Maximum entropy distribution under moments and quantiles constraints

• Proposed is a new method of evaluation of the maximum entropy distribution.
• This method allows for the simultaneous inclusion of the moments and quantiles.
• Presented framework can be used to extend methodologies introduced in BIPM GUM.

When the results of a measurement are transferred from one stage in the chain of traceability to the next, the information gathered about the measurement is summarised. The summary involves, for example, details about applied measurement methods, environmental conditions, and measurement results including measurement uncertainty. The information about uncertainty usually takes the form of summary statistics such as an estimate, a standard deviation and a coverage interval specified by two quantiles. The information is used to construct a probability distribution for a given property or characteristic of an artefact, which is needed when the artefact is used as a reference in a subsequent stage. But in order to ensure impartiality in the process to establish the probability distribution, a general rule should be applied, for example, the principle of maximum entropy. In this paper, the application of this principle to establish a probability distribution when the mentioned summary statistics are available will be discussed, and its extension to moment constraints to satisfy the requirements of metrology will be introduced.