Abstract: The _distributional simplicity bias_ (DSB) posits that neural networks learn low-order moments of the data distribution first, before moving on to higher-order correlations. In this work, we present compelling new evidence for the DSB by showing that networks automatically learn to perform well on maximum-entropy distributions whose low-order statistics match those of the training set early in training, then lose this ability later. We also extend the DSB to discrete domains by proving an equivalence between token $n$-gram frequencies and the moments of embedding vectors, and by finding empirical evidence for the bias in LLMs. Finally we use optimal transport methods to surgically edit the low-order statistics of one class to match those of another, and show that early-training networks treat the edited samples as if they were drawn from the target class.