4. The Typology of Uncertainty

Learning Objectives. By the end of this chapter, you will be able to distinguish Aleatory, Parametric Epistemic, Structural Epistemic, and Knightian uncertainty; use the Law of Total Variance and the Stirling Matrix to diagnose where ignorance lives; apply Pierre-Simon Laplace’s Rule of Succession to prevent dogmatic zero-probability estimates; explain parameter pseudo-convergence and Marc Kennedy and Anthony O’Hagan’s (2001) Gaussian Process structural discrepancy ($\delta(x)$); and combine Yakov Ben-Haim’s Info-Gap Decision Theory ($\hat{\alpha}, \hat{\beta}$) with Moshe Sniedovich’s Global Maximin (Wald) Safeguard to avoid catastrophic failure in “No-Man’s Land.”

Step 1 — Intuition and the Real-World Paradox

The Christmas Turkey and the Three Coins

Bertrand Russell once told the parable of an inductive chicken, later retold by Nassim Nicholas Taleb as the Christmas Turkey. Every morning at 9:00 a.m. sharp, the farmer walks into the barn and fills the trough with corn.

• On Day 1, the turkey is cautious.
• By Day 20, the turkey has formed a statistical hypothesis: “The appearance of the farmer at 9:00 a.m. causes breakfast.”
• By Day 100, the turkey has a hundred consecutive confirmations and zero counterexamples. Using standard textbook confidence intervals, the turkey’s estimated probability of being fed tomorrow morning approaches $100%$, and its standard error has shrunk to almost nothing.

Day 101 is the Wednesday before Thanksgiving (or Christmas Eve). The farmer walks in at 9:00 a.m., not with corn, but with an axe.

Why did one hundred days of real empirical data make the turkey more vulnerable rather than safer? Because the event that killed the turkey was never in the turkey’s list of hypotheses. More data only shrink uncertainty inside a correctly specified model; when the model’s hypothesis space omits the true mechanism of the world, accumulating more data merely builds false confidence around the wrong answer.

To see why treating all “uncertainty” as a single blurry concept is fatal, compare four situations:

1. The Known Fair Coin (Aleatory Uncertainty — $L_1$): You are about to toss a physically verified, symmetric coin once. Will it land Heads or Tails? Nobody on Earth can predict the single toss, and collecting a billion historical coin tosses will not make the next toss any closer to $100%$ Heads. The probability stays $50/50$ forever. This is irreducible physical randomness.
2. The Bent Coin of Unknown Bias (Parametric Epistemic Uncertainty — $L_2$): Someone hands you a bent coin from a workshop. It has a fixed physical bias $\theta \in (0, 1)$, but you do not know the number $\theta$. After $3$ tosses, you are very uncertain about $\theta$; after $10\text{,}000$ tosses, the Law of Large Numbers pins $\theta$ down to within a hundredth of a percent. Here, more data cure ignorance.
3. The Straight Ruler on a Curved Planet (Structural Epistemic Uncertainty — $L_2$): An engineer fits a linear friction formula $y = \theta x$ to predict wind drag on an aircraft wing across high-subsonic speeds, where shockwaves actually bend the physics nonlinearly. As the engineer collects more measurements, the least-squares fit forces the single knob $\theta$ to twist into an unphysical number just to compromise between the low-speed and high-speed points. The confidence interval around $\theta$ shrinks toward zero while the formula itself remains structurally wrong.
4. The Turkey’s Morning (Knightian Uncertainty — $L_3$): A decision-maker faces regimes, strategic reactions, or structural breaks where neither a unique probability distribution nor an exhaustive list of states can be trusted.

Step 2 — Foundations and Mathematical Formalization

4.1 The Law of Total Variance and the Stirling Matrix

Separating Aleatory Noise from Epistemic Ignorance

Mathematics gives us an exact accounting identity to separate the randomness of the World ($L_1$) from the parameter ignorance of the Model ($L_2$). Suppose an outcome $Y$ is governed by an unknown parameter vector $\theta$. By the Law of Total Variance (also known as Eve’s Law):

$$\mathrm{Var}(Y) = \underbrace{\mathbb{E}_\theta\bigl[\mathrm{Var}(Y \mid \theta)\bigr]}_{\text{Aleatory Uncertainty ($L_1$)}} + \underbrace{\mathrm{Var}_\theta\bigl(\mathbb{E}[Y \mid \theta]\bigr)}_{\text{Parametric Epistemic Uncertainty ($L_2$)}}$$

