How do we estimate internal, hidden state variables through noisy measurement data?

The solution to state estimation is related to the specific Motion Equation and Observation Equation of a robot and the noise probability distribution. Based on the motion/observation equations, and whether the noise is Gaussian, state estimation is divided into linear/non-linear and Gaussian/non-Gaussian systems.

Notes from: Kalman Filter from The Ground Up by Alex Becker

Basic State Estimation

Kalman Filtering

Kalman Filtering Basics

Multivariate Kalman Filters