--- title: Interval Confidence Uncertainty url: https://www.emergentmind.com/topics/interval-confidence-uncertainty type: topic --- # Interval Confidence Uncertainty Searching arXiv for recent and foundational papers on interval confidence uncertainty and related interval-based uncertainty quantification. “Interval confidence uncertainty” is introduced as a probability-mass notion of localization: uncertainty is measured by the minimal length of a single interval that contains a prescribed probability, rather than by variance or entropy [2605.04484]. In this sense, the object of interest is not the spread of an entire distribution, but the smallest interval support needed to guarantee confidence level \( \theta \). Closely related literatures in statistics, simulation, inverse problems, and machine learning likewise treat intervals, coverage, and interval-valued predictions as primary uncertainty objects, though the meanings of “confidence interval,” “uncertainty band,” and “interval-valued prediction” differ substantially across domains [2510.26995, 2310.06923, 2602.04775]. ## 1. Formal definition and scope In the quantum formulation, the paper distinguishes two quantities for position and momentum: the general confidence uncertainty \( \Delta^{c}x(\theta_x) \) and the interval confidence uncertainty \( \Delta^{I}x(\theta_x) \) [2605.04484]. The general version allows any measurable support set, \[ \Delta^{c}x(\theta_x)=\inf\!\left\{\mu(X):\int_{X}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}, \] whereas the interval version restricts the support to a single interval, \[ \Delta^{I}x(\theta_x)=\inf\!\left\{x_2-x_1:\int_{x_1}^{x_2}|{\psi}(x)|^{2}\,d x\ge\theta_x\right\}. \] The same definitions apply in momentum space, yielding \( \Delta^{c}p(\theta_p) \) and \( \Delta^{I}p(\theta_p) \) [2605.04484]. The interval restriction is geometrically significant. Because \( \Delta^c \) may use unions of disjoint sets while \( \Delta^I \) must use one interval, the paper states that \[ \Delta^{c}x\le\Delta^{I}x, \] and that both quantities are monotone non-decreasing in the confidence level [2605.04484]. The resulting uncertainty measure is therefore explicitly tied to containment probability and support geometry, rather than to second moments. This usage differs from classical statistical confidence intervals. In the quantum paper, the interval is a support set for probability mass of a fixed state; in frequentist statistics, a confidence interval is a random interval intended to cover an unknown parameter with prescribed probability over repeated sampling. Related machine-learning papers use yet other interval notions, including predictive intervals, interval-valued scores, and uncertainty bands [2605.04484, 2003.11566, 2601.18897]. ## 2. Phase transition and uncertainty inequalities The defining structural result is a phase transition at the line \[ \theta_x+\theta_p=1. \] For \[ \theta_x+\theta_p\le 1, \] the paper proves that no nontrivial uncertainty product lower bound exists: for every \( \varepsilon>0 \), one can find a state such that both confidence uncertainties are smaller than \( \varepsilon \) [2605.04484]. A special case is \[ \theta_x=\theta_p=\frac12, \] which implies that position and momentum can be simultaneously localized with confidence at least \( 50\% \) [2605.04484]. For \[ \theta_x+\theta_p>1, \] a strictly positive lower bound appears. The general confidence-uncertainty product satisfies \[ \Delta^{c}x(\theta_x)\cdot\Delta^{c}p(\theta_p)\;\ge\;2\pi\hbar\, \Bigl(\sqrt{\theta_x\theta_p}-\sqrt{(1-\theta_x)(1-\theta_p)}\Bigr)^{\!2}, \] derived by combining Lenard’s projection inequality with the Donoho–Stark operator-norm bound [2605.04484]. For the interval version, the paper gives the sharp implicit Landau–Pollak bound \[ \lambda_{0}\!