin et al., 2014, Walley, 1991]. Credal sets are also very appealing from an ML perspective for representing uncertainty, as they can represent both aleatoric and epistemic uncertainty (as opposed to a single probability measure). Numerous scholars emphasized the utility of representing uncertainty in ML via credal sets, e.g., credal classification [Zaffalon, 2002, Corani and Zaffalon, 2008] based on the Imprecise Dirichlet Model (IDM) [Walley, 1996], generalizing Bayesian networks to credal classifiers [Corani et al., 2012], or building credal decision-trees [Abellán and Moral, 2003].

Uncertainty representation via credal sets also requires a corresponding quantification of the underlying uncertainty, referred to as credal uncertainty quantification (CUQ). The task of (credal) uncertainty quantification translates to finding a suitable measure that can accurately reflect the uncertainty inherent to a credal set. In many ML applications, such as active learning [Settles, 2009] or classification with abstention, there is a need to quantify (predictive) uncertainty in a scalar way. Appropriate measures of uncertainty are often axiomatically justified [Bronevich and Klir, 2008, 2010].

Contributions. In this work, we consider the volume of the geometric representation of a credal set on the label space as a quite obvious and intuitively plausible measure of EU. We argue that this measure is indeed meaningful if we are in a binary classification setting. However, in a multi-class setting, the volume exhibits shortcomings that make it unsuitable for quantifying EU associated with a credal set.

Structure of the paper. The paper is divided as follows. Section 2 formally introduces the framework we work in, and Section 3 discusses the related literature. Section 4 presents our main findings, which are further discussed in Section 5. Proofs of our theoretical results are given in Appendix A, and (a version of) Carl-Pajor’s theorem, intimately related to Theorem 1, is stated in Appendix B.

Uncertainty is a crucial concept in many academic and applied disciplines. However, since its definition depends on the specific context a scholar works in, we now introduce the formal framework of supervised learning within which we will examine it.

Let $(\mathcal{X},\sigma(\mathcal{X}))$, and $(\mathcal{Y},\sigma(\mathcal{Y}))$ be two measurable spaces, where $\sigma(\mathcal{X})$, and $\sigma(\mathcal{Y})$ are suitable $\sigma$-algebras. We will refer to $\mathcal{X}$ as instance space (or equivalently, input space) and to $\mathcal{Y}$ as label space. Further, the sequence $\{({x}_{i},y_{i})\}_{i=1}^{n}\in(\mathcal{X}\times\mathcal{Y})^{n}$, is called training data. The pairs $(x_{i},y_{i})$ are realizations of random variables $(X_{i},Y_{i})$, which are assumed independent and identically distributed (i.i.d.) according to some probability measure $P$ on $(\mathcal{X}\times\mathcal{Y},\sigma(\mathcal{X}\times\mathcal{Y}))$.

Note that in Definition 1, the assumption of convexity is quite natural and considered to be rational (see, e.g., Levi [1980]). It is also mathematically appealing, since, as shown by Walley [1991, Section 3.3.3], the “lower boundary” $\underline{P}$ of $\mathcal{P}$, defined as $\underline{P}(A)\coloneqq\inf_{P\in\mathcal{P}}P(A)$, for all $A\in\mathcal{A}$ and called the lower probability associated with $\mathcal{P}$, is coherent [Walley, 1991, Section 2.5].

Further, in a supervised learning setting, we assume a hypothesis space $\mathcal{H}$, where each hypothesis $h\in\mathcal{H}$ maps a query instance $\mathbf{x_{q}}$ to a probability measure $P$ on $(\mathcal{Y},\sigma(\mathcal{Y}))$. We distinguish between different “degrees” of uncertainty-aware predictions, which are depicted in Table 1.

We denote by $\text{Cr}(\mathcal{Y})$ the set of all credal sets on $(\mathcal{Y},\sigma(\mathcal{Y}))$. While probabilistic predictions $h(\mathbf{x_{q}})=\hat{y}$ fail to capture the epistemic part of the (predictive) uncertainty, predictions in the form of credal sets $h(\mathbf{x_{q}})=\mathcal{P}\subseteq\mathcal{M}(\mathcal{Y},\sigma(\mathcal{Y}))$ account for both types of uncertainty. It should also be remarked that representing uncertainty is not restricted to the credal set formalism. Another possible framework to represent AU and EU is that of second-order distributions; they are commonly applied in Bayesian learning and have been recently inspected in the context of uncertainty quantification by Bengs et al. [2022].