Examine the two additive pieces on the right-hand side:

• The first term, $\mathbb{E}_\theta[\mathrm{Var}(Y \mid \theta)]$, is Aleatory Uncertainty: Even if an oracle whispered the exact true parameter $\theta$ into your ear, the outcome $Y$ would still fluctuate around $\mathbb{E}[Y \mid \theta]$ with variance $\mathrm{Var}(Y \mid \theta)$. Because this variability belongs to the physical generating process in $L_1$, collecting more data never reduces it to zero. You cannot “research away” aleatory risk; you mitigate it by buying insurance, building safety margins, capping exposure (Chapter 3), or redesigning the physical system.
• The second term, $\mathrm{Var}_\theta(\mathbb{E}[Y \mid \theta])$, is Parametric Epistemic Uncertainty: It measures how much the expected outcome $\mathbb{E}[Y \mid \theta]$ shifts across the plausible values of $\theta$ that your finite sample has not yet ruled out. As sample size $n \to \infty$ in an identifiable model, the posterior distribution of $\theta$ concentrates on a single point, and this second term shrinks to zero.

The Stirling Matrix: Crossing Outcomes with Probabilities

What happens when even the list of possible outcomes or the functional form of the model is in doubt? Science-policy theorist Andy Stirling organized uncertainty into a $2 \times 2$ diagnostic grid according to how well we know (1) the possible outcomes and (2) the probabilities attached to them:

In 1921, economist Frank Knight drew the classic boundary between measurable Risk (Column 1) and unmeasurable Uncertainty (Column 2 and Row 2). In the Stochology Protocol (crates/stochology-core/src/uncertainty.rs and card.rs), every Epistemological Card must declare its dominant uncertainty class from four formal types:

4.2 Parametric Epistemic Uncertainty: Laplace’s Rule of Succession

Suppose an insurer examines $n = 36$ consecutive months of fire-reinsurance records (as happened in the 1986–1988 window of danish_fire.csv) and observes $s = 0$ claims exceeding $50\text{ million DKK}$. What is the probability $p$ that a catastrophic claim will strike next month?

A naive frequentist plug-in estimator computes:

$$\hat{p}_{\text{naive}} = \frac{s}{n} = \frac{0}{36} = 0.00$$

Reporting $\hat{p} = 0$ is an epistemological crime (Prohibition 4, Cromwell’s Rule): it confuses “it did not happen in this 36-month window” with “it is physically impossible.” If an underwriter believes $p = 0$, any finite premium looks like free money, encouraging infinite leverage and guaranteeing eventual ruin (which arrived in August 1989 with a $152.4\text{M DKK}$ fire loss).

How do we prevent small samples from collapsing to dogmatic $0%$ or $100%$? In 1774, Pierre-Simon Laplace asked that exact question—famously framing it as the probability that the sun will rise tomorrow given that it has risen $n$ times in a row. Starting from a uniform $\mathrm{Beta}(1, 1)$ prior over the unknown probability $\theta \in [0, 1]$ and observing $s$ successes in $n$ independent Bernoulli trials, the posterior distribution is $\mathrm{Beta}(s + 1, n - s + 1)$, whose expectation is Laplace’s Rule of Succession:

$$\hat{p}_{\text{Laplace}} = \frac{s + 1}{n + 2}$$

Look at what this simple, transparent formula achieves:

• Before any data are collected ($n = 0, s = 0$), $\hat{p} = \frac{1}{2}$ (maximum entropy over a binary outcome).
• After $n = 36$ clean months with $s = 0$ hits, $\hat{p} = \frac{0 + 1}{36 + 2} = \frac{1}{38} \approx 0.0263$ ($2.63%$ per month)—never zero!
• As sample size grows large ($n \to \infty$), the $+1$ and $+2$ pseudocounts wash out and $\frac{s+1}{n+2} \to \frac{s}{n}$ smoothly.

4.3 Structural Epistemic Uncertainty: Kennedy-O’Hagan Discrepancy ($\delta(x)$)

The Trap of Parameter Pseudo-Convergence

Now consider a much deeper danger in $L_2$. Suppose an engineer or economist builds a mechanistic computer simulator $\eta(x, \theta)$ to predict a physical system $y(x)$ across control inputs $x$, where $\theta$ represents a physical constant (for instance, thermal conductivity, aerodynamic drag, or risk aversion).

