Analytic Certainty of Transition Edges (Dirichlet-Multinomial)

Closed-form Bayesian alternative to bootstrap_lsa() for the transition-probability edges of an lsa fit. Each state's outgoing transitions are modelled as Dirichlet-Multinomial: with a Jeffreys prior the posterior for a row is Dirichlet(count + prior), so each edge probability is marginally Beta(a, b) and its posterior mean, standard deviation, credible interval and stability decision are available analytically. No resampling, so it runs in microseconds.

Usage

certainty_lsa(
  fit,
  prior = 0.5,
  level_alpha = 0.95,
  inference = c("stability", "threshold"),
  consistency_range = c(0.75, 1.25),
  edge_threshold = NULL
)

Arguments

fit: An lsa fit from lsa(), or an lsa_group.
prior: Numeric > 0. Dirichlet prior concentration added to every cell. Default 0.5 (the Jeffreys prior).
level_alpha: Numeric in (0, 1). Credible-interval level. Default 0.95 (a 95% interval), matching bootstrap_lsa().
inference: "stability" (default) flags an edge whose posterior keeps it within a multiplicative consistency_range of its observed probability; "threshold" flags an edge whose posterior mass lies above edge_threshold.
consistency_range: Length-2 multiplicative bounds for stability inference. Default c(0.75, 1.25).
edge_threshold: Numeric or NULL. Fixed threshold for inference = "threshold"; NULL uses the 0.10 quantile of non-zero edge probabilities.

Value

An object of class c("lsa_certainty", "lsa_bootstrap", "list") with an edges data frame (from, to, prob_observed, prob_mean, prob_se, prob_ci_low, prob_ci_high, p_value, stable, plus adj_res_observed/adj_res_stable for plotting), the posterior matrices (mean, sd, ci_lower, ci_upper), and call metadata (prior, level_alpha, inference, ...). For an lsa_group, a named list of these (class lsa_certainty_group).

Details

The result carries class c("lsa_certainty", "lsa_bootstrap") and an edges table with the columns plot_forest() and as.data.frame() expect, so it is a drop-in for a bootstrap result (use metric = "prob").

Certainty vs bootstrap. Both answer "how precisely is this edge pinned down?". They agree on homogeneous data. The Dirichlet posterior treats transitions as independent, so on strongly heterogeneous data (a mixture of latent classes with long sequences) it reports more certainty than the sequence bootstrap – prefer bootstrap_lsa() then.

References

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

Examples

# \donttest{
fit  <- lsa(engagement)
cert <- certainty_lsa(fit)
cert
#> <lsa_certainty>  (analytic Dirichlet-Multinomial)
#>   engine:        classical
#>   prior:         Dirichlet(0.50)
#>   CI level:      95%  |  inference: stability
#>   certain edges: 7 of 9
head(as.data.frame(cert))
#>         from      to observed prob_observed prob_mean    prob_se prob_ci_low
#> 1     Active  Active      459     0.6975684 0.6967400 0.01788572  0.66113410
#> 2    Average  Active      153     0.2037284 0.2039867 0.01467978  0.17597523
#> 3 Disengaged  Active       39     0.1200000 0.1209801 0.01801983  0.08793233
#> 4     Active Average      176     0.2674772 0.2676270 0.01722641  0.23454702
#> 5    Average Average      458     0.6098535 0.6093023 0.01777441  0.57420054
#> 6 Disengaged Average      129     0.3969231 0.3966309 0.02703206  0.34428578
#>   prob_ci_high      p_value stable adj_res_observed adj_res_stable
#> 1    0.7312156 5.560385e-20   TRUE        21.662811           TRUE
#> 2    0.2334883 6.063196e-04   TRUE       -12.906038           TRUE
#> 3    0.1584213 9.448149e-02  FALSE       -10.549486          FALSE
#> 4    0.3020418 1.205186e-04   TRUE       -11.319132           TRUE
#> 5    0.6438538 3.141020e-17   TRUE        12.452615           TRUE
#> 6    0.4501757 2.232689e-04   TRUE        -1.736461           TRUE
# }