On the choice of a noninformative prior for Bayesian inference of discretized normal observations

Abstract

We consider the task of Bayesian inference of the mean of normal observations when the available data have been discretized and when no prior knowledge about the mean and the variance exists. An application is presented which illustrates that the discretization of the data should not be ignored when their variability is of the order of the discretization step. We show that the standard (noninformative) prior for location-scale family distributions is no longer appropriate. We work out the reference prior of Berger and Bernardo, which leads to different and more reasonable results. However, for this prior the posterior also shows some non-desirable properties. We argue that this is due to the inherent difficulty of the considered problem, which also affects other methods of inference. We therefore complement our analysis by an empirical Bayes approach. While such proceeding overcomes the disadvantages of the standard and reference priors and appears to provide a reasonable inference, it may raise conceptual concerns. We conclude that it is difficult to provide a widely accepted prior for the considered problem.

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

“…Therefore, requisite (i) holds for the pdf (15), at least for this example. In [7] the approach described in section 3 was applied to the analysis of the data in table 1. Using the posterior obtained with prior (6) the results were x = 7.484 with u(x) = 0.021 for case 1 and x = 7.467 with u(x) = 0.016 for case 2.…”

“…In this paper, the resolution problem has been addressed anew. Previous efforts have shown that in this inverse problem there seems to be some inherent difficulty that precludes obtaining a reasonably behaved posterior for the value of a quantity based solely on discretized measurement data [7]. These data refer indirectly to the quantity, because they have been filtered in a way that implies a loss of information.…”

“…This means that requisite (iv) fails to be met and that requisite (v) holds not for a very large number of identical observations but for as few as two of them. Since the prior (5) does not produce a well-behaved posterior, Elster and Lira [7] undertook to use the methodology in [8] to derive the applicable 'reference' prior. The result was…”

“…where σ y is chosen according to a specific rule, established in [7], that depends on the data y-thus the appellative 'empirical'. The posterior that follows from this prior conforms with requisites (i) through (iv).…”

“…This conclusion is not sensible, for it is then perfectly possible that the true value is in one of the adjacent intervals or even beyond, especially if the number of indications is small and the underlying unknown variance is large. More recently, Elster and Lira [7] placed the Bayesian treatment of the resolution problem under further scrutiny. They showed that the anomalous behaviour of the posterior when the variability of the data is of the order of the resolution interval can be traced to the use of the standard non-informative prior for the location-scale family of distributions.…”