Density approximation by summary statistics: an information-theoretic approach (Q2742764)

The following problem is considered. Let \(A\) be a set, \(\Lambda\) be a \(\sigma\)-field on \(A\) and let \(\nu\) be a \(\sigma\)-finite measure on \(\Lambda.\) Let \(\{ S, \Omega, P\}\) be a probability space. Let the function \(X\) from \(S\) to \(A\) be a random variable relative to \(\Omega\) and \(\Lambda\), and the distribution of \(X\) have density \(f\) with respect to \(V.\) The problem is to find a prediction of \(f\) by use of a probability density \(g\), where \(g\) is a non-negative measurable function on \(A\) and \(\int g dv =1.\) Let \(J(f, g) = \int f\log (g) dv.\) Optimal predictors can be obtained relative to a family \(G\) of densities in the following way: a member \(g^\ast\) of \(G\) is an optimal predictive density of \(f\) relative to \(G\) if \(J(f, g^\ast) = J(f, G)\), where \(J(f,G) = \sup_{g \in G} J(f, g).\)NEWLINENEWLINENEWLINEThe authors study conditions for the existence of optimal density approximations and properties of the optimal predictive densities. They consider in more detail the following case. Let \(f\) be unknown but \(E(y_j(x))\) be known, \(\int y_j f dv =c_j\) for \(1\leq j \leq k\), where \(c_j\) are real constants. Let \(G\) be the family of all possible probability densities and let \(H\) be the family of densities \(h\) such that \(y_{jh}\) is \(\nu\)-integrable and \(\int y_j h dv = c_j\) for \(1\leq j \leq k.\) The authors consider the problem how to predict \(f\) if all that is known is that \(f\) is in \(H.\) Applications of the proposed approach are illustrated and methods for the estimation of densities are provided in the case of simple random sampling.