Monte Carlo evaluation of uncertainty (GUM Supplement 1)

JCGM 101 describes the propagation of distributions by Monte Carlo. Each input quantity is assigned a probability distribution; a large number of trials (10⁵–10⁶) draws a value from each distribution, evaluates the model, and stores the result. The distribution of results yields the best estimate, its standard uncertainty and a coverage interval at the desired probability.

When it is worth it

• A dominant rectangular component, where the output is not normal.
• Non-linear models such as humidity from dew point, or MKT from a temperature series.
• Asymmetric inputs, e.g. a truncating display or one-sided limits.
• Validating an analytical budget: if the GUM result and the Monte Carlo interval agree, the linear approximation holds.

The method demands the same care in defining inputs; garbage distributions produce a precise-looking garbage interval.

What the method actually does

The analytical GUM approach propagates uncertainties: it linearises the model at the estimates, multiplies each input's standard uncertainty by a sensitivity coefficient, and combines in quadrature. JCGM 101 propagates distributions instead. Each input is given a probability distribution; a large number of trials draws one value from each, evaluates the model, and stores the result. The histogram of those results is the output distribution, and the coverage interval is read off it directly — no assumption that the output is normal, and no linearisation.

The dominant-rectangular case is the one worth internalising, because it is common in temperature work. A budget dominated by a coarse display resolution has an output much closer to rectangular than normal, and multiplying u_c by 2 then produces an interval covering closer to 100 % than 95 % — over-conservative rather than dangerous, but wrong, and the wrongness is invisible from the analytical result alone.

Running one properly

• Use 10⁵ to 10⁶ trials. JCGM 101 gives an adaptive procedure that keeps drawing until the interval is stable to the number of figures you intend to report, which is better than picking a round number.
• Take the 95 % coverage interval as the shortest interval containing 95 % of the results, not automatically the symmetric 2.5 % to 97.5 % quantiles — they differ when the output is skewed, and the shortest one is the meaningful answer.
• Use a generator with a long period and record the seed, so the run is reproducible. A result that cannot be reproduced is not a technical record.
• Validate against the analytical result in a case where both should agree. If they disagree in the easy case, the model implementation is wrong, not the method.
• Record the distributions assigned to each input and why. This is where the judgement lives, and a Monte Carlo run hides it behind a confident-looking histogram.

The method demands the same care in defining inputs as the analytical one, and offers more places to hide a mistake. Garbage distributions produce a precise-looking garbage interval, and it looks more authoritative than a garbage budget does.