Conjugate update of a normal mixture prior across replicates

Applies the normal-normal conjugate update to every replicate at once. This is the vectorised equivalent of calling RBesT::postmix() once per replicate: for a prior component with weight \(w_k\), mean \(\mu_k\) and standard deviation \(s_k\), and an observation \((m_i, se_i)\), the posterior component has variance \(1/(1/s_k^2 + 1/se_i^2)\), mean \(v_{ik}(\mu_k/s_k^2 + m_i/se_i^2)\) and weight proportional to \(w_k \, N(m_i; \mu_k, s_k^2 + se_i^2)\).

Usage

normal_mixture_posterior(weights, means, sds, estimate, standard_error)

Arguments

weights

Prior component weights. Either a vector of length n_components (a prior shared by every replicate) or a n_replicates x n_components matrix (one prior per replicate, as needed by empirical Bayes methods).

means

Prior component means, shaped like weights.

sds

Prior component standard deviations, shaped like weights.

estimate

Vector of per-replicate treatment effect estimates.

standard_error

Vector of per-replicate standard errors.

Value

A list of three n_replicates x n_components matrices: weights, means and sds of the posterior mixture.