In an array of random variables, each row can be regarded as a single, sequence-valued random variable. In this way, the array is seen as a sequence of sequences. Such an array is said to be row exchangeable if each row is an exchangeable sequence, and the entire array, viewed as a sequence of sequences, is exchangeable. We give a representation theorem, analogous to those of Aldous and Hoover, which characterizes row exchangeable arrays. We then use this representation theorem to address the problem of performing Bayesian inference on row exchangeable arrays.