Uncertainty in Theory and Practice

• Uncertainty is a multifaceted concept defined by formal measures that distinguish between randomness, ignorance, ambiguity, and epistemic limitations.
• Theoretical frameworks, including the aleatoric/epistemic split and majorization-based approaches, reveal distinct trade-offs in quantum measurements and machine learning predictions.
• Operational methods, from set-based to hierarchical formulations, extend uncertainty analysis to complex systems, enhancing practical insights across diverse applications.

Uncertainty denotes a family of formal and operational notions rather than a single invariant quantity. In quantum theory it may denote the limits on simultaneously sharp values of non-commuting observables, the uncertainty of a quantum state relative to a fixed projective measurement, or the uncertainty contained in the probability distribution induced by a chosen observable (Busch et al., 2019, Zhao et al., 2024, Gudder, 2024). In machine learning and inference it denotes the degree of confidence or lack thereof in a model’s predictions, and in broader uncertainty theory it can be represented by sets, capacities, credal families, or hierarchies of higher-order uncertainty rather than by a single probability measure (Weytjens et al., 7 Oct 2025, Cuzzolin, 2021, Adachi, 2023). Across these domains, uncertainty is treated as a property of distributions, states, measurements, feasible sets, or communicative acts, depending on whether the central issue is randomness, ignorance, ambiguity, incompatibility, or explanation (Sornette, 18 Oct 2025, Sourati et al., 27 Mar 2025).

1. Conceptual distinctions and uncertainty measures

A recurrent distinction separates risk from second-order or Knightian uncertainty. A broad survey of uncertainty theories argues that probability theory is “far from being the most general mathematical theory of uncertainty,” because classical Kolmogorov probability is tailored to first-order uncertainty while many applications involve uncertainty about the law, the model, or the probabilities themselves (Cuzzolin, 2021). A related distinction appears in complex-systems work that explicitly separates randomness from uncertainty: randomness is treated as intrinsic variability, whereas uncertainty is framed as ignorance, incomplete models, institutional blindness, or cognitive bias (Sornette, 18 Oct 2025).

A second recurrent distinction is the aleatoric/epistemic split. In machine learning, epistemic uncertainty is model uncertainty caused by lack of data or insufficient coverage of the domain, whereas aleatoric uncertainty is caused by noise, randomness, or inherent variability in the data (Weytjens et al., 7 Oct 2025). In visualization, aleatory uncertainty is attributed to random fluctuation and epistemic uncertainty to lack of knowledge (Mehta, 2022). In scientific communication, the distinction is reformulated as objective/statistical uncertainty, expressed through $p$-values or confidence intervals, versus subjective or language-based uncertainty, expressed through hedges, modal expressions, and cautious phrasing (Sourati et al., 27 Mar 2025).

A third distinction concerns the formal status of an uncertainty measure. Measure-independent approaches require invariance under relabelling and monotonicity under uncertainty-increasing transformations. One majorization-based formulation states that Schur-concave functions are the most general uncertainty quantifiers under invariance under mere relabelling of measurement outcomes (Friedland et al., 2013). An operational information-theoretic formulation defines a measure of uncertainty as a function monotone under both symmetry and recovery classes of doubly stochastic transformations (1505.02223). In a finite-dimensional state-based setting, uncertainty functions $f:P_d\to[0,1]$ are characterized by four axioms: zero at maximal certainty, one at maximal uncertainty, symmetry under permutation, and concavity (Gudder, 2024). This suggests that, across otherwise different theories, uncertainty is frequently formalized by symmetry and monotonicity principles rather than by a single privileged formula.

2. Quantum uncertainty in observables, regions, and measurement

For non-commuting observables, the canonical preparation uncertainty relations are the Robertson and Schrödinger inequalities. In one standard form,

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$

with Schrödinger’s refinement

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$

These express the familiar trade-off in standard deviations arising from non-commutativity (Li et al., 2020).

A more geometric viewpoint replaces a single inequality by an uncertainty region. For a pair $A,B$, the preparation uncertainty region is

$${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$

This viewpoint emphasizes that many textbook relations are coarse, may become trivial for bounded observables, and can miss upper bounds, asymmetries, non-convexity, or multiple boundary segments. In qubit systems, some regions can be characterized exactly; for qutrits, the Schrödinger inequality need not determine the full region (Busch et al., 2019).

