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We consider the problem of learning a linear control policy for a linear dynamical system, from demonstrations of an expert regulating the system. The standard approach to this problem is policy fitting, which fits a linear policy by minimizing a loss function between the demonstrations and the policy's outputs plus a regularization function that encodes prior knowledge. Despite its simplicity, this method fails to learn policies with low or even finite cost when there are few demonstrations. We propose to add an additional constraint to policy fitting, that the policy is the solution to some LQR problem, i.e., optimal in the stochastic control sense for some choice of quadratic cost. We refer to this constraint as a Kalman constraint. Policy fitting with a Kalman constraint requires solving an optimization problem with convex cost and bilinear constraints. We propose a heuristic method, based on the alternating direction method of multipliers (ADMM), to approximately solve this problem. Numerical experiments demonstrate that adding the Kalman constraint allows us to learn good, i.e., low cost, policies even when very few data are available.