[sekulovskin]

Description

Fixes how the SBM edge prior sets the inclusion probabilities of a newly opened
cluster. They were drawn from the prior instead of from their posterior given the
variable's edges. As a result, the sampler did not target the correct posterior
over partitions.

Problem / Motivation

In the allocation step (block_allocations_mfm_sbm), a variable that leaves a
shared cluster to start a new one is scored with the correct integrated
likelihood. But the new cluster's between-cluster inclusion probabilities were
then drawn from the prior Beta(alpha_between, beta_between), ignoring the
variable's edges. Variables updated later in the same sweep used these random
values to decide whether to join the new cluster.

Under the default sbm_prior() this biases the posterior over partitions
(cluster-count probabilities, co-clustering, clustering Bayes factors). With
strongly informative priors such as sbm_prior(8, 1, 1, 8) the bias is
negligible.

# Two triangles joined by one edge; large n pins the graph down.
p <- 6
edges <- rbind(c(1, 2), c(1, 3), c(2, 3), c(4, 5), c(4, 6), c(5, 6), c(3, 4))
omega <- diag(p)
omega[edges] <- -0.3
omega[edges[, 2:1]] <- -0.3
set.seed(20261023)
x <- matrix(rnorm(3000 * p), 3000, p) %*% chol(solve(omega))
fit <- bgm(x, variable_type = "continuous", edge_prior = sbm_prior(),
           iter = 100000, warmup = 2000, chains = 2, seed = 20261023)
alloc <- do.call(rbind, fit$raw_samples$allocations)
table(apply(alloc, 1, function(z) length(unique(z)))) / nrow(alloc)
# Before the fix: P(1 cluster) ~ 0.36; exact posterior (all 203 partitions) ~ 0.33

Proposed Changes / New Functionality

In the non-singleton branch of block_allocations_mfm_sbm
(src/priors/sbm_edge_prior.cpp), each between-cluster probability of the new
cluster is now drawn from Beta(alpha_between + s_ib, beta_between + n_b - s_ib).
Here s_ib is the number of edges from the variable to cluster b, and n_b is the
size of cluster b. The within-cluster entry keeps its prior draw, because the new
cluster has no within-cluster pairs yet.

This affects the sampling algorithm in src/ only. There are no changes to prior
specifications, summary outputs, or R functions.

Fixes #

Type of Change

• Bug fix
• New feature
• Breaking change
• Documentation update
• Performance optimization
• Statistical / methodological update

Documentation and Release Notes

• Updated function documentation for any signature or behaviour changes
• Regenerated man/*.Rd files if roxygen comments changed
• Added or updated NEWS.md entry if the change affects users

Testing and Validation

• Compared the sampler with exact enumeration of all partitions (6 variables,
  averaged over the sampled graphs). Before the fix: total variation distance
  0.107, largest deviation 11.8 Monte Carlo SEs. After: 0.018 and 1.8 SEs, i.e.
  within Monte Carlo error. sbm_prior(8, 1, 1, 8) is unaffected (TV 0.0009).

• Added tests/testthat/test-sbm-allocation.R (nightly tier,
  BGMS_RUN_SLOW_TESTS=true, ~8 s). It checks P(number of clusters) against
  exact enumeration within 0.02: the gap is 0.004 after the fix and 0.036 before,
  so the test fails on the old code.

• Full test suite: 8,514 passed, 0 failed. The slow tiers of
  test-sbm-correction.R and test-mixed-correction.R also pass.

• Added or updated tests in tests/testthat/ when needed

• Ran the project styler

source("inst/styler/bgms_style.R")
styler::style_pkg(style = bgms_style)

• Ran lintr::lint_package()
• Ran roxygen2::roxygenise() if roxygen comments changed
• Ran rcmdcheck::rcmdcheck(args = c("--no-manual", "--as-cran")) for non-trivial changes
• Verified numerical behaviour where relevant

Additional Notes

• Results under the default sbm_prior() will change numerically (partition
  probabilities, cluster counts, clustering Bayes factors). Results under strongly
  informative Beta priors are essentially unchanged.
• The singleton branch is unchanged; it was already correct.
• Follow-up: the "joint" branch (block_allocations_mfm_sbm_corrected) has the
  same prior draw. Its correct conditional includes the normalizer tilt, so it
  needs a separate fix.