In reality, no mathematical model $\eta(x, \theta)$ captures $100%$ of the real world’s physics. There is always a missing physical term—a structural discrepancy $\delta(x)$ separating the true generating process of the World ($L_1$) from the idealized simulator ($L_2$).

What happens if we ignore $\delta(x)$ and calibrate the model by minimizing ordinary least squares $\sum_{i=1}^n \bigl(y_i - \eta(x_i, \theta)\bigr)^2$?

1. Biased Physical Parameters (Pseudo-Convergence): The optimizer forces the physical knob $\theta$ away from its true physical value $\theta_{\text{true}}$ toward a distorted value $\hat{\theta}_{\text{naive}}$ that tries to soak up the missing curve $\delta(x)$ inside the training domain.
2. Delusional Confidence: As $n$ grows, the standard error around $\hat{\theta}_{\text{naive}}$ shrinks as $1/\sqrt{n}$. The analyst becomes $99.9%$ confident in a wrong parameter!
3. Extrapolation Collapse: The moment the decision-maker uses $\eta(x, \hat{\theta}_{\text{naive}})$ at a new operational point $x_{\text{extrap}}$ outside the training range, the distorted parameter $\hat{\theta}_{\text{naive}}$ and the unmodeled discrepancy $\delta(x_{\text{extrap}})$ stop canceling each other out, producing a catastrophic prediction error with tiny nominal error bars.

Marc Kennedy and Anthony O’Hagan’s (2001) Solution

In a landmark 2001 paper in the Journal of the Royal Statistical Society, Marc Kennedy and Anthony O’Hagan solved this problem by promoting the model’s structural flaw to a first-class mathematical object:

$$y(x) = \underbrace{\eta(x, \theta)}_{\text{Simulator ($L_2$)}} + \underbrace{\delta(x)}_{\text{Structural Discrepancy}} + \underbrace{\varepsilon}_{\text{Aleatory Noise ($L_1$)}}, \qquad \delta(x) \sim \mathcal{GP}\bigl(m(x), k(x, x’)\bigr)$$

What does $\delta(x) \sim \mathcal{GP}(m(x), k(x, x’))$ mean from first principles? A Gaussian Process ($\mathcal{GP}$) is simply a probability distribution over smooth unknown functions. Instead of assuming that the missing physics $\delta(x)$ is a specific polynomial, a $\mathcal{GP}$ specifies only a prior mean function $m(x)$ (typically $0$) and a covariance kernel $k(x, x’)$, such as the squared-exponential kernel:

$$k(x, x’) = \sigma_\delta^2 \exp\left(-\frac{(x - x’)^2}{2\ell^2}\right)$$

This kernel says something completely intuitive:

• If two inputs $x$ and $x’$ are close together ($|x - x’| \ll \ell$), the simulator’s structural errors $\delta(x)$ and $\delta(x’)$ must be similar (highly correlated).
• If a new query point $x_{\text{extrap}}$ moves far away from all previously observed training data ($|x_{\text{extrap}} - x_i| \gg \ell$), the correlation drops toward zero and the predictive variance automatically inflates back to $\sigma_\delta^2$, honestly warning the decision-maker that the model is extrapolating into the dark!

Why Boundary Anchoring Is Essential

There is one subtle identifiability trap in the raw Kennedy-O’Hagan equation: if $\delta(x)$ can be any curve, how can the data tell the difference between changing $\theta$ inside $\eta(x, \theta)$ and changing $\delta(x)$? Without an identifiability constraint, $\theta$ and $\delta(x)$ are confounded (Brynjarsdóttir & O’Hagan, 2014).

In KennedyOHaganDiscrepancy (crates/stochology-core/src/uncertainty.rs), the protocol enforces anchor constraints (known physical boundary regimes, such as low-excitation anchor points $x_0$ where the idealized law $\eta(x, \theta)$ holds closely and $\delta(x_0) \approx 0$). Anchoring breaks the collinearity between $\eta(x, \theta)$ and $\delta(x)$:

• The anchor regime pins down the true physical parameter $\theta$ without distortion;
• The Gaussian Process $\delta(x)$ absorbs the nonlinear structural departure away from the anchor;
• And out-of-domain predictions carry honest, widened uncertainty bands.

