Propagating Uncertainty Through the Model

Inputs carry uncertainty; a rigorous result carries it all the way to the output rather than dropping it at the first step.

Uncertainty in, uncertainty out

A model with uncertain inputs produces an uncertain output, and the whole job is to carry that uncertainty faithfully rather than collapsing inputs to point values and reporting a false-precise answer. We propagate by linearized sensitivities where the response is smooth and by sampling (Monte Carlo) where it is not.

Linear propagation and its Jacobian

For small input variations the output covariance follows from the sensitivity (Jacobian) matrix acting on the input covariance. Off-diagonal terms encode how inputs move together; ignoring them can hide or invent uncertainty. A schematic covariance mapping makes the bookkeeping explicit:

Reading it plainly: each row is how one output (scientific gain Q, or fusion power P) responds to each input (density n, temperature T). The output uncertainty is this matrix applied to the input uncertainties, so a small, well-known input with a large sensitivity can dominate the budget.

When linear is not enough

Near thresholds, ignitions, or stress limits the response is nonlinear and linear propagation understates the tails. There we sample the full model. This matters for the burner plug: because it sits far outside measured ground, the output distribution is wide and skewed, and a single central value would misrepresent it.

Propagation is what turns a deterministic design point into an honest one. The frozen values -- Q_sci 3.424, plasma current 9.86 MA -- are central estimates; the propagated uncertainty around them is part of the same result. Carrying uncertainty end to end is more work than quoting a point value, and it is the difference between a design number a reader can weigh and a number that merely looks precise while hiding how little is actually pinned down.