Quantum uncertainty is also analyzed in three operational versions: state preparation, joint measurement, and disturbance. In the language of positive operator measures, sharp observables are jointly measurable iff they commute, whereas noncommuting unsharp observables can be jointly measurable. For approximate position and momentum observables $M_1,M_2$, a joint-measurement tradeoff of the form

$d(Q,M_1)\, d(P,M_2) \ge Ch$

is presented as an operational Heisenberg relation (Busch, 2010).

Measurement-theoretic analyses refine the disturbance version further. In a position measurement model with linear post-measurement transformations, the relation

$$\epsilon(x_t)\,(\Delta p)_{\mathrm{dis}} \ge \frac{\hbar}{2}$$

holds when the uncertainty is the measurement error of the post-measurement position $x_t$. For the pre-measurement position $f:P_d\to[0,1]$0, the relevant uncertainty in the measurement result is not $f:P_d\to[0,1]$1 but

$f:P_d\to[0,1]$2

with the distinction attributed to wave-packet reduction (Kosugi, 2010). A common misconception is therefore that a single “measurement error” always plays the role of position uncertainty; this formulation explicitly rejects that identification.

3. Generalized quantum uncertainty and state-based formulations

One generalized formulation treats the Heisenberg relation as only the leading term of a broader hierarchy. The cumulant-generating function for an observable $f:P_d\to[0,1]$3 is

$f:P_d\to[0,1]$4

and for two observables

$f:P_d\to[0,1]$5

In this framework, $f:P_d\to[0,1]$6 is the covariance-like linear correlation term, while higher cross cumulants such as $f:P_d\to[0,1]$7 encode higher-order nonlinear dependence. The main generalized uncertainty relation is derived from Cauchy–Schwarz applied to $f:P_d\to[0,1]$8 and $f:P_d\to[0,1]$9, and combines classical/statistical dependence from cumulants with quantum/noncommutative dependence from the Baker–Campbell–Hausdorff expansion (Li et al., 2020). The familiar Heisenberg/Schrödinger structure is then the second-order, linear-dependence sector; higher-order relations, including a third-order skewness uncertainty relation, probe distribution asymmetry and nonlinear dependence.

A different line of work treats uncertainty as an intrinsic property of a quantum state relative to a fixed projective measurement $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$0, where the relevant data are the diagonal probabilities $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$1. In this setting the free states are the basis states $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$2, uncertainty-preserving operations are exactly those whose Kraus operators have the form

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$3

and maximally uncertain states are precisely the states with uniform diagonal,

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$4

The same framework gives a universal decomposition

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$5

where $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$6 is a coherence-type quantity interpreted as quantum uncertainty and $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$7 as classical uncertainty, together with

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$8

It also defines a geometric uncertainty

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left|\langle [X,Y]\rangle\right|^2,$$9

linking uncertainty directly to fidelity-based geometric coherence (Zhao et al., 2024).

A related finite-dimensional formulation defines state uncertainty contextually as

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$0

where $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$1 is an uncertainty function on the probability simplex. Within this axiomatic framework, four examples are singled out: variance uncertainty, entropy uncertainty, geometric uncertainty, and sine uncertainty (Gudder, 2024). This suggests a systematic split between a quantum component, namely the observable-induced distribution, and a classical component, namely the scalar functional applied to that distribution.

Uncertainty is also tied to complementarity. For a $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$2-path interferometer, a complete relation is given by

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$3

where $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$4 is total quantum uncertainty, $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$5 total classical uncertainty, and $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$6 linear predictability. The total quantum uncertainty is shown to be exactly the Wigner–Yanase coherence,

$$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$7

while the classical part becomes a quantum-correlation quantifier for subsystems of bipartite pure states (Basso et al., 2020).

4. Beyond precise probability: operational, set-based, and hierarchical frameworks

General uncertainty theory contains several clusters that extend or relax classical probability. These include probability robustified, behavioural probability, generalising set theory, generalising measure theory, generalising Bayesian reasoning, set-valued probability, and measures on functional spaces (Cuzzolin, 2021). Within this landscape, capacities, credal sets, lower and upper probabilities, possibility measures, belief functions, and random sets are not treated as isolated alternatives but as interconnected formalisms with different levels of generality.