In this paper, we restrict our attention to the credal set representation. Given a credal prediction set, it remains to properly quantify the uncertainty encapsulated in it using a suitable measure. Credal set representations are often illustrated in low dimensions (usually $d=2$ or $d=3$). Examples of such geometrical illustrations can be found in the context of machine learning in [Hüllermeier and Waegeman, 2021] and in imprecise probability theory in [Walley, 1991, Chapter 4]. This suggests that a credal set and its geometric representation are strictly intertwined. We will show in the following sections that this intuitive view can have disastrous consequences in higher dimensions and that one should exercise caution in this respect. Furthermore, it remains to be discussed whether a geometric viewpoint on (predictive) uncertainty quantification is in fact sensible.

In this section we examine some axiomatically defined properties of (credal) uncertainty measures. For a more detailed discussion of various (credal) uncertainty measures in machine learning and a critical analysis thereof, we refer to Hüllermeier et al. [2022].

Let $S$ denote the Shannon entropy [Shannon, 1948], whose discrete version is defined as

A suitable measure of credal uncertainty $U:\text{Cr}(\mathcal{Y})\rightarrow\mathbb{R}$ should satisfy the following axioms proposed by Abellán and Klir [2005], Jiroušek and Shenoy [2018]:

• A1

  Non-negativity and boundedness:

  • (i)

    $U(\mathcal{P})\geq 0$, for all $\mathcal{P}\in\text{Cr}(\mathcal{Y})$;

  • (ii)

    there exists $u\in\mathbb{R}$ such that $U(\mathcal{P})\leq u$, for all $\mathcal{P}\in\text{Cr}(\mathcal{Y})$.

• A2

  Continuity: $U$ is a continuous functional.

• A3

  Monotonicity: for all $\mathcal{Q},\mathcal{P}\in\text{Cr}(\mathcal{Y})$ such that $\mathcal{Q}\subset\mathcal{P}$, we have $U(\mathcal{Q})\leq U(\mathcal{P})$.

• A4

  Probability consistency: for all $\mathcal{P}\in\text{Cr}(\mathcal{Y})$ such that $\mathcal{P}=\{P\}$, we have $U(\mathcal{P})=S(P)$.

• A5

  Sub-additivity: Suppose $\mathcal{Y}=\mathcal{Y}_{1}\times\mathcal{Y}_{2}$, and let $\mathcal{P}$ be a joint credal set on $\mathcal{Y}$ such that $\mathcal{P}^{\prime}$ is the marginal credal set on $\mathcal{Y}_{1}$ and $\mathcal{P}^{\prime\prime}$ is the marginal credal set on $\mathcal{Y}_{2}$, respectively. Then, we have

  $$U(\mathcal{P})\leq U(\mathcal{P}^{\prime})+U(\mathcal{P}^{\prime\prime}).$$

  (1)

• A6

  Additivity: If $\mathcal{P}^{\prime}$ and $\mathcal{P}^{\prime\prime}$ are independent, (1) holds with equality.

In axiom A6, independence refers to a suitable notion for independence of credal sets, see e.g. Couso et al. [1999]. An axiomatic definition of properties for uncertainty measures is a common approach in the literature [Pal et al., 1992, 1993]. Examples of credal uncertainty measures that satisfy some of the axioms A1–A6 are the maximal entropy [Abellan and Moral, 2003] and the generalized Hartley measure [Abellán and Moral, 2000].

Recall that the lower probability $\underline{P}$ of $\mathcal{P}$ is defined as $\underline{P}(A)\coloneqq\inf_{P\in\mathcal{P}}P(A)$, for all $A\in\sigma(\mathcal{Y})$, and call upper probability its conjugate $\overline{P}(A)\coloneqq 1-\underline{P}(A^{c})=\sup_{P\in\mathcal{P}}P(A)$, for all $A\in\sigma(\mathcal{Y})$. Since we are concerned with the fundamental question of whether the volume functional is a suitable measure for epistemic uncertainty, we replace A4 with the following axiom that better suits our purposes.

