Bayesian Inference for the Wishart Distribution Parameters

Posterior inference for the shape parameter alpha and mean matrix mu in the model X_i ~ Wishart_p(2*alpha, Sigma), under both an improper prior and a proper Gamma/inverse-Wishart prior. The posterior mode is found via a Newton-within-EM algorithm and joint samples are drawn via rejection sampling.

wishartinference

Bayesian inference for the shape parameter of the Wishart distribution — and its p = 1 reduction, the Gamma distribution.

Given data $X_1, \dots, X_n \overset{iid}{\sim} \text{Wishart}_p(2\alpha, \Sigma)$, this package provides posterior inference for the shape parameter $\alpha$ and the mean matrix $\mu = \alpha\Sigma$. The posterior mode is found via a Newton-within-EM algorithm, and joint samples of $(\alpha, \mu)$ are drawn via rejection sampling with a Gamma covering density. The core computation is implemented in C++ (RcppArmadillo + Boost) for speed.

The package has two independent backends, covered in their own sections below — there is no unifying dispatcher, so use whichever matches your data:

• wishart_inference() — matrix data, p >= 2
• gamma_inference() — scalar data, p = 1 (the exact p = 1 reduction of the same model, so scalar observations never need to be wrapped in trivial 1 x 1 matrices)

A full walkthrough of both backends — improper and proper priors, the manual step-by-step workflow, and the package's error-handling conventions — is in demo/wishart_demo.R.

Installation

install.packages(
  "wishartinference",
  repos = c("https://sunnyhq-shi.r-universe.dev", "https://cloud.r-project.org")
)

This installs a precompiled binary where available — no local C++ compiler or Rtools setup required.

Wishart model (p >= 2)

• Improper prior: $$p(\alpha, \mu) \propto (\alpha - (p-1)/2)^{-1} \cdot |\mu|^{-(p+1)/2}$$

• Proper prior: $$(\alpha - (p-1)/2) \sim \text{Gamma}(\beta, \text{rate} = \beta\eta), \qquad \mu \mid \alpha \sim \text{inv-Wishart}(2\kappa\alpha, 2\kappa\alpha\mu_0)$$

Quick start

library(wishartinference)

# Simulate data from a known truth
set.seed(6)
p <- 3; nu <- 10; Sigma <- diag(p); n <- 10
X <- rwishart(n, p, nu, Sigma)

# Full pipeline: sufficient stats, EM mode, rejection sampling
res <- wishart_inference(X, nsamp = 5000)

res$results$ahat                                    # posterior mode
quantile(res$results$alpha_samples, c(0.025, 0.975)) # 95% interval

Improper vs. proper prior

The prior type is auto-detected from the arguments you supply. beta = 0, eta = 1, kappa = 0 (the function defaults) is not merely a placeholder flag — it is mathematically identical to the improper prior above.

beta  <- 50
eta   <- (beta - 1) / (beta * (3 - (p - 1) / 2))  # places the prior mode at alpha = 3
kappa <- 1
mu0   <- diag(10, p)

res_pro <- wishart_inference(X, mu0 = mu0, beta = beta, eta = eta,
                              kappa = kappa, nsamp = 5000)

Manual step-by-step workflow

For full control over each stage, or to reuse the mode/sampler independently:

stat <- wishart_stats(X)                               # sufficient statistics
ahat <- mode_alphaEM(n, p, stat$xbar, stat$ldetxbarg)   # posterior mode

lambda   <- max(n * (ldet(stat$xbar) - stat$ldetxbarg), 1e-10)
nu_star  <- ahat[1] * lambda + 1

samp <- rejection_sampler(ahat[1], ahat[2], lambda, nu_star,
                           p, n, stat$xbar, stat$ldetxbarg)

Exported functions

Constraints

• p >= 2 and n >= 2 are required.
• Under the improper prior, n * (log|xbar| - mean(log|X_i|)) must be strictly positive; fails only when all n observations are numerically identical.

Gamma model (p = 1)

The exact p = 1 case of the model above: $X_i \sim \text{Gamma}(\alpha, \alpha/\mu)$. Same priors, same EM-then-rejection-sampling approach, just scalar inputs/outputs instead of matrices.

Quick start

set.seed(11)
n <- 20; a <- 3
x <- rgamma(n, a, a)   # mean 1

res <- gamma_inference(x, nsamp = 5000)

res$results$ahat                                    # posterior mode
quantile(res$results$alpha_samples, c(0.025, 0.975)) # 95% interval

# Proper prior
res_pro <- gamma_inference(x, mu0 = 1, beta = 5, eta = 0.5, kappa = 2, nsamp = 5000)

Exported functions

Differences from the matrix case

• kappa = 0 is allowed even with beta > 0 here (proper prior on alpha, flat prior on mu) — the matrix backend requires kappa >= 1 whenever the prior isn't fully improper.
• Structural edge case: at n = 2 with a fully improper prior (beta = 0, kappa = 0), the posterior has no interior mode for any dataset — its supremum sits exactly at the boundary alpha = 0. gamma_inference() raises an informative error here rather than returning a degenerate result. Any beta > 0, any kappa > 0, or n > 2 resolves it. (The matrix model has no analogous failure — an interior mode is always guaranteed there at n >= 2.)

Error handling

Bad inputs (invalid nsamp, malformed priors, non-positive-definite matrices, degenerate data, etc.) raise an informative R-level error immediately, for both backends. Genuine numerical non-convergence is instead caught internally by wishart_inference() / gamma_inference(), which return list(error = "...") rather than stopping — useful when iterating over many datasets in a batch job. The manual-workflow functions (mode_alphaEM(), rejection_sampler(), and their _gamma counterparts) always raise directly.

r <- wishart_inference(X, nsamp = 100, max_em_iter = 1)
if (!is.null(r$error)) {
  message("Inference failed: ", r$error)
}

Authors

• Phil Everson — [email protected]
• Hanqi Shi (maintainer) — [email protected]

Developed at Swarthmore College.

License

MIT — see LICENSE.