  Linear Quadratic Gaussian (LQG) control is a framework first introduced in
control theory that provides an optimal solution to linear problems of
regulation in the presence of uncertainty. This framework combines Kalman-Bucy
filters for the estimation of hidden states with Linear Quadratic Regulators
for the control of their dynamics. Nowadays, LQG is also a common paradigm in
neuroscience, where it is used to characterise different approaches to
sensorimotor control based on state estimators, forward and inverse models.
According to this paradigm, perception can be seen as a process of Bayesian
inference and action as a process of optimal control. Recently, active
inference has been introduced as a process theory derived from a variational
approximation of Bayesian inference problems that describes, among others,
perception and action in terms of (variational and expected) free energy
minimisation. Active inference relies on a mathematical formalism similar to
LQG, but offers a rather different perspective on problems of sensorimotor
control in biological systems based on a process of biased perception. In this
note we compare the mathematical treatments of these two frameworks for linear
systems, focusing on their respective assumptions and highlighting their
commonalities and technical differences.
