Rasch models with exchangeable rows and columns

The article studies distributions of doubly infinite binary matrices with exchangeable rows and columns which satify the further property that the probability of any $m \times n$ submatrix is a function of the row- and column sums of that matrix. We show that any such distribution is a (unique) mixture of random Rasch distributions. The non-degenerate elements of these distributions were introduced by Rasch (1960). We investigate the relationship between these random Rasch distributions and a problem in visual perception, the characters of a certain Abelian semigroup, and the problem of existence of measures with given marginals.