The relation between information theory and the differential geometry approach to statistics

?encov has shown that the Riemannian metric on the probability simplex ∑x_i = 1 defined by $\text{(ds)}{}^2\text{=}\text{∑(dx}{}_{\mathrm{i}}\text{)}{}^2\text{x}{}_{\mathrm{i}}$ has an invariance property under certain probabilistically natural mappings. No other Riemannian metric has the same property. The geometry associated with this metric is shown to lead almost automatically to measures of divergence between probability distributions which are associated with Kullback, Bhattacharyya, and Matusita. Certain vector fields are associated in a natural way with random variables. The integral curves of these vector fields yield the maximum entropy or minimum divergence estimates of probabilities. Some other consequences of this geometric view are also explored.