The Fitness Gradient

The Lande equation is a cornerstone of quantitative genetics: the change in mean phenotype per generation equals the additive genetic covariance matrix times the selection gradient. Vanchurin shows this is covariant gradient ascent on a fitness landscape.

The derivation starts from maximum entropy. If you know only the mean and covariance of a population’s trait distribution, the maximum entropy distribution is Gaussian. The inverse of its covariance matrix defines a metric tensor on phenotype space. This identification — inverse covariance as metric — is the key step. It turns phenotype space into a Riemannian manifold where distances reflect statistical distinguishability.

With the metric in place, the Lande equation becomes the gradient of mean fitness with respect to the metric. The covariance matrix, which in classical genetics is an empirical observation, becomes the inverse metric tensor — a geometric object that determines how the population navigates the fitness landscape. Evolution is gradient ascent, and the “learning rate” is determined by the functional relationship between the metric and the noise covariance of evolutionary changes.

The framework works consistently in both genotype and phenotype spaces. A coordinate transformation between the two preserves the covariant structure. This means the same equation describes evolution whether you track allele frequencies or trait values, with the metric transforming appropriately.

The unmeasured quantity is specific: the covariance of evolutionary changes across generations. The metric (the inverse genotypic covariance matrix) has been characterized extensively. The noise covariance has not. The relationship between the two determines what kind of learning algorithm evolution is running — gradient descent, natural gradient, or something else. Measuring it is the experimental challenge.

The landscape metaphor that pervades evolutionary biology was always a geometric statement. Now it has the actual geometry.