The Supplement 1 (S1) to the GUM (Guide to the Expression of Uncertainty in Measurement) describes a general numerical approach, known as Monte Carlo method (MCM) for estimating measurement uncertainty, based on the principle of propagation of distributions. The MCM applies to a measurement model that has a single output quantity where the input quantities are characterized by specified PDFs (probability density function). When an input quantity has Type A uncertainty that is estimated with a limited number of observations (a sample), the GUM-S1 recommends using the scaled and shifted t distribution conditioned on the sample mean and sample standard deviation as the PDF of the input quantity. However, we recently revealed that the scaled and shifted t-distribution is inappropriate for MCM because of the so-called t transformation distortion. Detailed discussion on this subject can be found on https://authors.elsevier.com/a/1YMxcxsQa4NsV
It should be pointed out that MCM does not require the Bayesian view; it can be implemented purely based on the frequentist view. The GUM-S1 seems to generate a misconception that MCM must be interpreted based on the Bayesian statistics. This misconception may be partially due to the use of the scaled and shifted t-distribution that is derived from the Bayesian approach, but mainly due to GUM-S1’s interpretation of PDF based on the Bayesian probability to represent a state of knowledge. In fact, Monte Carlo simulation is a frequentist method; it does not need a Bayesian interpretation. Monte Carlo simulation is also known as ‘sampling-domain simulation’, which is a counterpart of the ‘probability-domain simulation’ (Huang and Fergen 1995). A coverage interval, generated from the output PDF, is a probability interval, not a Bayesian credible interval, and it is not a confidence interval (CI) either.
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