Exact Cholesky Factorization and Empirical Bayes Type-II Maximum Likelihood

How do we select the Gaussian Process hyperparameters—the correlation length-scale $\ell$ and the discrepancy amplitude $\sigma_\delta$—without ad-hoc hand-tuning? In crates/stochology-core/src/uncertainty.rs, the covariance matrix $\Sigma = K_\delta(X, X; \ell, \sigma_\delta) + \sigma_n^2 I$ is factorized via exact lower-triangular Cholesky decomposition $\Sigma = L L^\top$ (cholesky_decompose, cholesky_solve, cholesky_log_det). Given residual vector $r = y - \eta(X, \hat{\theta})$, KennedyOHaganDiscrepancy::log_marginal_likelihood evaluates the exact Gaussian Process Log Marginal Likelihood (Type-II MLE):

$$\ln p(y \mid X, \hat{\theta}, \ell, \sigma_\delta) = \underbrace{-\frac{1}{2} r^\top (K_\delta + \sigma_n^2 I)^{-1} r}_{\text{Data Fit}} \underbrace{- \sum_{i=1}^n \ln L_{ii}}_{\text{Occam’s Razor Complexity Penalty } (-\frac{1}{2}\ln|\Sigma|)} - \frac{n}{2}\ln(2\pi)$$

Notice how the Cholesky log-determinant term $-\frac{1}{2}\ln|\Sigma| = -\sum_{i=1}^n \ln L_{ii}$ acts as an automatic Bayesian Occam’s Razor, penalizing overly wiggly kernels ($\ell \to 0$) while the quadratic form $-\frac{1}{2}r^\top\Sigma^{-1}r$ penalizes underfitting (KennedyOHaganDiscrepancy::calibrate_empirical_bayes).

4.4 Knightian Uncertainty: Info-Gap Decision Theory and Sniedovich’s Safeguard

Foundations: Decision Criteria When Probabilities Cannot Be Trusted

What should an agent ($L_3$) do when operating in the right-hand column of the Stirling Matrix, where probability distributions over future states of the world $u \in \mathcal{U}$ do not exist or cannot be trusted?

Classical decision theory offers four foundational rules, illustrated famously by David van Dantzig (1956) after the catastrophic 1953 North Sea flood prompted the Netherlands to redesign the height of its coastal dikes:

Yakov Ben-Haim’s Info-Gap Decision Theory

In engineering design and operational resilience, Yakov Ben-Haim proposed Info-Gap Decision Theory as a way to navigate severe uncertainty when you have a nominal (best-guess) model $\tilde{u}$ that you know may be wrong, and a critical survival requirement $r_c$ (the minimum reward $R(q, u) \ge r_c$ the system needs to avoid disaster).

Instead of fixing a single uncertainty set, Info-Gap surrounds the nominal model $\tilde{u}$ with a nested family of uncertainty balls $\mathcal{U}(\alpha, \tilde{u})$ indexed by an uncertainty horizon $\alpha \ge 0$:

$$\mathcal{U}(\alpha, \tilde{u}) = \bigl\lbrace u : \lVert u - \tilde{u} \rVert \le \alpha \bigr \rbrace$$

When $\alpha = 0$, the ball contains only the nominal guess $\tilde{u}$. As $\alpha$ grows, the ball expands to allow larger and larger departures from $\tilde{u}$. Info-Gap then defines two horizons for each candidate decision $q$:

1. The Robustness Horizon ($\hat{\alpha}(q, r_c)$ — Immunity to Failure): $$\hat{\alpha}(q, r_c) = \max\left\lbrace \alpha \ge 0 : \min_{u \in \mathcal{U}(\alpha, \tilde{u})} R(q, u) \ge r_c \right\rbrace$$ In plain words: How wrong can our nominal model $\tilde{u}$ be (how large can $\alpha$ get) before decision $q$ risks violating the survival threshold $r_c$? A larger $\hat{\alpha}$ means more robustness.

2. The Opportuneness Horizon ($\hat{\beta}(q, r_w)$ — Propensity for Windfall): $$\hat{\beta}(q, r_w) = \min\left\lbrace \alpha \ge 0 : \max_{u \in \mathcal{U}(\alpha, \tilde{u})} R(q, u) \ge r_w \right\rbrace$$ In plain words: What is the smallest departure $\alpha$ from the nominal model needed to make an ambitious windfall reward $r_w > r_c$ possible? A smaller $\hat{\beta}$ means windfall requires less luck.