Operationally measure-independent approaches begin from transformations rather than formulas. For a single classical variable with distribution $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$8, uncertainty is a real-valued function $$\Delta X^2 \Delta Y^2 \ge \frac{1}{4}\left| \langle [X,Y]\rangle \right|^2 + \frac{1}{4}\left| \langle \{X,Y\}\rangle - 2\langle X\rangle \langle Y\rangle \right|^2.$$9 that should be invariant under the relevant symmetry group and monotone under symmetry-based and recovery-based doubly stochastic transformations. Joint uncertainty is then a function $A,B$0 satisfying

$A,B$1

and need not be tied to a product distribution or direct-sum construction. One example is

$A,B$2

for which a qubit relation

$A,B$3

is derived for two rank-1 projective measurements (1505.02223). A complementary majorization-based approach yields universal uncertainty relations of the form

$A,B$4

from which every scalar relation generated by a Schur-concave uncertainty quantifier follows (Friedland et al., 2013).

Set-based inference replaces distributions by feasible sets. An uncertainty variable is defined as

$A,B$5

where $A,B$6 is the uncertainty set. Conditional uncertainty is represented by a set-valued map $A,B$7, Bayes-like updating takes the form

$A,B$8

and independence is characterized by factorization,

$A,B$9

This framework supports analogues of Bayes’ law, the law of total probability, conditional independence, and Bayesian networks, yielding Bayesian uncertainty networks whose local and global conditional independence properties parallel those of ordinary Bayesian networks (Talak et al., 2019).

Hierarchical formalisms extend this set-based and capacity-based perspective to multiple layers. An uncertainty space is

$${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$0

with $${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$1 a non-empty set of capacities on $${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$2. A U-sequence is a hierarchy $${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$3 such that

$${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$4

This yields uncertainty over uncertainty, and then uncertainty over that higher-order uncertainty. The associated universal uncertainty space is constructed as an inverse limit of iterated liftings, and is proposed as a basis for multi-layer uncertainty theory (Adachi, 2023).

The relation between information and uncertainty is not monotone without additional conditions. For conditional variance

$${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$5

the intuition that shrinking an interval $${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$6 must reduce uncertainty is stated to be false in general. For absolutely continuous $${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$7, partial monotonicity of $${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$8 is characterized by log-concavity conditions on double-integral transforms of the distribution function; if the CDF is log-concave on an interval $${\rm PUR}_\Delta(A,B)=\Bigl\{(\Delta A,\Delta B)\in\mathbb{R}^2 \;\big|\; \exists \rho\in\mathcal S(\mathcal H): \Delta A=\Delta_\rho A,\ \Delta B=\Delta_\rho B\Bigr\}.$$9, then

$M_1,M_2$0

for intervals $M_1,M_2$1 (Chen, 2011). A common simplification—“more information always reduces uncertainty”—therefore requires explicit structural assumptions.

5. Statistical, inferential, and machine-learning uncertainty

In machine learning, predictive uncertainty is commonly decomposed into total, aleatoric, and epistemic components. One formulation defines

$M_1,M_2$2

$M_1,M_2$3

with MC dropout used to approximate the Bayesian predictive distribution (Roberts et al., 5 Mar 2025). A broader practical treatment distinguishes epistemic uncertainty as reducible model uncertainty and aleatoric uncertainty as generally irreducible data noise, and develops uncertainty quantification for linear regression, random forests, neural networks, and conformal prediction (Weytjens et al., 7 Oct 2025).

These methods differ in their primitives. In ordinary least squares, uncertainty is expressed by confidence intervals based on residual variance and $M_1,M_2$4-critical values. In random forests, total uncertainty is the Shannon entropy of the ensemble class distribution,

$M_1,M_2$5

while conditional entropy and mutual information separate aleatoric and epistemic components. In neural networks, Bayesian neural networks place distributions over weights, and inference-time dropout approximates posterior predictive means and variances. Conformal prediction provides prediction sets or intervals with finite-sample coverage guarantees,

$M_1,M_2$6

under exchangeability of calibration and test data (Weytjens et al., 7 Oct 2025).

Uncertainty is also treated as something to be explained. A concept-based framework states that uncertainty is “not just a number to be estimated, but also something whose causes can be explained,” especially in high-dimensional settings where raw feature attributions are too local. The method computes uncertainty for each input, splits the data into certain and uncertain groups by a two-component Gaussian Mixture Model, extracts concepts separately for the groups by NMF on a nonnegative latent space, and estimates concept importance with Sobol-based sensitivity analysis, yielding both local and global explanations (Roberts et al., 5 Mar 2025). A plausible implication is that uncertainty explanation is becoming a distinct layer above uncertainty estimation.

