Learning from Quasi Perfect Observations Under Prior Ignorance

The imprecise Beta model (IBM) of Bernard (1996) and Walley (1996) is the most popular model for learning about a binomial random variable under prior ignorance. Piatti et al. (2005) show that there is a fundamental issue with the interpretation of results produced by the IBM in applications. When the possibility that data may contain errors can be excluded, the IBM is able to learn from each sequence of observations of the variable of interest. However, in the more realistic case in which observations may be affected by errors, the IBM is unable to learn. In this paper, we propose a modified approach that allows to learn from imperfect observations under a weak specification of prior knowledge if the probability of error is small. The approach is based on an additional assumption that seems natural and acceptable in applications with moderate probabilities of observation errors. We show that the results produced by the modified model are arbitrarily close to those produced by the IBM, when the probability of observation errors is smaller than a pre-specified threshold that depends on the desired accuracy level. This last finding yields a possible explanation for the usefulness of the IBM in applications characterized by a small probability of observation errors.