Propagate uncertainty without hiding dependence

Try

An uncertain input is not automatically a random variable with a known distribution. Sometimes only an admissible range is justified. For Q=a/b with a between 6 and 10 and b between 2 and 5, a worst-case range can be calculated without assigning probabilities.

Because b is positive, Q increases with a and decreases with b. This monotonicity identifies the extreme corners of the full rectangle, avoiding an unsupported claim based on a few sampled points.

Predict

Predict the smallest and largest ratios. Then suppose a and b cannot vary independently because a=2b. Would the rectangle’s extreme values still be attainable? The joint admissible set matters as much as the marginal ranges.

Experiment

For the full rectangle, the range is [6/5,5]. Under a=2b and the same marginal restrictions, every admissible ratio is 2.

Assumption	Possible ratio range
Every rectangle combination allowed	[6/5,5]
Exact relation a=2b	Single value 2

Neither range is a 95% interval. They describe possibilities under different assumptions. Assigning a coverage percentage would require a probabilistic construction and an interpretation of repeated uncertainty or prediction.

Understand

For random quantities, dependence enters through covariance. Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y). If both variances are 1, perfect positive dependence gives variance 4 for the sum, independence gives 2 and Y=−X gives zero. Ignoring covariance can therefore overstate or understate uncertainty.

For a differentiable scalar output g(θ), linearization near θ₀ gives approximate variance ∇gᵀΣ∇g, where Σ is the parameter covariance matrix. This delta-method calculation assumes the uncertainty is sufficiently concentrated for the linear approximation to be useful. Strong curvature, boundaries or a denominator near zero can make it misleading. Compare with direct bounded calculations or a justified simulation when those issues arise.

Separate uncertainty sources. Process variability concerns random outcomes even with known parameters. Parameter uncertainty comes from limited information about those parameters. Structural uncertainty concerns whether the chosen equations and observation mechanism are adequate. Monte Carlo error is the finite-simulation error in estimating a quantity under the already chosen model. More simulated paths reduce that last error, not the other three automatically.

A simulation report must state how inputs were sampled, including dependence, and how many independent replications were used. A seed supports reproducibility but does not validate the sampling assumptions. Repeated draws from an implausible distribution can estimate the wrong model’s output very precisely.

Apply

For one model output, provide an interval analysis and a probabilistic analysis only if you can justify the required distributions. Explain why their interpretations differ. Include covariance or a dependence scenario, and identify the uncertainty source most likely to change the decision. State which new observation could reduce that uncertainty.

Continue

Validation tests predictions against observations excluded from fitting. It cannot be replaced by an uncertainty calculation that assumes the model is already correct.