A central theorem of Info-Gap is the Zero-ing Property: if you demand a critical reward $r_c$ equal to what the nominal model predicts ($r_c = R(q, \tilde{u})$), your robustness horizon is identically zero ($\hat{\alpha}(q, R(q, \tilde{u})) = 0$). Optimizing for the nominal model buys zero robustness against model error! To gain positive robustness ($\hat{\alpha} > 0$), you must step back from nominal optimality and satisfice.

Moshe Sniedovich’s Critique (“No-Man’s Land”) and the Global Wald Safeguard

Why does the Stochology Protocol refuse to use raw Info-Gap alone?

In a series of incisive mathematical papers (2007–2012), operations researcher Moshe Sniedovich proved a fatal flaw in unguarded Info-Gap analysis. Because $\hat{\alpha}(q, r_c)$ expands outward from a single local point $\tilde{u}$ and stops looking the instant a curve crosses $r_c$, Info-Gap is completely blind to what happens at $\alpha > \hat{\alpha}(q, r_c)$—a region Sniedovich called “No-Man’s Land.”

Look at how easily raw Info-Gap can recommend a Christmas Turkey:

• Suppose Option A (Fragile Turkey) performs brilliantly near the nominal guess $\tilde{u}$ and stays above $r_c = 100$ up to $\hat{\alpha}_A = 1.60$, but at $\alpha = 2.0$ its performance plunges off a cliff to $-450$ (total systemic ruin).
• Suppose Option B (Robust Engineering) has a slightly smaller local crossing or a modest trade-off near $\tilde{u}$, or stays above $r_c$ up to $\hat{\alpha}_B = 3.50$, and even under the worst conceivable state in the entire physical universe $\mathcal{U}_{\text{global}}$, its reward never drops below $+45.0$.

If $\tilde{u}$ is severely misspecified—which is the very definition of Knightian uncertainty!—reality may easily lie outside $\mathcal{U}(\hat{\alpha}, \tilde{u})$. An algorithm that checks only local robustness around a wrong guess $\tilde{u}$ is performing a local maximin around a fiction while ignoring global catastrophe.

To cure this defect permanently, InfoGapOption (crates/stochology-core/src/uncertainty.rs) enforces Sniedovich’s Global Maximin Safeguard:

Every decision under Knightian uncertainty must pass three simultaneous checks:

1. Zero-ing Audit: $\hat{\alpha}(q, R(q, \tilde{u})) = 0$ is verified algebraically;
2. Positive Local Robustness: $\hat{\alpha}(q, r_c) > 0$ at the declared critical threshold $r_c$;
3. Global Wald Maximin Safeguard: Across the entire physically admissible state space $\mathcal{U}_{\text{global}}$ (including all states outside $\mathcal{U}(\hat{\alpha}, \tilde{u})$), the global worst-case payoff must stay above the catastrophic ruin floor: $$\min_{u \in \mathcal{U}_{\text{global}}} R(q, u) \ge r_{\text{Wald, min}}$$

Any option that violates the global Wald floor in No-Man’s Land is automatically rejected (Gate G6 and --defect sniedovich).

Step 3 — Auditable Laboratory in Rust and Exercises

Run Laboratory 2 from your terminal to see both the Kennedy-O’Hagan Gaussian Process calibration and the Info-Gap + Sniedovich safeguard in action:

cargo run --release --bin labs -- --uncertainty

=== LAB 2: KENNEDY-O'HAGAN GP DISCREPANCY AND INFO-GAP + SNIEDOVICH ===
  [Kennedy-O'Hagan GP] theta_true=0.800 | without_GP=0.6454 (bias=0.1546, extrap_err=0.2022) | with_GP=0.7865 (bias=0.0135, extrap_err=0.0324) | EmpBayes(ell*=3.60, sigma*=1.00, LML=+27.27)
  [Info-Gap + Sniedovich] fragile_nominal_turkey: alpha_hat=1.60, zero_ing=0.0, Wald=-450.0, approved=false | robust_satisficing_wald: alpha_hat=3.50, beta_hat=2.13, Wald=45.0, approved=true

