Bayesian Transition Entropy

Bayesian estimation of the transition entropy quantities of transition_entropy and the edge-level decomposition of entropy_network. Each row of the transition matrix gets an independent Dirichlet posterior (counts + prior); Monte Carlo draws propagate count uncertainty into the entropy rate, the per-state branching entropies, and every edge's entropy contribution, yielding posterior means and credible intervals.

The practical purpose is to exclude unstable estimates: an edge whose contribution rests on a handful of observations has a wide posterior, and is flagged non-credible unless it credibly accounts for at least min_share of the process entropy. $model is the entropy network with non-credible edges zeroed - the stable entropy skeleton.

Usage

entropy_bayes(
  x,
  prior = 0.5,
  draws = 4000,
  ci = 0.95,
  min_share = 0.01,
  base = 2,
  seed = NULL
)

# S3 method for class 'net_entropy_bayes'
print(x, digits = 3, ...)

# S3 method for class 'net_entropy_bayes_group'
print(x, ...)

# S3 method for class 'net_entropy_bayes'
summary(object, ...)

# S3 method for class 'net_entropy_bayes'
plot(x, top = 25, title = "Bayesian edge entropy contributions", ...)

Arguments

x: A frequency netobject (build_network(method = "frequency")), any netobject that carries its $data (counts are rebuilt automatically), a count matrix, or a wide sequence data.frame. Group dispatch on netobject_group. For the print() and plot() methods: an object of class net_entropy_bayes or net_entropy_bayes_group.
prior: Numeric. Dirichlet prior concentration added to every cell of the count matrix (default 0.5, Jeffreys).
draws: Integer. Number of Monte Carlo posterior draws (default 4000).
ci: Numeric in (0, 1). Credible interval mass (default 0.95).
min_share: Numeric in [0, 1). An edge is credible when the lower bound of the credible interval of its share of the entropy rate exceeds this value (default 0.01: the edge credibly accounts for at least 1% of process entropy).
base: Numeric. Logarithm base (default 2, bits).
seed: Integer or NULL. RNG seed for reproducibility.
digits: Integer. Digits to round numeric output. Default 3.
...: In plot.net_entropy_bayes(), print.net_entropy_bayes(), print.net_entropy_bayes_group() and summary.net_entropy_bayes(): Ignored.
object: For the summary() method: an object of class net_entropy_bayes.
top: Integer. Show at most this many edges, by posterior mean contribution (default 25).
title: Character. Plot title.

Value

An object of class "net_entropy_bayes" with:

summary: Tidy data.frame - one row per chain-level quantity (entropy_rate, stationary_entropy, redundancy), with posterior mean, sd, ci_lower, ci_upper.
states: Tidy data.frame - one row per state: posterior mean/CI of the row entropy and of the stationary probability.
edges: Tidy data.frame - one row per observed transition: posterior mean/sd/CI of the contribution (bits), posterior mean/CI of its share of the entropy rate, and the credible flag.
network: netobject - posterior-mean entropy network (all edges), carrying the entropy house style.
model: netobject - the pruned entropy network: posterior means where credible, 0 elsewhere.
draws_entropy_rate: Numeric vector of posterior entropy-rate draws (for further analysis or plotting).
prior, draws, ci, min_share, base, states_names: Call metadata.

For a netobject_group the result is a "net_entropy_bayes_group": a named list holding one such object per group.

In print.net_entropy_bayes() and print.net_entropy_bayes_group(): x invisibly.

In summary.net_entropy_bayes(): The tidy edge table (data.frame), one row per observed transition, sorted by posterior mean contribution, returned invisibly. The chain-level and per-edge tables are printed as a side effect.

In plot.net_entropy_bayes(): A ggplot object.

Details

With prior > 0 the posterior puts mass on every transition, so contribution draws are strictly positive and a naive "CI excludes zero" rule would flag every edge as credible. The share criterion is used instead: an edge is stable when it credibly carries at least min_share of $h(P)$. Unobserved transitions (count 0) get only prior mass and are never credible under any sensible min_share.

