**5. Discussion**

The GUM recommends that uncertainty information be presented in such a way that permits it to be used in a subsequent uncertainty calculation. On a calibration certificate, information about a quantity is frequently provided in the form of an estimate of the quantity and either an associated standard uncertainty or expanded uncertainty corresponding to a specified coverage probability. If the quantity is to be used in another calculation, it is common, in the absence of any additional information, to assign a Gaussian (or normal) probability distribution to the quantity. This assignment may be made even if the true probability distribution for the quantity is significantly different. Consequently, the results of the subsequent calculation may be unreliable.

Even if uncertainty evaluation has been undertaken using a Monte Carlo method, it is common for only the summary information mentioned above to be presented on the calibration certificate. The reasons are understandable, e.g., limiting the number of pages in the certificate to a reasonable value, the difficulty in putting to practical use the full set of values returned by the Monte Carlo method.

Digital calibration certificates (DCCs) provide two key benefits that immediately aid the reporting and use of a complete set of Monte Carlo results. First, the presentation of information in a fully machine-readable form. Second, through the application of the Digital SI (D-SI) data model, the potential to include much greater volumes of data than is currently practical in a paper-based or electronic (e.g., PDF-A) certificate.

For a subsequent uncertainty calculation, a second Monte Carlo approach can be implemented. The sample values for the first quantity can be read from the DCC and samples from (an approximation to) the probability distribution for the first quantity can be obtained by randomly sampling from that set of values. The storage and use in calculations of a large number (e.g., 106) of sample values is facilitated in modern personal computers and mathematical software packages.

DCCs therefore provide the means to transfer uncertainty information that is encapsulated in a set of Monte Carlo samples. However, one should be aware of the circumstances under which the use of a Monte Carlo approach does not support transferability of results, e.g., as discussed in [24].

Although the D-SI was not designed with a Monte Carlo approach to uncertainty evaluation explicitly in mind, the data components in the D-SI are readily suitable for reporting the full set of results from a Monte Carlo calculation. As its use becomes more widespread, it is anticipated that the D-SI will be subject to updates, e.g., to take account of feedback from users from various metrology domains. One potential update could be the development of a specific component in the D-SI that allows all information from a Monte Carlo calculation to be encapsulated. That information could include details of the approach (standard or adaptive) and the number of Monte Carlo trials implemented. It may also be useful to provide all relevant information on the input quantities and their probability distributions as well as the measurement model (in an appropriate format).

**Author Contributions:** Conceptualization, I.S.; methodology, I.S. and D.H.; software, I.S.; validation, D.H. and Y.L.; formal analysis, I.S.; investigation, I.S.; resources, I.S.; data curation, I.S.; writing—original draft preparation, I.S.; writing—review and editing, D.H. and Y.L.; visualization, I.S.; supervision, I.S.; project administration, I.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been carried out as part of the European Metrology Programme for Innovation and Research (EMPIR) project 17IND02. The EMPIR initiative is co-funded by the European Union's Horizon 2020 research and innovation programme and the EMPIR Participating States.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
