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Active inference unifies intentional and conflict-resolution imperatives of motor control

Fig 5

Derivation, under the Laplace approximation, of the variational free energy (VFE), , for the active inference agent.

See S1 Appendix for the detailed derivation of Eq F.1, in which is expressed in terms of the Laplace encoded energy, and C is a term assumed here to be a constant that encodes the optimal variance (omitted for clarity as it will not be used for computing the gradients as it does not depend on the internal state μ nor the action A). p(s_p|μ) and p(s_v|μ) are the proprioception and visual likelihood given the internal belief μ_θ. is the joint probability of the internal state vector up to 2^nd order, which can be expressed in terms of conditional probabilities on the system state as expected from the internal dynamical model. Both the sensory state likelihood and the conditional probabilities are approximated as Gaussians centred on the expected sensory state and the expected value of the dynamics at the different orders (from Eq M.3 in Fig 3) respectively. Please see the main text for a more detailed explanation.

doi: https://doi.org/10.1371/journal.pcbi.1010095.g005