if, at non-equilibrium evolutionary steady-state, the likelihood of an agent’s genotype is proportional to the likelihood of its phenotypic trajectory (where \ denotes exclusion), then the following holds (Equation 12).
11
An agent’s autonomous dynamics can be cast as a gradient descent on a Lagrangian, whose path integral (i.e., action) corresponds to negative fitness. This Lagrangian depends upon the flow parameters (and initial states) supplied by the genotype. The agent’s genotype can then be cast as a stochastic gradient descent on negative fitness (i.e., genetic drift)
12
The existence of a nonequilibrium evolutionary steady-state solution to the density dynamics (at both scales) allows us to express the fast and slow dynamics of agents and their autonomous states in terms of Helmholtz decompositions.
13
The gradients of surprisal at the slow scale, with respect to any given agent’s ‘kind’ or genotype, are the gradients of action
14
In the limit of small fluctuations, the autonomous paths become the paths of least action; i.e., when the fluctuations take their most likely value of zero.
15
the genotypic state of the n-th agent
the phenotypic state of the n-th agent
the action (i.e., negative fitness) scoring the accumulated evidence. This evidence is also known as a marginal likelihood because it marginalises over external dynamics; i.e., other agents.
a phenotype’s generative model
the extended genotype covers both the genetic and epigenetic specification of developmental trajectories and the initial conditions necessary to realise those trajectories, including external states conditions necessary for embryogenesis.