Log normalizing constant for the MRF prior (per edge)

Computes the log normalizing constant \(\log C(\Theta, \nu)\) for the Markov random field prior on graph edge indicators, as defined in Peterson et al. (2015), Equation 3.2: $$C(\Theta, \nu) = \sum_{g \in \{0,1\}^K} \exp(\nu \cdot \mathbf{1}'g + g'\Theta g)$$ This is the computational hot spot in the MRF prior evaluation and is implemented in C++ via RcppArmadillo for performance.

Usage

calc_mrf_logC(Theta, nu)

Arguments

Theta

Numeric matrix (K x K); symmetric graph similarity parameters. Off-diagonal entry \(\Theta_{km}\) controls the tendency for edge inclusion to be shared between groups k and m.

nu

Numeric vector of length 1 or more; edge-specific log-odds parameters \(\nu_{ij}\). Each value controls the baseline edge inclusion probability for one edge across all groups.

Value

Numeric vector of the same length as nu, where each entry is \(\log C(\Theta, \nu_{ij})\) for the corresponding \(\nu_{ij}\).

Details

The computation requires summing over all \(2^K\) binary vectors in \(\{0,1\}^K\), which is feasible for moderate K (the method is designed for K up to about 20). For each binary vector \(g\), the term \(\nu \cdot \mathbf{1}'g + g'\Theta g\) is computed and the log-sum-exp trick is used for numerical stability.

See also

multiggm_mcmc()

Examples

# Two groups, no similarity
Theta <- matrix(0, 2, 2)
calc_mrf_logC(Theta, nu = c(-1, 0, 1))
#> [1] 0.6265234 1.3862944 2.6265234

# Two groups with similarity theta = 0.5
Theta[1,2] <- Theta[2,1] <- 0.5
calc_mrf_logC(Theta, nu = c(-1, 0, 1))
#> [1] 0.7436684 1.7436684 3.2779784