\left(\frac{\Delta^{I}x\cdot\Delta^{I}p}{4\hbar}\right)\ge \Bigl(\sqrt{\theta_x\theta_p}-\sqrt{(1-\theta_x)(1-\theta_p)}\Bigr)^{\!2}, \] equivalently \[ \Delta^{I}x\cdot\Delta^{I}p\;\ge\;4\hbar\,\lambda_{0}^{-1}\!\Bigl((\sqrt{\theta_x\theta_p}-\sqrt{(1-\theta_x)(1-\theta_p)})^{2}\Bigr). \] Here \( \lambda_0(c) \) is the largest prolate-spheroidal eigenvalue [2605.04484]. The paper also records an earlier elementary interval bound, valid only when \( 2\theta_x+\theta_p>2 \), \[ {A}\Big|_{LW=\Delta^{I}\!x\Delta^{I}\!p}\;\ge\;\frac{\pi}{\theta_{x}\!\left[\theta_{p}-2\sqrt{(\theta_{x}+\theta_{p}-1)(1-\theta_{x})}\right]}, \] and explicitly states that it is weaker than the Landau–Pollak result [2605.04484]. The statement that joint \( 50\% \) localization does not contradict Heisenberg is part of the paper’s interpretive framework: the Heisenberg–Kennard bound concerns variances, whereas confidence uncertainty concerns concentration of probability mass in chosen intervals or sets [2605.04484]. ## 3. Sharpness, saturating states, and asymptotics The sharp interval bound is reduced to classical Slepian–Pollak–Landau concentration theory [2605.04484]. The defining operator is \[ (K_c f)(u)=\int_{-1}^{1}\frac{\sin c(u-v)}{\pi(u-v)}f(v)\,d v, \] whose principal eigenvalue is \( \lambda_0(c) \) [2605.04484]. The key identity is \[ {P_X Q_P}^{2}=\lambda_{0}\!\left(\frac{|X||P|}{4\hbar}\right), \] with \( P_X \) and \( Q_P \) the position- and momentum-space projectors for intervals \( X \) and \( P \) [2605.04484]. The interval bound is saturated. The saturating state is the leading Slepian function, namely the principal eigenfunction of the prolate-spheroidal kernel [2605.04484]. The paper describes this as the standard Lenard-equality configuration, so the bound is not merely existential but attained by a concrete optimal state. Asymptotically, the largest Slepian eigenvalue satisfies \[ \lambda_{0}(c)\simeq\frac{2c}{\pi},\qquad c\to0^{+}, \] and \[ 1-\lambda_{0}(c)\simeq 4\sqrt{\pi c}\,e^{-2c},\qquad c\to\infty. \] Hence \[ \lambda_0^{-1}(\theta)\simeq\frac{\pi\theta}{2}\quad(\theta\to0), \qquad \lambda_0^{-1}(\theta)\simeq-\tfrac{1}{2}\ln(1-\theta)\quad(\theta\to1) \] [2605.04484]. The high-confidence regime therefore diverges logarithmically. For \( \theta_x=1 \), \[ \Delta^{I}x\cdot\Delta^{I}p\;\sim\;-2\hbar\ln(1-\theta_p),\qquad\theta_p\to 1^{-} \] [2605.04484]. The numerical evaluation reported in the paper includes - \( c=1.5 \): \( \lambda_0(c)=0.763 \), - \( c=4 \): \( \lambda_0(c)=0.996 \), - \( c=10 \): \( 1-\lambda_0(c)\approx 4.4\times10^{-8} \) [2605.04484]. The paper compares the resulting product to several classical uncertainty relations. Heisenberg–Kennard gives \[ \Delta x\,\Delta p\ge \frac{\hbar}{2}, \] the Białynicki-Birula–Mycielski inequality gives \[ h(x)+h(p)\ge\log(\pi e\hbar), \] and the Donoho–Stark concentration bound gives \[ |T|\,|W|\;\ge\;2\pi\hbar\bigl(1-\epsilon_{T}-\epsilon_{W}\bigr)^{2}_{+}. \] The quantum confidence-uncertainty paper stresses that these control different objects: variance, entropy, and concentration are not interchangeable with confidence-mass localization [2605.04484]. ## 4. Interval information and the partial order on uncertainty A distinct statistical literature asks whether conditioning on a smaller interval necessarily reduces uncertainty. The paper "A Partial Order on Uncertainty and Information" shows that this proposition is not generally true [1108.1505]. For an absolutely continuous or integer-valued random variable, the uncertainty measure is conditional variance under interval information, \[ \operatorname{VAR}(X\mid X\in A), \] and the desired