state space model

A state space model is a partially observed Markov model:

• ${\mathbf{z}}_t$ are the latent states
• ${\mathbf{y}}_t$ are the observations
• ${\mathbf{u}}_t$ are exogenous inputs

Using the Markov property, the above probabilistic graphical model forms the joint distribution

$p({\mathbf{y}}_{1:T},{\mathbf{z}}_{1:T}|{\mathbf{u}}_{1:T})=[p({\mathbf{z}}_1|{\mathbf{u}}_{1)}{\prod }_i^T\underset{\text{dynamics model}}{\underbrace{p({\mathbf{z}}_t|{\mathbf{z}}_{t-1},{\mathbf{u}}_t)}}][{\prod }_i^T\underset{\text{observation model}}{\underbrace{p({\mathbf{y}}_t|{\mathbf{z}}_t,{\mathbf{u}}_t)}}]$

When the dynamics model is categorical so that we have discrete latent states, then we have hidden Markov models.

When the dynamics model and the observation model are Gaussian, we have a linear gaussian ssm, which can be efficiently solved using kalman filtering and smoothing.

See ssm resources.