# Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)

## Abstract

The iterative proportional fitting procedure (IPFP), introduced in 1937 by Kruithof, aims to adjust the elements of an array to satisfy specified row and column sums. Thus, given a rectangular non-negative matrix X0 and two positive marginals a and b, the algorithm generates a sequence of matrices (Xn)n≥0 starting at X0, supposed to converge to a biproportional fitting, that is, to a matrix Y whose marginals are a and b and of the form Y = D1X0D2, for some diagonal matrices D1 and D2 with positive diagonal entries.

When a biproportional fitting does exist, it is unique and the sequence (Xn)n≥0 converges to it at an at least geometric rate. More generally, when there exists some matrix with marginal a and b and with support included in the support of X0, the sequence (Xn)n≥0 converges to the unique matrix whose marginals are a and b and which can be written as a limit of matrices of the form D1X0D2.

In the opposite case (when there exists no matrix with marginals a and b whose support is included in the support of X0), the sequence (Xn)n≥0 diverges but both subsequences (X2n)n≥0 and (X2n+1)n≥0 converge.

In the present paper, we use a new method to prove again these results and determine the two limit-points in the case of divergence. Our proof relies on a new convergence theorem for backward infinite products ⋯M2M1 of stochastic matrices Mn, with diagonal entries Mn(i, i) bounded away from 0 and with bounded ratios Mn(j, i)∕Mn(i, j). This theorem generalizes Lorenz’ stabilization theorem. We also provide an alternative proof of Touric and Nedić’s theorem on backward infinite products of doubly-stochastic matrices, with diagonal entries bounded away from 0. In both situations, we improve slightly the conclusion, since we establish not only the convergence of the sequence (MnM1)n≥0, but also its finite variation.

## Keywords

Infinite products of stochastic matrices Contingency matrices Distributions with given marginals Iterative proportional fitting Relative entropy I-divergence

## Subject Classifications

15B51 62H17 62B10 68W40

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