State-Dependent Observation Noise Reintroduces Epistemic Value in Linear-Gaussian Active Inference
Recent work established that under active inference, linear-Gaussian state-space models lose their epistemic drive (any incentive to act so as to gain information) "under any circumstances". The epistemic term of the Expected Free Energy becomes constant: the agent flattens to a Kalman filter whose gain sequence is fixed in advance, regardless of action. The minimal departure that restores the drive is unknown; the only established route is control entering the dynamics multiplicatively; the observation side of this boundary is unexplored. We show that state-dependent observation noise is such a departure: a covariance R(x) that varies with the state x, representing a sensor's accuracy degrading with range. The agent runs the standard first-order Gaussian filter of this literature, R evaluated at the predicted mean. Coupling R(x) to a controllable latent mean makes the posterior covariance, and hence the effective Kalman gain, depend on the action. Consequently, no fixed linear-Gaussian filter reproduces the agent and, under a mild rank condition on the observation map and a non-degeneracy condition on R(x), epistemic value is no longer constant; for scalar observations, reachable non-constancy alone is needed. This is a minimal constructive instance of the Bar-Shalom-Tse dual effect in the agent's maintained covariance: actions now influence the quality of future estimates, not merely the state. Our library cpomdp detects the incompatibility from model specification alone and raises a typed IncompatibleLinearizationError. The theorem ships with an executable witness: exhibiting any fixed filter that reproduced the agent's beliefs would refute both theorem and witness at once. Together this offers a precise, observation-side characterisation of curiosity in a Gaussian agent, bridging dual control and active inference.