Random draws from the posterior distribution with Normal-Independent-Inverse-Wishart (NIIW) prior.
Given iid d-dimensional niche indicators X = (X_1,…,X_N) with X_i \sim N(μ, Σ), this function generates random draws from p(μ,Σ | X) for the Normal-Independent-Inverse-Wishart (NIIW) prior.
niiw.post(nsamples, X, lambda, Omega, Psi, nu, mu0 = lambda, burn)
nsamples |
the number of posterior draws. |
X |
a data matrix with observations along the rows. |
lambda |
mean of mu. See Details. |
Omega |
variance of mu. Defaults to |
Psi |
scale matrix of Sigma. Defaults to |
nu |
degrees of freedom of Sigma. Defaults to |
mu0 |
initial value of mu to start the Gibbs sampler. See Details. |
burn |
burn-in for the MCMC sampling algorithm. Either an integer giving the number of initial samples to discard, or a fraction with |
The NIIW distribution p(μ, Σ | λ, κ, Ψ, ν) is defined as
Σ \sim W^{-1}(Ψ, ν), \quad μ | Σ \sim N(λ, Ω).
The default value Omega = 0 uses the Lebesque prior on μ: p(μ) \propto 1. In this case the NIW and NIIW priors produce identical resuls, but niw.post is faster.
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 Omega = 0 and Psi = 0 makes E[μ|X]=\code{colMeans(X)} and E[Σ | X]=\code{var(X)}.
Random draws are obtained by a Markov chain Monte Carlo (MCMC) algorithm; specifically,
a Gibbs sampler alternates between draws from p(μ | Σ, X) and p(Σ | μ, X), which are Normal and Inverse-Wishart distributions respectively.
Returns a list with elements mu and Sigma of sizes c(nsamples, length(lambda)) and c(dim(Psi), nsamples).
# simulate data
d <- 4
mu0 <- rnorm(d)
Sigma0 <- matrix(rnorm(d^2), d, d)
Sigma0 <- Sigma0 %*% t(Sigma0)
N <- 100
X <- rmvnorm(N, mean = mu0, sigma = Sigma0)
# prior parameters
# flat prior on mu
lambda <- 0
Omega <- 0
# informative prior on Sigma
Psi <- crossprod(matrix(rnorm(d^2), d, d))
nu <- 5
# sample from NIIW posterior
nsamples <- 2e3
system.time({
siiw <- niiw.post(nsamples, X, lambda, Omega, Psi, nu, burn = 100)
})
# sample from NIW posterior
kappa <- 0
system.time({
siw <- niw.post(nsamples, X, lambda, kappa, Psi, nu)
})
# check that posteriors are the same
# p(mu | X)
clrs <- c("black", "red")
par(mar = c(4.2, 4.2, 2, 1)+.1)
niche.par.plot(list(siiw, siw), col = clrs, plot.mu = TRUE, plot.Sigma = FALSE)
legend(x = "topright", legend = c("NIIW Prior", "NIW Prior"), fill = clrs)
# p(Sigma | X)
par(mar = c(4.2, 4.2, 2, 1)+.1)
niche.par.plot(list(siiw, siw), col = clrs, plot.mu = FALSE, plot.Sigma = TRUE)
legend(x = "topright", legend = c("NIIW Prior", "NIW Prior"), fill = clrs)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.