Bayesian update of a Gaussian mixture on a noisy linear observation

Conditions a Gaussian mixture g on a single noisy linear observation \(y = A x + b + \epsilon\), \(\epsilon \sim \mathcal{N}(0, R)\). Per component, applies the Kalman gain $$K_k = \Sigma_k A^\top S_k^{-1}, \qquad S_k = A \Sigma_k A^\top + R,$$ and updates $$\mu'_k = \mu_k + K_k (y - A \mu_k - b), \qquad \Sigma'_k = (I - K_k A) \Sigma_k.$$ Component weights are multiplied by the marginal evidence \(\pi_k \mathcal{N}(y; A \mu_k + b, S_k)\) and renormalised. This is the finite-mixture analogue of a Kalman update step.

Usage

gmm_observe(g, A, y, noise_cov, b = 0, ridge_eps = 1e-06)

Arguments

g

A gmm (or gmm_fit) in R^p.

A

An m by p numeric matrix.

y

A length-m numeric vector (the observation).

noise_cov

An m by m SPD numeric matrix (the observation noise covariance R). Required.

b

Numeric scalar or length-m vector. Default 0.

ridge_eps

Tiny ridge added to updated covariances for numerical hygiene. Set to 0 to disable.

Value

A gmm in R^p with the same number of components and the reweighted component weights.

Details

If the marginal evidence vanishes at every component (e.g. y is many standard deviations from every component), the function issues a warning and returns g unchanged with metadata$gmm_observe_no_update = TRUE.

See also

Other operators: gmm_affine(), gmm_aggregate(), gmm_convolve(), gmm_counterfactual(), gmm_filter(), gmm_intervene(), gmm_missing(), gmm_mix(), gmm_product(), gmm_reduce()

Examples

g <- gmm(weights = c(0.5, 0.5),
         means = list(c(-1, 0), c(1, 0)),
         covariances = list(diag(2), diag(2)))
A <- matrix(c(1, 0), nrow = 1L)
gmm_observe(g, A = A, y = 0.8, noise_cov = matrix(0.25, 1, 1))
#> <observe(gmm)>: K = 2 components in p = 2 dimensions
#>   [1] w = 0.2176, |mu| = 0.4400, tr(Sigma) = 1.2000
#>   [2] w = 0.7824, |mu| = 0.8400, tr(Sigma) = 1.2000