Limits of Learning from Imperfect Observations under Prior Ignorance: the Case of the Imprecise Dirichlet Model

Consider a relaxed multinomial setup, in which there may be mistakes in observing the outcomes of the process—this is often the case in real applications. What can we say about the next outcome if we start learning about the process in conditions of prior ignorance? To answer this question we extend the imprecise Dirichlet model to the case of imperfect observations and we focus on posterior predictive probabilities for the next outcome. The results are very surprising: the posterior predictive probabilities are vacuous, irrespectively of the amount of observations we do, and however small is the probability of doing mistakes. In other words, the imprecise Dirichlet model cannot help us to learn from data when the observational mechanism is imperfect. This result seems to rise a serious question about the use of the imprecise Dirichlet model for practical applications, and, more generally, about the possibility to learn from imperfect observations under prior ignorance.