Sample Precision Matrices from the GGM Prior

Draws precision matrices \(K\) from the prior of a Gaussian graphical model using the same constrained NUTS sampler that drives bgm for continuous data. The likelihood is omitted (\(n = 0\), \(S = 0\)), so the chain targets the prior alone.

Usage

sample_precision_prior(
  p,
  n_samples,
  n_warmup = 2000,
  interaction_prior = cauchy_prior(scale = 2.5),
  precision_scale_prior = gamma_prior(shape = 1, rate = 1),
  step_size = 0.1,
  max_depth = 10L,
  seed = 1L,
  verbose = TRUE,
  edge_indicators = NULL
)

Arguments

p

Integer. Dimension of the precision matrix (\(p \ge 2\)).

n_samples

Integer. Number of post-warmup draws to keep.

n_warmup

Integer. NUTS warmup iterations. Default 2000.

interaction_prior

A bgms_parameter_prior for the off-diagonal entries. Use cauchy_prior() or normal_prior(); beta_prime_prior() is not supported here. Default: cauchy_prior(scale = 2.5).

precision_scale_prior

A bgms_scale_prior for the diagonal entries of \(K\). Use gamma_prior() or exponential_prior(). Default: gamma_prior(1, 1).

step_size

Positive numeric. Initial NUTS step size used to seed dual-averaging adaptation. Default 0.1.

max_depth

Integer. Maximum NUTS tree depth. Default 10.

seed

Integer. RNG seed for the chain. Default 1L.

verbose

Logical. If TRUE (default), print a progress bar.

edge_indicators

Optional integer \(p \times p\) matrix with 1 = edge included, 0 = excluded. Must be symmetric with 1s on the diagonal. Default: full graph (all edges included).

Value

A list with components

K_offdiag

Numeric matrix of size n_samples x p * (p - 1) / 2 containing the upper-triangle off-diagonal entries of \(K\) for each draw, in column-major order \((K_{12}, K_{13}, K_{23}, K_{14}, \ldots)\). Excluded edges are returned as 0.

K_diag

Numeric matrix of size n_samples x p containing the diagonal entries \(K_{11}, \ldots, K_{pp}\).

offdiag_names

Character vector of length p * (p - 1) / 2 naming the columns of K_offdiag (e.g. "K_1_2").

diag_names

Character vector of length p naming the columns of K_diag.

step_size

The (initial) NUTS step size used.

edge_indicators

The integer edge-indicator matrix used (full graph if not supplied).

Details

Off-diagonals are placed on the association scale \(K_{yy,ij} = -K_{ij}/2\) and assigned the supplied interaction_prior. A normal_prior(scale = s) therefore constrains \(K_{yy}\) with standard deviation \(s\), equivalent to a \(\textrm{Normal}(0, 2s)\) prior on \(K_{ij}\) itself. The diagonals \(K_{ii}\) are drawn from the supplied precision_scale_prior. When edge_indicators is supplied, off-diagonals at excluded positions are constrained to zero throughout the chain.

See also

cauchy_prior, normal_prior, gamma_prior, exponential_prior, bgm

Examples

# \donttest{
# Default Cauchy(0, 2.5) off-diagonal, Gamma(1, 1) diagonal, p = 4.
draws = sample_precision_prior(
  p = 4, n_samples = 200, n_warmup = 200,
  verbose = FALSE
)
dim(draws$K_offdiag) # 200 x 6
#> [1] 200   6
colnames(draws$K_offdiag) = draws$offdiag_names
head(draws$K_offdiag)
#>          K_1_2      K_1_3      K_1_4       K_2_3       K_2_4
#> [1,] -1.233684 -1.9146341  0.7934043  2.45108392 -1.99732141
#> [2,]  1.285579 -0.5028680  1.0132068  0.59625222 -0.17293213
#> [3,] -1.037669  0.2762245 -1.4285318 -0.64342766  0.87019167
#> [4,]  1.297038 -0.4459275  1.6381413 -0.07162156  0.65340822
#> [5,]  1.664220 -0.3653032  1.3650031  0.48766110 -0.03671729
#> [6,] -1.240221  0.3221408 -0.5824119 -0.33342054 -0.83995087
#>            K_3_4
#> [1,] -1.04512815
#> [2,] -2.55826117
#> [3,]  1.45388788
#> [4,] -0.01859332
#> [5,] -1.40563154
#> [6,] -0.79436832

# Sparser graph: drop the (1, 4) edge.
E = matrix(1L, 4, 4)
E[1, 4] = E[4, 1] = 0L
draws = sample_precision_prior(
  p = 4, n_samples = 200, n_warmup = 200,
  edge_indicators = E, verbose = FALSE
)
colnames(draws$K_offdiag) = draws$offdiag_names
all(draws$K_offdiag[, "K_1_4"] == 0) # TRUE
#> [1] TRUE
# }