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PC + LKJ hyperprior for a random-effect covariance

Source: R/nested_laplace_re_cov.R
re_cov_pc_lkj_prior.Rd

Construct the default weakly-informative hyperprior used by tulpa_re_cov_nested() for one covariance block: independent Penalized-Complexity (PC) priors on the marginal standard deviations sigma_i together with an LKJ prior on the correlation matrix R (correlated block) or no correlation (diagonal block), returned as a log_prior_theta function in the block's integration coordinates.

Usage

re_cov_pc_lkj_prior(
  n_coefs,
  prior_sigma = c(3, 0.05),
  eta = 2,
  correlated = TRUE
)

Arguments

n_coefs

Number of coefficients c in the RE block.

prior_sigma

c(U, alpha) giving P(sigma_i > U) = alpha (default c(3, 0.05)), applied independently to every marginal SD.

eta

LKJ shape (default 2). eta = 1 is uniform on correlation matrices; larger values favour weaker correlations. Ignored for a diagonal block.

correlated

TRUE (default) for a full covariance block (log-Cholesky coordinates, LKJ prior); FALSE for a diagonal / uncorrelated block (log-SD coordinates, no correlation). For n_coefs = 1 the two coincide.

Value

A function(theta) returning the scalar log prior density in the block's integration coordinates, suitable for one block of the log_prior_theta argument of tulpa_re_cov_nested().

Details

For a correlated block the prior is specified on the natural scale, p(sigma, R) = LKJ(R | eta) * prod_i PC(sigma_i), then pushed to the log-Cholesky coordinates theta of Sigma = L L' by the exact change-of-variables Jacobian. For a diagonal (uncorrelated) block the LKJ factor drops and theta_i = log sigma_i with Jacobian sum_i theta_i.

PC prior (Simpson et al. 2017) on each marginal SD: exponential with rate lambda = -log(alpha) / U, so P(sigma_i > U) = alpha – the prior_sigma = c(U, alpha) convention also used by the SPDE prior in tulpa.

LKJ prior (Lewandowski et al. 2009) on the correlation matrix: p(R) proportional to det(R)^(eta - 1). eta = 1 is uniform over correlation matrices; eta > 1 concentrates toward the identity. The normalizing constant is dropped (constant across the grid, so it cancels when the integration weights are renormalized).

Jacobian (correlated block): with theta packing log L_ii on the diagonal and the raw strict-lower entries of L, the change of variables from (sigma, R) to theta adds sum_i (c + 2 - i) * log L_ii - c * sum_i log sigma_i to log p(sigma, R). (Composition of the log-diagonal map, the standard Cholesky-to-covariance Jacobian 2^c prod_i L_ii^(c+1-i), and the covariance-to-(sigma, R) Jacobian; verified against numerical differentiation in test-re-cov-prior.R.)

See also

tulpa_re_cov_nested()