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nicheROVER (version 1.0)

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)

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.

Value

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

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 \(E[\mu|X]=\code{colMeans(X)}\) and \(E[\Sigma | X]=\code{var(X)}\).

See Also

rniw, niw.mom, niw.post.

Examples

Run this code
# NOT RUN {
# 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)
# }

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