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 $[-\sqrt{2},\sqrt{2}{]}^2$. 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 $(\sqrt{2},0),(-\sqrt{2},0),(0,\sqrt{2}),(0,-\sqrt{2})$ each have probability $1/4$. Both are bounded, have zero expected value, and satisfyFor 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 $-\sqrt{2},+\sqrt{2}$ with equal probabilities. Consequently,Even a linear output's threshold risk is not fixed by its variance. For the nonlinear output $F=X^2{Y}^2$, 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.^[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.