L1-based probabilistic correction applied to misclassification in health care statistical matching

In recent contributions it has been proposed an efficient procedure for correcting inconsistent (i.e. incoherent) probability assessments based on $L1$ distance minimization and encoded in mixed integer programming (MIP) problems. The procedure is particular apt to deal with assessments stemming from different sources of information, and the so named statistical matching problem is one of those cases. Albeit the statistical matching problem is based on conditional probabilities estimates and different distance minimizations are reasonable, always in \cite{vantstatmatch} it has been proven that inconsistencies can appear only among assessments given on the same conditioning values, hence a correction instance can be splitted in a finite set of unconditional correction instances where the $L1$-based correction can efficiently operate. The problem has been recently enriched with the possibility of a misclassification setting \cite{DiZioVantaggi} breaking the aforementioned segmentation possibility. If marginal assessments on the conditioning variable are taken for good, the only possible correction are the closest Fr\'echet -- Hoeffding bounds for the misclassification probabilities. On the contrary, if also the marginal probabilities are allowed to be modified, the $L1$-based procedure can be applied by a straightforward translation in a MIP problem, albeit the set of consistent solutions turns out to be not convex and hence potential disconnected solutions can appear. Such procedure is intended to be practically applied by conducting an analysis of the relationships between economic aspects and socio-demographic determinants of the health care, in particular to health care expenditures and health conditions data observed in two different data sources .