Fusion of data and estimation by entropy maximization

Data fusion as discussed here means to create a set of data on not jointly observed variables from two different sources. Suppose for instance that observations are available for (X,Z) on a set of individuals and for (Y,Z) on a different set of individuals. Each of X, Y and Z may be a vector variable. The main purpose is to gain insight into the joint distribution of (X,Y) using Z as a so-called matching variable. At first however, it is attempted to recover as much information as possible on the joint distribution of (X,Y,Z) from the distinct sets of data. Such fusions can only be done at the cost of implementing some distributional properties for the fused data. These are conditional independencies given the matching variables. Fused data are typically discussed from the point of view of how appropriate this underlying assumption is. Here we give a different perspective. We formulate the problem as follows: how can distributions be estimated in situations when only observations from certain marginal distributions are available. It can be solved by applying the maximum entropy criterium. We show in particular that data created by fusing different sources can be interpreted as a special case of this situation. Thus, we derive the needed assumption of conditional independence as a consequence of the type of data available.