The Approximation of a Discrete Joint Probability Distribution by Two Independent Ones

When implementing the Pearson chi-squared test for independence for a probability matrix P, we test against two independent one dimensional probability distributions, namely the Pearson approximants. These are formed by the row - sum vector u and the column - sum vector v of the matrix P. Even though an averaging process is used, the resulting approximation of a given matrix P need not be optimal in the least squares sense. In our presentation we classify the probability matrices P for which the Pearson approximants are optimal in the least squares sense.