Kalman filters have several assumptions

1. the belief state is Gaussian
2. Linear kinematics/dynamics However, they are great since they are computationally efficient

We can rewrite $\overline{Bel}(x_t)$, the belief state after applying the motion model, as a linear operation on the normal distribution

We can now express the new belief distribution as a weighted product of 2 normal distribution

that weight $k_t$ is known as the Kalman gain

Kalman Filter and its relation to the Algebraic Riccati Equation(ARE) and more general optimization problems such as LQR

The following post discusses the following questions

1. How did we arrive at $K_t$?
2. What is the ARE and how do we use it to solve for $K_t$

$x_t = Ax_{t-1} +Bu_t + \epsilon_t \ \ \epsilon_t \sim N(0, R) \\y_t= Cx_t+\delta_t \ \ \delta_t \sim N(0, Q)$

We reiterate the mean and covariance update in Kalman Filter from lecture

From prediction step, we write $x_t$ as a normal distribution with

$$ \overline{\mu}t = A{\mu}{t-1} + Bu_t\\ \overline{\Sigma}_t = A{\Sigma}_tA^T + R $$

$$ \mu_t = \overline{\mu}_t + K_t(z_t- C\overline{\mu}_t)\\ \Sigma_t = (I-K_tC)\overline{\Sigma}_t $$

And from the correction step, our new belief state is modelled by the normal distribution w/ parameters of

where $K_t = \overline{\Sigma_t}C^T(C\overline{\Sigma}_tC^T + Q)^{-1}$

We expand the definitions of $\mu_t$ and $\Sigma_t$

$$ \mu_t = A{\mu}_{t-1} + Bu_t + K_t(z_t- C\overline{\mu}_t)\\ \Sigma_t = (I-K_tC)(A{\Sigma}_tA^T + R) $$