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Summary Statistics for rho Distribution

summary_rho.Rd

Computes comprehensive summary statistics for the co-clustering probability rho under the hierarchical prior alpha ~ Gamma(a, b).

Usage

summary_rho(
  a,
  b,
  M = .QUAD_NODES_DEFAULT,
  M_verify = NULL,
  abs_tol = 1e-10,
  rel_tol = 1e-08,
  strict = FALSE
)

Arguments

a

Numeric; shape parameter of the Gamma prior on alpha (a > 0).

b

Numeric; rate parameter of the Gamma prior on alpha (b > 0).

M

Integer; selected quadrature order. Default is 80.

M_verify

Optional independent verification order. When omitted and available, the package-wide minimum refinement order is used.

abs_tol, rel_tol

Non-negative selected-versus-verification tolerances.

strict

Logical; if TRUE, require verified rho moments.

Value

A list of class "rho_summary" containing:

mean

E(rho | a, b)

var

Var(rho | a, b)

sd

SD(rho | a, b) = sqrt(Var)

cv

Coefficient of variation SD/mean

status, usable, verified, message

Numerical verification state

estimand, label, units

Explicit estimand metadata

params

List of input parameters (a, b)

alpha_prior

Summary of the alpha prior (mean, sd, cv)

conditional_at_alpha_mean

Conditional moments evaluated at E(alpha)

verification, provenance

Quadrature comparison and method metadata

Details

The co-clustering probability rho indicates how likely two randomly chosen observations are to belong to the same cluster a priori.

The conditional_at_alpha_mean component provides a "plug-in" estimate for comparison: what the moments would be if alpha were fixed at its prior mean. The function reports numerical values and verification status only; scientific action thresholds must be supplied explicitly by an analysis policy rather than inferred from uncalibrated labels.

See also

mean_rho, var_rho, summary_w1

Examples

summary_rho(a = 2, b = 1)
#> Co-Clustering Probability (rho) Summary
#> ================================================== 
#> 
#> Status:      CONVERGED
#> Verified:    yes
#> Estimand:    rho (Conditional pairwise co-clustering probability)
#> Method:      gauss-laguerre-marginal-moments
#> 
#> Gamma prior: alpha ~ Gamma(2.0000, 1.0000)
#> E[alpha] = 2.0000, SD(alpha) = 1.4142, CV(alpha) = 70.7%
#> 
#> Marginal distribution of rho:
#> ----------------------------------- 
#>   Mean:   0.4037
#>   SD:     0.2397
#>   CV:     59.4%
#> 
#> Conditional at E[alpha] = 2.0000 (plug-in):
#> ----------------------------------- 
#>   E[rho | E[alpha]]:   0.3333
#>   Var(rho | E[alpha]): 0.0222
summary_rho(a = 1.6, b = 1.22)
#> Co-Clustering Probability (rho) Summary
#> ================================================== 
#> 
#> Status:      CONVERGED
#> Verified:    yes
#> Estimand:    rho (Conditional pairwise co-clustering probability)
#> Method:      gauss-laguerre-marginal-moments
#> 
#> Gamma prior: alpha ~ Gamma(1.6000, 1.2200)
#> E[alpha] = 1.3115, SD(alpha) = 1.0368, CV(alpha) = 79.1%
#> 
#> Marginal distribution of rho:
#> ----------------------------------- 
#>   Mean:   0.5084
#>   SD:     0.2664
#>   CV:     52.4%
#> 
#> Conditional at E[alpha] = 1.3115 (plug-in):
#> ----------------------------------- 
#>   E[rho | E[alpha]]:   0.4326
#>   Var(rho | E[alpha]): 0.0344