Pointwise product of two Gaussian mixtures

The normalised pointwise product of two mixture densities – the conjugate Bayes update when one mixture plays the prior and the other the likelihood. The product of two Gaussian mixtures is again a Gaussian mixture with \(K_1 K_2\) components: $$p(x)\, q(x) \propto \sum_{ij} w_i v_j z_{ij}\, \mathcal{N}\!\left(x;\ \mu_{ij},\ \Sigma_{ij}\right),$$ with \(\Sigma_{ij} = (\Sigma_i^{-1} + S_j^{-1})^{-1}\), \(\mu_{ij} = \Sigma_{ij} (\Sigma_i^{-1} \mu_i + S_j^{-1} m_j)\), and \(z_{ij} = \mathcal{N}(\mu_i;\ m_j,\ \Sigma_i + S_j)\) evaluated in the log domain. The overall normaliser \(\int p\, q = \sum_{ij} w_i v_j z_{ij}\) is returned in the result's metadata as log_integral (it is the marginal evidence of the update).

Usage

gmm_product(g1, g2, reduce = NULL)

Arguments

g1, g2

Two gmm (or gmm_fit) objects of the same ambient dimension.

reduce

Optional positive integer: reduce the product to at most this many components via gmm_reduce(). Default NULL (no reduction).

Value

A gmm with K1 * K2 components (or at most reduce), with metadata$log_integral recording \(\log \int p\, q\).

Details

Because the component count multiplies, chained products grow geometrically; pass reduce to cap the count via gmm_reduce() after the product (the standard assumed-density trick), or manage the budget yourself.

See also

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

Examples

prior <- gmm(weights = c(0.5, 0.5), means = list(-2, 2),
             covariances = list(matrix(1), matrix(1)))
lik <- gmm(weights = 1, means = list(0.5), covariances = list(matrix(0.5)))
post <- gmm_product(prior, lik)
post@weights
#> [1] 0.2086085 0.7913915