Continuity of f-projections and applications to the iterative proportional fitting procedure

Abstract

This paper proves continuity of f-projections and the continuous dependence of the limit matrix of the iterative proportional fitting procedure (IPF procedure) on the given matrix as well as the given marginals under certain regularity constraints. For discrete spaces, the concept of f-projections of finite measures on a compact and convex set is introduced and continuity of f-projections is proven. This result is applied to the IPF procedure. Given a nonnegative matrix as well as row and column marginals the IPF procedure generates a sequence of matrices, called the IPF sequence, by alternately fitting rows and columns to match their respective marginals. If the IPF sequence converges, the application of the previous result yields the continuous dependence of the limit matrix on the given matrix. By generalized convex programming and under some constraints, it is shown that the limit matrix of the IPF sequence continuously depends not only on the given matrix but also on the marginals.Keywords Continuity of f-projections · Iterative proportional fitting · Limit matrix · Dependence on input matrix and marginals · f-divergence · f-projection · I-divergence · I-projection Mathematics Subject Classification (2000) 68W40 · 62B10 · 62H17 · 90C25 *

Statistical Matching Analysis for Complex Survey Data With Applications

“…Proof of Proposition 2. From Theorem 4.2 in Gietl and Reffel (2013), it follows that if IPF converges than the solution matrix of IPF continuously depends on the starting matrix and on the marginals. Hence, using Proposition 1 and C3, p * (y l 1 1 , y l 2 2 |x) converges in probability to the solution of IPF procedure that uses p ST (y l 1 1 , y l 2 2 |x)s as entries of the starting matrix, and with marginals p h (•|x), i.e.…”