Kalman Filters
Here is a historical treatment on the topic: https://jwmi.github.io/ASM/6-KalmanFilter.pdf. Kalman Filters are defined as follows: We start with a variable $X_{0} \sim \mathcal{N}(\mu, \Sigma)$, then we have a motion model and a sensor model: $$ \begin{cases} X_{t + 1} = FX_{t} + \varepsilon_{t} & F \in \mathbb{R}^{d\times d}, \varepsilon_{t} \sim \mathcal{N}(0, \Sigma_{x})\\ Y_{t} = HX_{t} + \eta_{t} & H \in \mathbb{R}^{m \times d}, \eta_{t} \sim \mathcal{N}(0, \Sigma_{y}) \end{cases} $$Inference is just doing things with the Gaussians. One can interpret the $Y$ to be the observations and $X$ to be the underlying beliefs about a certain state. We see that the Kalman Filters satisfy the Markov Property, see Markov Chains. These independence properties allow a easy characterization of the joint distribution for Kalman Filters: ...