ELIR effective sample size of a normal mixture, per replicate

Vectorised equivalent of RBesT::ess(mix, method = "elir"). The ELIR effective sample size is \(\sigma^2 \int p(\theta) \, i(\theta) \, d\theta\), where \(i(\theta) = -\partial^2_\theta \log p(\theta)\) is the Fisher information of the prior.

RBesT evaluates that integral with Gauss-Hermite quadrature centred on each mixture component, doubling the node count until successive estimates agree and falling back to adaptive quadrature if they never do. That fallback is the usual outcome for a robust mixture prior, whose informative and vague components differ in scale by a factor of around twenty: nodes spaced for the vague component step over the sharp peak the informative one puts in \(i(\theta)\), so the estimate never settles.

This function instead integrates \(p\,i\) directly, over panels split at every component's centre and tails, with Gauss-Legendre quadrature on each panel. Every component then gets panels matched to its own width, whatever the spread of scales, and the result agrees with RBesT to around 1e-10 while running on all replicates at once.

Several cases short-circuit the quadrature. A single-component mixture has constant Fisher information, so its ELIR is exactly \(\sigma^2/\mathrm{variance}\); so does any row whose other components all have weight zero, as when the Egidi mixture selects no weak component. A prior shared by every replicate is integrated once and rescaled, since ELIR is proportional to \(\sigma^2\). A prior that varies across replicates through one component's standard deviation alone, as the empirical Bayes robust mixture's vague component does, is integrated at Chebyshev nodes over the range of that standard deviation and interpolated - see interpolate_smooth_function() - to a relative error below 1e-9.

Usage

normal_mixture_elir_ess(weights, means, sds, sigma, n_nodes = 40L, spread = 9)

Arguments

weights

Mixture weights: a vector when the prior is shared by every replicate, or a n_replicates x n_components matrix.

means

Component means, shaped like weights.

sds

Component standard deviations, shaped like weights.

sigma

Reference scale, either a single value or one value per replicate.

n_nodes

Number of Gauss-Legendre nodes per panel.

spread

How many standard deviations each component's panels reach.

Value

Vector of ELIR effective sample sizes, one per replicate.