**Some basic background**

The Kalman filter is a (linear) state estimation algorithm that presumes that there is some sort of uncertainty (optimally Gaussian) in the state observations of the dynamical system. The Kalman filter has been extended to nonlinear systems, constrained for cases where direct state observation is impossible, etc. As one should know, the algorithm essentially performs a prediction step based on prior state knowledge, compares that prediction to measurements, and updates the knowledge of the state based on a state covariance estimate that is also updated at every step.

In short, we have a state-space vector usually represented by a vector in $\mathbb{R}^n$. This vector is operated on by a plant matrix, added to a control term (which is operated on by a control matrix), operated on by a covariance matrix, etc. There are usually not intrinsic constraints on the state-space vector beyond those which may be embedded in the description of the dynamical system.

**My question**

Can we describe the Kalman filter using the language of category theory? In other words, can we generalize the algebra of what is happening with the Kalman filter for a dynamical system to develop Kalman-like estimators for other processes with vector-like components derived from some known (or partially known) model?

**A motivating example**

A repeated matrix game involves two or more players choosing strategies according to some payoff function -- which is a function of the game's state space and each player's action -- typically with some notion of equilibrium (e.g. Nash equilibrium). A repeated game with a finite action space implies that strategies are discrete probability distributions representable by a vector in $\mathbb{R}^N$ such that every element of the strategy vector is non-negative and the vector sums to unity.

This is a more restrictive case than a general dynamical system because we have constraints on the vector representation of the strategy. But, given input (i.e. actions taken), output (state measurement of the game's state space), and a payoff structure, it might make sense to attempt a Kalman-like estimator for the strategy vector in the game.

The challenge, of course, is developing a compatible notion of covariance. It's not clear what covariance would mean in relation to probability distributions, since we don't treat the quantity $P(\textrm{action} = \textrm{index})$ as random variable. But if we had a category theoretical view of the Kalman filter, could we derive a similar structure based on its algebraic properties in such a way that we can guarantee that our estimate of the opponent's strategy vector converges in some way?

**Does what I'm asking even make sense?**