niw.coeffs: Posterior coefficients of the Normal-Inverse-Wishart distribution with its conjugate prior.

Description

Given iid $d$-dimensional niche indicators $X = (X_1,\ldots,X_N)$ with $X_i \sim N(\mu, \Sigma)$, this function calculates the coefficients of the Normal-Inverse-Wishart (NIW) posterior $p(\mu, \Sigma | X)$ for a conjugate NIW prior. Together with niw.mom(), this can be used to rapidly compute the point estimates $E[\mu | X]$ and $E[\Sigma | X]$.

Usage

niw.coeffs(X, lambda, kappa, Psi, nu)

Value

Returns a list with elements lambda, kappa, Psi, nu corresponding to the coefficients of the NIW posterior distribution $p(\mu, \Sigma | X)$.

Arguments

X: A data matrix with observations along the rows.
lambda: Location parameter. See 'Details'.
kappa: Scale parameter. Defaults to kappa = 0. See 'Details'.
Psi: Scale matrix. Defaults to Psi = 0. See 'Details'.
nu: Degrees of freedom. Defaults to nu = ncol(X)+1. See 'Details'.

Details

The NIW distribution $p(\mu, \Sigma | \lambda, \kappa, \Psi, \nu)$ is defined as $$ \Sigma \sim W^{-1}(\Psi, \nu), \quad \mu | \Sigma \sim N(\lambda, \Sigma/\kappa). $$ The default value kappa = 0 uses the Lebesque prior on $\mu$: $p(\mu) \propto 1$.

The default value Psi = 0 uses the scale-invariant prior on $\Sigma$: $p(\Sigma) \propto |\Sigma|^{-(\nu+d+1)/2}$.

The default value nu = ncol(X)+1 for kappa = 0 and Psi = 0 makes $E[\mu|X]=`colMeans(X)`$ and $E[\Sigma | X]=`var(X)`$.

Examples

Run this code

# NIW prior coefficients
d <- 3
lambda <- rnorm(d)
kappa <- 5
Psi <- crossprod(matrix(rnorm(d^2), d, d))
nu <- 10

# data
data(fish)
X <- fish[fish$species == "ARCS",2:4]

# NIW posterior coefficients
post.coef <- niw.coeffs(X, lambda, kappa, Psi, nu)

# compare
mu.mean <- niw.mom(post.coef$lambda, post.coef$kappa, post.coef$Psi, post.coef$nu)$mu$mean
mu.est <- rbind(prior = niw.mom(lambda, kappa, Psi, nu)$mu$mean,
                data = colMeans(X),
                post = mu.mean)
round(mu.est, 2)