Gibbs sampler for the Bradley–Terry Stochastic Block Model (BT–SBM)

Runs a Gibbs sampler for the BT–SBM with optional DP/PY/DM/GN priors on the partition. Returns raw draws plus minimal summaries; n_ij is computed internally as w_ij + t(w_ij).

Usage

gibbs_bt_sbm(
  w_ij,
  a = 4,
  b = exp(digamma(a)),
  prior = c("DP", "PY", "DM", "GN"),
  alpha_PY = NA_real_,
  sigma_PY = NA_real_,
  beta_DM = NA_real_,
  K_DM = NA_integer_,
  gamma_GN = NA_real_,
  T_iter = 2000,
  T_burn = 1000,
  init_x = NULL,
  store_z = FALSE,
  verbose = TRUE
)

Arguments

w_ij

Integer or numeric square matrix \(n \times n\) of directed wins (i over j). Must be nonnegative with zero diagonal. The function builds \(n_{ij} = w_{ij} + w_{ji}\) internally.

a, b

Positive shape and rate parameters for the Gamma prior \(\lambda_k \sim \mathrm{Gamma}(a,b)\). By default, \(b = \exp(\psi(a))\), so that \(\mathbb{E}[\log \lambda_k] = 0\) a priori.

prior

Character scalar, one of "DP", "PY", "DM", "GN".

alpha_PY, sigma_PY

Hyperparameters for Pitman–Yor / Dirichlet Process. For prior="DP" use alpha_PY (with sigma_PY ignored). For prior="PY" use both alpha_PY and sigma_PY in (0,1).

beta_DM, K_DM

Hyperparameters for the finite Dirichlet–Multinomial prior. K_DM is the maximum number of allowed clusters.

gamma_GN

Hyperparameter for the Gnedin prior.

T_iter, T_burn

Integers: total iterations and burn-in. Must satisfy T_burn < T_iter.

init_x

Optional integer vector of length n with initial labels (1-based).

store_z

Logical; if TRUE, store latent Z draws (heavy).

verbose

Logical; if TRUE, prints progress every 1000 iterations.

Value

A list with:

• x_samples: integer matrix \(S \times n\) of raw labels (\(S = T_{\mathrm{iter}}-T_{\mathrm{burn}}\)).

• lambda_samples: list of length \(S\); each element is a numeric vector of length \(L_{\mathrm{cap}}\) for that draw, with NA at empty labels.

• K_per_iter: integer vector length \(S\) (occupied cluster count per saved draw).

• L_cap_per_iter: integer vector length \(S\) (label capacity trace).

• z_samples: if store_z=TRUE, a numeric array \(S \times n \times n\); otherwise NULL.

Details

Row names of w_ij (if present) are propagated to item-level outputs; otherwise items are named Item_1, ..., Item_n.

Examples

if (FALSE) { # \dontrun{
set.seed(1)
n <- 6L
w <- matrix(0L, n, n)
w[lower.tri(w)] <- rpois(sum(lower.tri(w)), 2)
diag(w) <- 0
rownames(w) <- colnames(w) <- paste0("P", seq_len(n))
fit <- gibbs_bt_sbm(
  w_ij = w, prior = "GN", gamma_GN = 0.5,
  T_iter = 200, T_burn = 100, verbose = FALSE
)
} # }