ORDER INDEPENDENCE AND FACTOR CONVERGENCE IN ITERATIVE SCALING

We prove two results concerning iterative scaling which have been claimed, but not proved, by other authors in the field. Iterative scaling is a procedure which begins with a probability measure Q on a finite set X and a finite set of partition constraints on X. It produces a sequence of iterates P0 = Q, P1,..., P(t),... in which each P(t) (t > 0) is an adjustment (scaling) of P(t-1) to fit a single set of prescribed partition constraints. Under fairly general hypotheses, the iterates P(t) converge to the unique probability P* which simultaneously satisfies the given constraints and is exponentially equivalent to the starting measure Q (or equivalently, which minimizes the I-divergence D(P(parallel-to)Q) over all probabilities P on X which satisfy the constraints). The first main result (Theorem 3.1) states that the convergence of iterative scaling to the unique limit is independent of the order in which the adjustments for the prescribed partition constraints are made, provided each adjustment is made infinitely many times. The second main result (Theorem 4.2) is that, under certain commonly satisfied conditions, each factor of the exponential form being built up within the algorithm (one factor for each partition constraint) converges separately. A third group of results (Sections 5,6) provides an account of the relationship between the probability measures which satisfy all of the constraints and those which are exponentially equivalent to Q.