Bayesian Dirichlet-Multinomial comparison of two transition networks

Compares two transition networks estimated by build_network (method "relative" or "frequency") using a Bayesian Dirichlet-Multinomial model. The outgoing transitions from each source state are modelled as a Multinomial draw with a Dirichlet prior on the transition probabilities. With a Jeffreys prior the posterior for the transitions out of state \(i\) is \(\mathrm{Dirichlet}(c_i + \alpha)\), where \(c_i\) are the observed outgoing counts. Each edge probability is then marginally Beta-distributed, so the posterior mean difference between the two networks is available in closed form and a credible interval is obtained by Monte Carlo.

This is a complement to permutation: the permutation test answers "is this difference more extreme than chance?"; the Bayesian comparison answers "what is the plausible range of the true difference, and how precisely is it estimated given the counts?". An edge with few outgoing transitions from its source state yields a wide credible interval even when its row-normalised probability looks decisive.

bayes_compare() also accepts two net_edge_betweenness objects (source method "relative" only). Edge betweenness is a nonlinear function of the whole transition matrix, so instead of Beta marginals the full transition matrix is drawn from each group's row-wise Dirichlet posterior and edge betweenness is recomputed on every draw - the Bayesian analogue of permutation()'s edge-betweenness dispatch. The result summarises the posterior of EB(x) - EB(y): diff is the posterior mean difference, prob_x/prob_y hold the posterior mean betweenness matrices, and observed_diff the plug-in difference of the two input networks. Both inputs must use the same invert setting.

Usage

bayes_compare(
  x,
  y = NULL,
  prior = 0.5,
  draws = 10000L,
  ci = 0.95,
  mean_threshold = 0.01,
  bound_threshold = 0.001,
  seed = NULL
)

Arguments

x: A netobject (from build_network), a netobject_group, an mcml object, or a net_edge_betweenness object. Must use a transition method ("relative" / "frequency" and their aliases).
y: A second object of the same kind as x, or NULL. When x is a netobject_group and y is NULL, all pairwise comparisons among the groups are returned.
prior: Numeric. Dirichlet prior concentration added to every cell (default 0.5, the Jeffreys prior). Use 1 for a uniform (Laplace) prior.
draws: Integer. Number of Monte Carlo posterior draws used for the credible intervals (default 10000).
ci: Numeric in (0, 1). Credible interval mass (default 0.95).
mean_threshold: Numeric. An edge is flagged significant only if the absolute posterior mean difference exceeds this (default 0.01).
bound_threshold: Numeric. An edge is flagged significant only if the credible-interval bound nearest zero exceeds this in absolute value (default 0.001). Guards against differences that are detectable but negligibly small.
seed: Integer or NULL. RNG seed for reproducible credible intervals.

Value

An object of class c("net_bayes", "netdifference", "net_permutation"). It carries the same fields as a permutation result, so it is a drop-in wherever a net_permutation is consumed, and also carries a netdifference difference matrix for cograph difference plotting, plus Bayesian extras:

x, y: The two input netobjects.
diff: Posterior mean difference matrix (prob_x - prob_y); the analogue of the permutation observed difference.
difference_matrix: Alias of diff for cograph netdifference helpers.
diff_sig: Difference where sig, else 0.
p_values: P-value matrix (the two-sided Bayesian p-equivalent).
effect_size: Posterior mean difference over its posterior SD.
ci_lower, ci_upper: Credible-interval bound matrices.
p_difference: Probability of the difference: the share of posterior mass on the dominant side of zero, in \([0.5, 1]\) (\(P(\mathrm{High} > \mathrm{Low})\) for a positive difference).
p_bayes: Alias of p_values (two-sided Bayesian p, \(2(1-\mathrm{p\_difference})\)).
prob_x, prob_y: Posterior mean transition-probability matrices.
sig: Logical significance matrix (CI excludes zero, mean and nearest bound exceed their thresholds).
summary: Long-format data frame whose columns are a superset of summary.net_permutation (from, to, weight_x, weight_y, diff, effect_size, p_value, sig) plus count_x, count_y, ci_lower, ci_upper, ci_width, p_difference.
method, iter, alpha, paired, adjust: Permutation-compatible settings (iter = draws, alpha = 1 - ci, paired = FALSE, adjust = "none").
prior, draws, ci, mean_threshold, bound_threshold: Bayesian settings.

References

Johnston, L. & Jendoubi, T. (2026). How Delivery Mode Reshapes Resource Engagement: A Bayesian Differential Network Analysis. TNA Workshop 2026.

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis (3rd ed.). CRC Press.

Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London A, 186(1007), 453-461.

Examples

s1 <- data.frame(V1 = c("A","B","C"), V2 = c("B","C","A"))
s2 <- data.frame(V1 = c("A","C","B"), V2 = c("C","B","A"))
n1 <- build_network(s1, method = "relative")
n2 <- build_network(s2, method = "relative")
bayes_compare(n1, n2, draws = 500, seed = 1)
#> Bayesian Dirichlet-Multinomial Comparison: Transition Network (relative probabilities) 
#>   Prior: Dirichlet(0.50)  |  Draws: 500  |  CI: 95%
#>   Thresholds: |mean diff| > 0.010, nearest CI bound > 0.001
#>   Nodes: 3  |  Edges compared: 6  |  Credibly different: 0