= Nonuniqueness of bounded uncertainty laws with a fixed covariance
<Nonuniqueness of bounded uncertainty laws with a fixed covariance> means that specifying finite bounds, <expected values> and a <covariance matrix> can still leave several different joint <probability distributions>. They agree on the <variance> of every <linear combination> of the errors, but can disagree on threshold probabilities and nonlinear outputs. A particular construction in <bounded errors with prescribed correlation> therefore requires a distributional assumption beyond its second moments.
Here is an original example with a common allowed box $[-\sqrt2,\sqrt2]^2$. In model A, $X$ and $Y$ are independent fair signs: the four points $(1,1),(1,-1),(-1,1),(-1,-1)$ each have <probability> $1/4$. In model B, the four points $(\sqrt2,0),(-\sqrt2,0),(0,\sqrt2),(0,-\sqrt2)$ each have <probability> $1/4$. Both are bounded, have zero <expected value>, and satisfy
$$
\mathbb E[X^2]=\mathbb E[Y^2]=1,\qquad
\mathbb E[XY]=0,\qquad
\operatorname{Cov}\begin{pmatrix}X\\Y\end{pmatrix}=I_2.
$$
For model B, each squared coordinate is two at two of the four points and zero otherwise, proving its unit <variance> directly. Its zero <covariance> does not make the coordinates <independent random variables>: when $X$ is nonzero, $Y$ must vanish. The common box is an allowed deterministic bound; its boundary need not be attained by every model.
The linear output $U=X+Y$ has <expected value> zero and <variance> two under either model. Yet under model A it takes $-2,0,2$ with probabilities $1/4,1/2,1/4$, while model B gives only $-\sqrt2,+\sqrt2$ with equal probabilities. Consequently,
$$
\mathbb P(|U|>1.5)=\begin{cases}1/2&\text{model A},\\0&\text{model B}.\end{cases}
$$
Even a linear output's threshold risk is not fixed by its <variance>. For the nonlinear output $F=X^2Y^2$, model A gives $F=1$ always and model B gives $F=0$ always. Thus the complete output law, and even a nonlinear output's <expected value>, can differ.
The <law of propagation of uncertainty> computes first-order output <variance> from <sensitivity coefficients> and input <covariance>; it does not identify all probabilities. NIST correspondingly requires distributional justification to assign a confidence level to an uncertainty interval.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A.3 and §§6.2–6.3]{ref} The finite counterexample above is an original exact calculation. For guaranteed linear limits, <robust linear constraints for bounded covariance models> must use the chosen <support of a probability distribution>, rather than treating a <covariance matrix> as a support description.
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