• A4’

  Probability consistency: $U(\mathcal{P})$ reduces to $0$ as the distance between $\overline{P}(A)$ and $\underline{P}(A)$ goes to $0$, for all $A\in\sigma(\mathcal{Y})$.

While A4’ addresses solely the epistemic component of uncertainty assoicated with the credal set $\mathcal{P}$, A4 incorporates the aleatoric uncertainty. Finally, we introduce a seventh axiom that subsumes a desirable property of $U$ proposed by Hüllermeier et al. [2022, Theorem 1.A3-A5].

• A7

  Invariance: $U$ is invariant to rotation and translation.

Call $d$ the cardinality of the label space $\mathcal{Y}$. In the next section, we will note that many of these axioms are satisfied by the volume operator in the case $d=2$ but can no longer be guaranteed for $d>2$.

As pointed out in Section 3, there is no unambiguous measure of (credal) uncertainty for machine learning purposes. In this section, we present a measure for EU rooted in the geometric concept of volume and show how it is well-suited for a binary classification setting, while it loses its appeal when moving to a multi-class setting.

Since we are considering a classification setting, we assume that $\mathcal{Y}$ is a finite Polish space so that $|\mathcal{Y}|=d$, for some natural number $d\geq 2$. We also let $\sigma(\mathcal{Y})=2^{\mathcal{Y}}$ to work with the finest possible $\sigma$-algebra of $\mathcal{Y}$; the results we provide still hold for any coarser $\sigma$-algebra.¹¹1Call $\tau$ the topology on $\mathcal{Y}$. The ideas expressed in this paper can be easily extended to the case where $\mathcal{Y}$ is not Polish. We require it to convey our results without dealing with topological subtleties. Because $\mathcal{Y}$ is Polish, $\mathcal{M}(\mathcal{Y},\sigma(\mathcal{Y}))$ is Polish as well. In particular, the topology endowed to $\mathcal{M}(\mathcal{Y},\sigma(\mathcal{Y}))$ is the weak topology, which – because we assumed $\mathcal{Y}$ to be finite – coincides with the topology $\tau_{\|\cdot\|_{2}}$ induced by the Euclidean norm. Consider a credal set $\mathcal{P}\subset\mathcal{M}(\mathcal{Y},\sigma(\mathcal{Y}))$, which can be seen as the outcome of a procedure involving an imprecise Bayesian neural network (IBNN) [Caprio et al., 2023a], or an imprecise neural network (INN) [Caprio et al., 2023b]; an ensemble-based approach is proposed by Shaker and Hüllermeier [2020].

Since $\mathcal{Y}=\{y_{1},\ldots,y_{d}\}$, each element $P\in\mathcal{P}$ can be seen as a $d$-dimensional probability vector, $P=(p_{1},\ldots,p_{d})^{\top}$, where $p_{j}=P(\{y_{j}\})$, $j\in\{1,\ldots,d\}$, $p_{j}\geq 0$, for all $j\in\{1,\ldots,d\}$, and $\sum_{j=1}^{d}p_{j}=1$. This entails that if we denote by $\Delta^{d-1}$ the unit simplex in $\mathbb{R}^{d}$, we have $\mathcal{P}\subset\Delta^{d-1}$, which means that $\mathcal{P}$ is a convex body inscribed in $\Delta^{d-1}$.²²2In the remaining part of the paper, we denote by $\mathcal{P}$ both the credal set and its geometric representation, as no confusion arises.