monotonicity for intervals \( A\subset B \) is \[ \operatorname{VAR}(X\mid X\in A)\le \operatorname{VAR}(X\mid X\in B) \] [1108.1505]. The paper calls this property partial monotonicity of the conditional variance and provides necessary and sufficient conditions. For an absolutely continuous \( X \) with cdf \( F(x) \), Theorem 2 states that \( \operatorname{VAR}\{X\mid a\le X\le b\} \) is increasing in \( b \) if and only if \[ \int_{a\le x\le y\le b}\{F(x)-F(a)\}\,dx\,dy \] is log-concave in \( b \), and decreasing in \( a \) if and only if \[ \int_{a\le x\le y\le b}\{F(b)-F(x)\}\,dx\,dy \] is log-concave in \( a \) [1108.1505]. If both hold on a convex set \( C \), then conditional variance is partially monotonic for intervals \( A\subset C \). A broad sufficient condition is log-concavity of the cdf. Theorem 3 states that if \( F(x) \) is log-concave on an interval \( C \), then \[ \operatorname{VAR}\{X\mid X\in A\}\le \operatorname{VAR}\{X\mid X\in B\} \] for any intervals \( A\subset B\subset C \) [1108.1505]. The paper also extends the framework to integer-valued variables via a continuous surrogate \( Y \), and derives an analogous monotonicity result for Shannon information of the difference variable \( U=X_1-X_2 \) under log-concavity assumptions [1108.1505]. This interval-conditioning theory is conceptually related to interval confidence uncertainty because both ask how uncertainty changes when one restricts admissible support. The underlying objects, however, are different: conditional variance under interval information on the one hand, and minimal interval length capturing prescribed probability on the other. ## 5. Interval-valued uncertainty in contemporary inference and machine learning Outside the quantum setting, interval-centered uncertainty appears in several non-equivalent forms, including self-reported confidence intervals, interval-valued classifier scores, propagated solution bands, and interval outputs of neural or fuzzy systems [2510.26995, 2602.04775, 2310.06923, 2502.02674, 2003.11566, 2601.18897]. | Domain | Interval object | Reported role | |---|---|---| | LLM numerical estimation | \([10^L,10^U]\) in exponent form | Coverage and sharpness evaluation | | Uncertainty-aware classification | \(I(x)=[L(x),U(x)]\) | Ranking with correct, incorrect, and overlap regions | | Deterministic PDEs | Joint CI \((L_Z,U_Z)\) over \( \Omega \) | Simultaneous domain-wide coverage | | Constrained inverse problems | Four constraint-aware CIs | Coverage with reduced conservatism | | Interval neural and fuzzy models | Learned lower and upper outputs; FOU | Width interpreted as uncertainty/confidence | In "LLMs are Overconfident: Evaluating Confidence Interval Calibration with FermiEval," models are prompted to output exponent-form intervals \( [10^L,10^U] \) in JSON for 1000 Science Olympiad Fermi questions [2510.26995]. The benchmark evaluates coverage, sharpness, and the Winkler interval score, \[ WinklerScore = \frac{1}{N}\sum_{i=1}^{N}\Big[ (U_i - L_i) + \frac{2}{\alpha}\,\bigl|\,y_i - \operatorname{proj}_{[L_i,U_i]}(y_i)\,\bigr| \Big]. \] The headline finding is that nominal \( 99\% \) intervals cover the true answer only \( 65\% \) of the time on average, indicating systematic overconfidence; a conformal prediction based adjustment restores accurate \( 99\% \) observed coverage and decreases the Winkler interval score by \( 54\% \) [2510.26995]. In uncertainty-aware classification, interval-valued predictions \[ I(x)=[L(x),U(x)] \] replace point scores [2602.04775]. A positive interval \( I_1 \) and negative interval \( I_0 \) can be confidently correct, confidently incorrect, or overlapping. The paper defines \[ AUC_L = P(L_1>U_0)=P(I_1>I_0), \qquad AUC_U = 1-P(U_1I_0)+P(I_1