Inferential uncertainty need not concern predictions alone. In ranking problems, the uncertainty in estimated ranks is determined by the uncertainty in the parameter estimates. Given interval estimates $M_1,M_2$7, the induced strict partial order

$M_1,M_2$8

determines the compatible rankings, which are exactly the linear extensions $M_1,M_2$9. The set estimator

$d(Q,M_1)\, d(P,M_2) \ge Ch$0

is a valid confidence set for the true ranking when the intervals have simultaneous coverage $d(Q,M_1)\, d(P,M_2) \ge Ch$1, and its normalized size $d(Q,M_1)\, d(P,M_2) \ge Ch$2 is proposed as a measure of ranking uncertainty (Rising, 2021).

6. Task-specific uncertainty, visualization, and scientific communication

Task-specific work often shows that uncertainty attached to an upstream model is not the uncertainty relevant to the downstream decision. In learning-based image registration, transformation uncertainty and appearance uncertainty are described as poor proxies for label-propagation error, and the problem is reformulated in terms of epistemic segmentation uncertainty and aleatoric segmentation uncertainty. The former is defined as the voxelwise entropy of mean propagated label probabilities; the latter is predicted by a compact auxiliary DNN from the absolute appearance difference $d(Q,M_1)\, d(P,M_2) \ge Ch$3 and feature maps from the registration network, trained with a $d(Q,M_1)\, d(P,M_2) \ge Ch$4-NLL loss. On 3D cardiac MRI data, the reported mean Pearson correlations with label propagation error were $d(Q,M_1)\, d(P,M_2) \ge Ch$5 for transformation uncertainty, $d(Q,M_1)\, d(P,M_2) \ge Ch$6 for appearance uncertainty, $d(Q,M_1)\, d(P,M_2) \ge Ch$7 for epistemic segmentation uncertainty, $d(Q,M_1)\, d(P,M_2) \ge Ch$8 for aleatoric segmentation uncertainty, and $d(Q,M_1)\, d(P,M_2) \ge Ch$9 for their combination (Chen et al., 2024). This directly challenges the common substitution of model-internal uncertainty for task-level uncertainty.

When uncertainty is communicated visually, the representation itself becomes part of the problem. A survey of uncertainty visualization argues that when visualizations are used for inference and decision-making, uncertainty must also be visualized. It organizes the visualization pipeline into data collection, preprocessing, visualization, and inference, and traces uncertainty to measurement error, bias, sampling error, nonresponse error, missing data, rounding, rescaling, resampling, and quantizing. The same survey notes that box plots and error bars are common encodings but often difficult for viewers to interpret, while hypothetical outcome plots and bubble treemaps are explored as alternatives (Mehta, 2022).

Scientific writing introduces another layer: uncertainty in language. A study of scientific abstracts distinguishes statistical uncertainty from subjective uncertainty, measures the latter at sentence level on conclusive sentences, compares five estimators, and reports that fine-tuned SciBERT and a 5-layer LSTM outperform the alternatives, with SciBERT slightly best. It further reports that subjective uncertainty varies across disciplines, years, and geographical locations; biology has the lowest subjective certainty on average among the ten fields examined, more quantitative and computational fields tend to express higher certainty, stronger echo-chamber effects are associated with higher subjective certainty in most STEM fields, and by the mid-2010s all disciplines show significantly negative correlations between certainty and citations (Sourati et al., 27 Mar 2025). A common misconception is therefore that uncertainty in scientific work is exhausted by formal statistical metrics; the linguistic presentation of findings is itself measurable and field-dependent.

In complex systems and policy, uncertainty is again reframed. One recent account argues that uncertainty should not be equated with randomness, that much of what is labeled uncertain is in fact epistemic, and that prediction should shift “from prophecy to diagnosis,” emphasizing feedbacks, critical thresholds, early-warning signals, adaptive leadership, transparent communication, and systemic learning (Sornette, 18 Oct 2025). This suggests a unifying theme across otherwise disparate literatures: uncertainty is often less a single object than a structured mismatch between what is known, what is represented, and what a task actually requires.