Assigning a probability density function for the value of a quantity based on discrete data: the resolution problem

Abstract

It often happens that knowledge about a particular quantity has to be reached by processing a series of resolution-limited indications. It is a well-established fact that if the variability of the data is large compared with the resolution interval, the effect of discretization can be ignored. Otherwise, it needs to be taken into account since it can then be an important source of uncertainty, sometimes more significant than randomness itself. The objective of this paper is to derive a probability density function (pdf) for the value of a quantity based on discretized data. This pdf allows the standard uncertainty associated with the best estimate of the quantity to be computed and, perhaps more importantly, it can be used as an input to evaluate a measurement model in which the quantity is involved. Bayesian concepts are used towards this goal. Although reaching an appropriate pdf has been attempted before, limited success has been attained, as the pdfs that have been obtained exhibit some undesirable characteristics. Herein a new approach is proposed. Unlike previous efforts, this time the quantity of interest is modelled as a sum of two other quantities, one that can only assume discrete values and the other that takes values within the resolution interval centred on zero. The resulting pdf exhibits a satisfactory behaviour, but further work would be required to provide firmer theoretical grounds for the employed prior.

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