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With increasing use of digital control it is natural to view control inputsand outputs as stochastic processes assuming values over finite alphabetsrather than in a Euclidean space. As control over networks becomes increasinglycommon, data compression by reducing the size of the input and output alphabetswithout losing the fidelity of representation becomes relevant. This requiresus to define a notion of distance between two stochastic processes assumingvalues in distinct sets, possibly of different cardinalities. If the twoprocesses are i.i.d., then the problem becomes one of defining a metric betweentwo probability distributions over distinct finite sets of possibly differentcardinalities. This is the problem addressed in the present paper. A metric isdefined in terms of a joint distribution on the product of the two sets, whichhas the two given distributions as its marginals, and has minimum entropy.Computing the metric exactly turns out to be NP-hard. Therefore an efficientgreedy algorithm is presented for finding an upper bound on the distance. Thisproblem also turns out to be NP-hard, so again a greedy algorithm isconstructed for finding a suboptimal reduced order approximation. Takentogether, all the results presented here permit the approximation of an i.i.d.process over a set of large cardinality by another i.i.d. process over a set ofsmaller cardinality. In future work, attempts will be made to extend this workto Markov processes over finite sets.

A Metric Between Probability Distributions on Finite Sets of Different Cardinalities and Applications to Order Reduction