# Posterior coefficients of the Normal-Inverse-Wishart distribution with its conjugate prior.

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### Description

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

### Usage

 1 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.

### Details

The NIW distribution p(μ, Σ | λ, κ, Ψ, ν) is defined as

Σ \sim W^{-1}(Ψ, ν), \quad μ | Σ \sim N(λ, Σ/κ).

The default value kappa = 0 uses the Lebesque prior on μ: p(μ) \propto 1. The default value Psi = 0 uses the scale-invariant prior on Σ: p(Σ) \propto |Σ|^{-(ν+d+1)/2}. The default value nu = ncol(X)+1 for kappa = 0 and Psi = 0 makes E[μ|X]=\code{colMeans(X)} and E[Σ | X]=\code{var(X)}.

### Value

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

rniw, niw.mom, niw.post.

### Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 # 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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