Active inference selects actions by minimising an expected free energy functional of predicted futures. However, the expectation over yet-unobserved outcomes strips the functional of its Kullback-Leibler structure which hinders message passing treatments of inference procedures. Adding the expectation over future outcomes to the numerator restores the Kullback-Leibler structure but ultimately means that its stationary point no longer includes a mutual information term, which eliminates the agent’s epistemic drive. We propose an alternative formulation based on a Bethe free energy functional, fully supporting inference by message passing. The epistemic drive is maintained by imposing an information constraint, next to normalisation, marginalisation and form constraints, insisting that the mutual information between future observations, states and parameters given actions must be at least as large as the entropy of the goal prior. For a specific value of the corresponding Lagrange multiplier, the stationary point of this constrained Bethe Lagrangian recovers the EFE solution. We show that, as the information demand is varied, the solved Lagrange multiplier moves through its inactive, interior, and saturated regimes. In the inactive regime the agent’s epistemic drive switches off entirely, while in the saturated regime it is maximal. We test the performance of the constrained Bethe agent in an experiment with information gathering on a gridworld and one in which salience and novelty have to be balanced.