Intuitively, the “larger” $\mathcal{P}$ is, the higher the credal uncertainty. A natural way of capturing the size of $\mathcal{P}$, then, appears to be its volume $\text{Vol}(\mathcal{P})$. Notice that $\text{Vol}(\mathcal{P})$ is a bounded quantity: its value is bounded from below by $0$ and from above by $\sqrt{d}/[(d-1)!]$, the volume of the whole unit simplex $\Delta^{d-1}$. The latter corresponds to the case where $\mathcal{P}=\Delta^{d-1}$, that is, to the case of completely vacuous beliefs: the agent is only able to say that the probability of $A$ is in $[0,1]$, for all $A\in\mathcal{F}$. In this sense, the volume is a measure of the size of set $\mathcal{P}$ that increases the more uncertain the agent is about the elements of $\mathcal{F}$. This argument shows that $\text{Vol}(\mathcal{P})$ is well suited to capture credal uncertainty. But why is it appropriate to describe EU?³³3The concept of volume has been explored in the imprecise probabilities literature, see e.g., Bloch [1996], [Cuzzolin, 2021, Chapter 17], and Seidenfeld et al. [2012], but, to the best of our knowledge, has never been tied to the notion of epistemic uncertainty. More in general, the geometry of imprecise probabilities has been studied, e.g., by Anel [2021], Cuzzolin [2021]. Think of the extreme case where EU does not exist, so that the agent faces AU only. In that case, they would be able to specify a unique probability measure $P\in\mathcal{M}(\mathcal{Y},\sigma(\mathcal{Y}))$ (or equivalently, $P\in\Delta^{d-1}$), and $\text{Vol}(\{P\})=0$. Hence, if $\text{Vol}(\mathcal{P})>0$, then this means that the agent faces EU. In addition, let $(\mathcal{P}_{n})_{n\in\mathbb{N}}$ be a sequence of credal sets on $(\mathcal{Y},\sigma(\mathcal{Y}))$ representing successive refinements of $\mathcal{P}$ computed as new data becomes available to the agent.⁴⁴4Clearly $|\mathcal{P}_{n}|=|\mathcal{P}|$, for all $n$. If, after observing enough evidence, the EU is resolved, that is, if $\lim_{n\rightarrow\infty}[\overline{P}_{n}(A)-\underline{P}_{n}(A)]=0$ for all $A\in\mathcal{F}$, we see that the following holds. Sequence $(\mathcal{P}_{n})_{n\in\mathbb{N}}$ converges – say in the Hausdorff metric – as $n\rightarrow\infty$ to $\mathcal{P}^{\star}\subset\mathcal{M}(\mathcal{Y},\sigma(\mathcal{Y}))$ such that $|\mathcal{P}^{\star}|=|\mathcal{P}_{n}|$, for all $n$, and all the elements of $\mathcal{P}^{\star}$ are equal to $P^{\star}$, the (unique) probability measure that encapsulates the AU.⁵⁵5Technically $\mathcal{P}^{\star}$ is a multiset, that is, a set where multiple instances for each of its elements are allowed. Through the learning process, we refine our estimates for the “true” underlying aleatoric uncertainty (pertaining to $P^{\star}$), which is left after all the EU is resolved. Then, the geometric representation of $\mathcal{P}^{\star}$ is a point whose volume is $0$. Hence, we have that the volume of $\mathcal{P}_{n}$ converges from above to $0$ (that is, it possesses the continuity property), which is exactly the behavior we would expect as EU resolves.

As we shall see, while this intuitive explanation holds if $d=2$, for $d>2$, continuity fails, thus making the volume not suited to capture EU in a multi-class classification setting. We also show in Theorem 1 that the volume lacks robustness in higher dimensions. Small perturbations to the boundary of a credal set make its volume vary significantly. This may seriously hamper the results of a study, leading to potentially catastrophic consequences in downstream tasks.

Credal sets provide a flexible and powerful formalism for representing uncertainty in various scientific disciplines. In particular, uncertainty representation via credal sets can capture different degrees of uncertainty and allow for a more nuanced representation of epistemic and aleatoric uncertainty in machine learning systems. Moreover, the corresponding geometric representation of credal sets as $d$-dimensional polytopes enables a thoroughly intuitive view of uncertainty representation and quantification.

In this paper, we showed that the volume of a credal set is a sensible measure of epistemic uncertainty in the context of binary classification, as it enjoys many desirable properties suggested in the existing literature. On the other side, the volume forfeits these properties in a multi-class classification setting, despite its intuitive meaningfulness.

In addition, this work stimulates a fundamental question as to what extent a geometric approach to uncertainty quantification (in ML) is sensible.

This is the first step toward studying the geometric properties of (epistemic) uncertainty in AI and ML. In the future, we plan to explore the geometry of aleatoric uncertainty and introduce techniques from high-dimensional geometry and high-dimensional probability to enhance and deepen the study of EU and AU in the contexts of AI and ML.

Yusuf Sale and Michele Caprio contributed equally to this paper.