linearGaussian

In statistics and signal processing, linear Gaussian models describe systems where variables are related by linear equations with additive Gaussian noise. A prominent instance is the linear Gaussian state-space model, where a latent state evolves linearly and observations are linear functions of the state, both contaminated by Gaussian noise.

Formally, the state transition is x_{t+1} = A x_t + w_t, with w_t ~ N(0, Q), and the observation is y_t = C x_t + v_t, with v_t ~ N(0, R). The initial state x_0 is often taken as x_0 ~ N(mu_0, Sigma_0). All noise terms are assumed independent. In a static variant, one observes y = B x + e with x ~ N(mu_x, Sigma_x) and e ~ N(0, R).

Because all conditional distributions are Gaussian, exact inference is available via the Kalman filter for online estimation and the Rauch–Tung–Striebel smoother for offline estimation. Parameter estimation can be performed by maximum likelihood using expectation–maximization or via Bayesian methods.

Applications include control, robotics, signal processing, econometrics, and time-series analysis. Limitations arise when the true dynamics are nonlinear or noise is non-Gaussian; in such cases extensions like the extended Kalman filter, unscented Kalman filter, or particle filters are used.

See also Kalman filter, Gaussian distribution, state-space model.