Walk through the two laboratory experiments:

1. In [Kennedy-O'Hagan GP], the true physical parameter is $\theta_{\text{true}} = 0.800$, but the physical world includes a nonlinear structural discrepancy $\delta(x)$ absent from the bare model $\eta(x, \theta)$.
   • Without the GP discrepancy, ordinary least squares twists the physical parameter down to $\hat{\theta} = 0.6454$ (a parameter bias of $0.1546$) and suffers an out-of-domain extrapolation error of $0.2022$.
   • With the anchored Kennedy-O’Hagan GP $\delta(x) \sim \mathcal{GP}(0, k)$, the estimator recovers $\hat{\theta} = 0.7865$ (cutting parameter bias more than $11$-fold to $0.0135$), reduces extrapolation error more than $6$-fold to $0.0324$, and selects Empirical Bayes hyperparameters $(\ell^{\ast} = 3.60,\ \sigma_\delta^{\ast} = 1.00)$ maximizing the Cholesky log marginal likelihood ($\mathrm{LML} = +27.27$).
2. In [Info-Gap + Sniedovich], the fragile_nominal_turkey option boasts a positive local robustness horizon $\hat{\alpha} = 1.60$, yet collapses to $\text{Wald} = -450.0$ in No-Man’s Land and is blocked (approved = false). The robust_satisficing_wald option achieves $\hat{\alpha} = 3.50$, opportuneness $\hat{\beta} = 2.13$, and a positive global Wald floor of $+45.0$, earning approval (approved = true).

Exercises

1. (Concept — Uncertainty Typing) For each scenario below, name both the ontological layer ($L_1, L_2, L_3$) and the uncertainty class (aleatory, epistemic_parameter, epistemic_specification, knightian):
   • (a) Thermal agitation (Johnson-Nyquist noise) inside an already-calibrated resistor.
   • (b) Estimating the mean service rate $\mu$ of a newly hired call-center agent after only $15$ calls.
   • (c) Using an M/M/1 queueing formula that assumes constant arrival rates $\lambda$ when real customer arrivals surge every Monday morning.
   • (d) Designing a power-grid reserve policy when an unmodeled cyber-attack or geomagnetic storm could disable transmission lines outside historical weather records.
2. (Calculation — Laplace’s Rule of Succession) A clinical trial tests a surgical procedure on $n = 18$ patients and observes $s = 18$ successful recoveries ($0$ complications).
   • (a) What is the naive plug-in complication probability, and why does Prohibition 4 ban it?
   • (b) What complication probability does Laplace’s Rule of Succession assign?
3. (Concept — Kennedy-O’Hagan) Explain in your own words why collecting $100\text{,}000$ additional observations worsens overconfidence when a mechanistic simulator $\eta(x, \theta)$ is calibrated without a structural discrepancy term $\delta(x)$.
4. (Laboratory — Sniedovich Negative Proof) Run cargo run --release --bin gates -- --defect sniedovich and verify that deploying an option that violates the global Wald floor in No-Man’s Land is blocked with exit code 1.

Chapter Summary

• Aleatory uncertainty ($L_1$) is irreducible noise of the World; Parametric epistemic uncertainty ($L_2$) is finite-sample parameter ignorance; Structural epistemic uncertainty ($L_2$) is model misspecification; Knightian uncertainty ($L_3$) is deep ignorance where probability distributions themselves are untrustworthy.
• The Law of Total Variance ($\mathrm{Var}(Y) = \mathbb{E}[\mathrm{Var}(Y\mid\theta)] + \mathrm{Var}(\mathbb{E}[Y\mid\theta])$) separates $L_1$ aleatory variance from $L_2$ parametric variance.
• Laplace’s Rule of Succession ($\hat{p} = \frac{s+1}{n+2}$) prevents small samples from collapsing to dogmatic $0%$ or $100%$.
• Kennedy-O’Hagan Structural Discrepancy ($y(x) = \eta(x, \theta) + \delta(x) + \varepsilon$ with anchored $\mathcal{GP}$) prevents parameter pseudo-convergence and extrapolation overconfidence.
• Info-Gap Decision Theory coupled with Sniedovich’s Global Maximin Safeguard measures local robustness ($\hat{\alpha}$) and opportuneness ($\hat{\beta}$) while guaranteeing a global Wald worst-case floor across No-Man’s Land.