We prove that Fisher-Rao natural gradient descent (FR-NGD) optimally approximates the continuous time replicator equation (an essential model of evolutionary dynamics), and term this correspondence "conjugate natural selection". This correspondence promises alternative approaches for evolutionary computation over continuous or high-dimensional hypothesis spaces. As a special case, FR-NGD also provides the optimal approximation of continuous Bayesian inference when hypotheses compete on the basis of predicting actual observations. In this case, the method avoids the need to compute prior probabilities. We demonstrate our findings on a non-convex optimization problem and a system identification task for a stochastic process with time-varying parameters.