We observe a random sample of $n$ observations from an unknown distribution having mean $\mu$. In this paper we consider the problem of making inferences about the unknown parameter $\mu$ and other charasteristics of the unknown distribution by adopting a non-parametric and objective minded framework, in which the observations are considered bounded and discrete (finite precision measurement). We first review the Bayesian approach based on Dirichlet priors and discuss the problems encountered by the usual vague priors. An alternative approach is then proposed which models prior ignorance by an imprecise Dirichlet model (IDM) with parameter $\ps$ (Walley, 1996). The comparison of inferences produced by the IDM with the ones from more common parametric approaches gives support for setting $\ps=2$ in the IDM. The new method of inference is illustrated on Darwin's maize data.
Keywords. Nonparametric inference, Bayesian inference, Dirichlet distribution, L-Dirichlet distribution, Prior ignorance, IDM, Upper and lower probabilities
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