A novel approach to Bayesian consistency (Q1684160)

Let \( X_1\), \(X_2,\dots,X_n\) be i.i.d. real-valued random variables from a true density \(p_0\). Let \(\mathcal L\) be the space of all Lebesgue densities on \((\mathbb R,\mathcal R)\) equipped with the total variation metric, and \(\Pi\) be a prior on \((\mathcal L,\mathfrak{L})\), where \(\mathcal R\) and \(\mathfrak{L}\) are Borel \(\sigma\)-algebras. For a (pseudo-)metric \(d\) on \(\mathcal L\), the posterior distribution \(\Pi(\cdot| X_1,\dots,X_n)\) is called \(d\)-consistent at \(p_0\) if \(\Pi(d(p_0,p )>\eta | X_1,\dots,X_n)\) converges to zero in probability for every \(\eta>0\). When \(d\) is the total variation (Lévy-Prokhorov, resp.), it is called stongly (weakly, resp.) consistent. Let \( K(p,q)=\int p \log(\frac{p}{q}) d\mu\) be the Kullback-Leibler (KL) divergence, where \(\mu\) is the Lebesgue measure. It is known that, if \(p_0\) lies in the KL support of \(\Pi\), i.e. \[ \Pi(p \in \mathcal L:K(p_0,p)<\delta)>0 \text{ for every }\delta>0, \] then the posterior distribution is weakly consistent at \(p_0\). This KL support condition is not sufficient for strong consistency. In the literature, along with the KL support condition various sufficient conditions for strong consistency have been studied. In this paper the authors present a new sufficient condition for strong consistency. They also give applications to nonparametic mixture models with heavy-tailed components, such as the Student-\(t\).