Séminaire de Probabilités XLIX pp 75-117 | Cite as

# Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices

## 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 *X*_{0} and two positive marginals *a* and *b*, the algorithm generates a sequence of matrices (*X*_{n})_{n≥0} starting at *X*_{0}, supposed to converge to a biproportional fitting, that is, to a matrix *Y* whose marginals are *a* and *b* and of the form *Y* = *D*_{1}*X*_{0}*D*_{2}, for some diagonal matrices *D*_{1} and *D*_{2} with positive diagonal entries.

When a biproportional fitting does exist, it is unique and the sequence (*X*_{n})_{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 *X*_{0}, the sequence (*X*_{n})_{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 *D*_{1}*X*_{0}*D*_{2}.

In the opposite case (when there exists no matrix with marginals *a* and *b* whose support is included in the support of *X*_{0}), the sequence (*X*_{n})_{n≥0} diverges but both subsequences (*X*_{2n})_{n≥0} and (*X*_{2n+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 ⋯*M*_{2}*M*_{1} of stochastic matrices *M*_{n}, with diagonal entries *M*_{n}(*i*, *i*) bounded away from 0 and with bounded ratios *M*_{n}(*j*, *i*)∕*M*_{n}(*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 (*M*_{n}⋯*M*_{1})_{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## Notes

### Acknowledgements

We thank A. Coquio, D. Piau, G. Geenens, F. Pukelsheim and the referee for their careful reading and their useful remarks.

## References

- 1.E. Aas, Limit points of the iterative scaling procedure. Ann. Oper. Res.
**215**(1), 15–23 (2014)Google Scholar - 2.M. Bacharach, Estimating nonnegative matrices from marginal data. Int. Econ. Rev.
**6**(3), 294–310 (1965)Google Scholar - 3.L.M. Bregman, Proof of the convergence of Sheleikhovskii’s method for a problem with transportation constraints. USSR Comput. Math. Math. Phys.
**7**(1), 191–204 (1967)Google Scholar - 4.J.B. Brown, P.J. Chase, A.O. Pittenger, Order independence and factor convergence in iterative scaling. Linear Algebra Appl.
**190**, 1–39 (1993)Google Scholar - 5.I. Csiszár, I-divergence geometry of probability distributions and minimization problems. Ann. Probab.
**3**(1), 146–158 (1975)Google Scholar - 6.S. Fienberg, An iterative procedure for estimation in contingency tables. Ann. Math. Stat.
**41**(3), 907–917 (1970)Google Scholar - 7.C. Gietl, F. Reffel, Accumulation points of the iterative scaling procedure. Metrika
**73**(1), 783–798 (2013)Google Scholar - 8.C.T. Ireland, S. Kullback, Contingency tables with given marginals. Biometrika
**55**(1), 179–188 (1968)Google Scholar - 9.J. Kruithof, Telefoonverkeers rekening (Calculation of telephone traffic). De Ingenieur 52, pp. E15–E25 (1937). Partial English translation: Krupp, R.S. http://wwwhome.ewi.utwente.nl/~ptdeboer/misc/kruithof-1937-translation.html
- 10.J. Lorenz, A stabilization theorem for dynamics of continuous opinions. Phys. A Stat. Mech. Appl.
**355**(1), 217–223 (2005)Google Scholar - 11.O. Pretzel, Convergence of the iterative scaling procedure for non-negative matrices. J. Lond. Math. Soc.
**21**, 379–384 (1980)Google Scholar - 12.F. Pukelsheim, Biproportional matrix scaling and the iterative proportional fitting procedure. Ann. Oper. Res.
**215**, 269–283 (2014)Google Scholar - 13.R. Sinkhorn, Diagonal equivalence to matrices with prescribed row and column sums. Am. Math. Mon.
**74**(4), 402–405 (1967)Google Scholar - 14.B. Touri, Product of random stochastic matrices and distributed averaging. Springer Thesis (2012)Google Scholar
- 15.B. Touri, A. Nedic, Alternative characterization of ergodicity for doubly stochastic chains, in
*Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC)*, Orlando, FL (2011), pp. 5371–5376Google Scholar - 16.B. Touri, A. Nedić, On ergodicity, infinite flow and consensus in random models. IEEE Trans. Autom. Control
**56**(7), 1593–1605 (2011)MathSciNetCrossRefGoogle Scholar - 17.B. Touri, A. Nedić, On backward product of stochastic matrices. Automatica
**48**(8), 1477–1488 (2012)MathSciNetCrossRefGoogle Scholar - 18.R. Webster,
*Convexity*(Oxford University Press, Oxford, 1994)zbMATHGoogle Scholar