The posterior mean entropy rate is typically slightly below the plug-in estimate on sparse data (Dirichlet smoothing pulls rows toward uniform but averages over uncertainty); the difference vanishes as counts grow.

Methods

plot.net_entropy_bayes(): Forest plot of the per-edge entropy contributions: posterior mean and credible interval, credible edges in Okabe-Ito blue, unstable (non-credible) edges in grey. The dashed line marks min_share of the posterior-mean entropy rate - the stability criterion.

References

Cover, T.M. & Thomas, J.A. (2006). Elements of Information Theory, 2nd ed. Wiley.

Examples

net <- build_network(group_regulation_long, method = "relative",
                     actor = "Actor", action = "Action", time = "Time")
eb <- entropy_bayes(net, draws = 1000, seed = 1)
eb
#> Bayesian Transition Entropy (9 states, bits; Dirichlet prior 0.5, 1000 draws)
#> 
#>   entropy_rate:       2.411 [2.396, 2.425]
#>    stationary_entropy: 2.783 [2.771, 2.796]
#>    redundancy:         0.373 [0.361, 0.384]
#> 
#> Edges: 78 observed; 31 credible (share of h(P) credibly > 1%).
#> Use summary() for the edge table, plot() for the posterior, and
#> $model for the pruned stable entropy network.
summary(eb)
#> Chain-level posterior:
#>            quantity   mean     sd ci_lower ci_upper
#>        entropy_rate 2.4109 0.0074   2.3963   2.4250
#>  stationary_entropy 2.7834 0.0066   2.7710   2.7957
#>          redundancy 0.3726 0.0060   0.3612   0.3845
#> 
#> Per-edge posterior (sorted by contribution):
#>        from         to count contribution     sd ci_lower ci_upper  share
#>   consensus       plan  2505       0.1320 0.0011   0.1297   0.1341 0.0547
#>        plan       plan  2304       0.1298 0.0017   0.1268   0.1330 0.0538
#>        plan  consensus  1788       0.1267 0.0017   0.1236   0.1300 0.0526
#>   consensus    discuss  1190       0.1130 0.0015   0.1099   0.1158 0.0469
#>   consensus coregulate  1188       0.1130 0.0014   0.1104   0.1156 0.0469
#>        plan    emotion   904       0.0993 0.0017   0.0959   0.1027 0.0412
#>     discuss  consensus  1269       0.0798 0.0012   0.0774   0.0821 0.0331
#>   consensus  consensus   519       0.0738 0.0021   0.0695   0.0778 0.0306
#>     discuss    discuss   770       0.0697 0.0016   0.0663   0.0727 0.0289
#>        plan    monitor   465       0.0689 0.0019   0.0652   0.0727 0.0286
#>   consensus    emotion   460       0.0685 0.0020   0.0645   0.0723 0.0284
#>        plan    discuss   418       0.0645 0.0019   0.0607   0.0682 0.0267
#>     discuss  synthesis   557       0.0604 0.0014   0.0578   0.0633 0.0251
#>     emotion   cohesion   923       0.0575 0.0010   0.0554   0.0594 0.0238
#>     emotion  consensus   909       0.0574 0.0010   0.0554   0.0594 0.0238
#>     discuss    emotion   418       0.0520 0.0015   0.0491   0.0547 0.0216
#>   consensus    monitor   295       0.0515 0.0020   0.0476   0.0554 0.0214
#>     discuss coregulate   333       0.0456 0.0015   0.0424   0.0485 0.0189
#>  coregulate    discuss   539       0.0419 0.0009   0.0401   0.0437 0.0174
#>     discuss      adapt   282       0.0412 0.0016   0.0383   0.0443 0.0171
#>  coregulate       plan   471       0.0404 0.0009   0.0386   0.0422 0.0168
#>     emotion    discuss   289       0.0366 0.0013   0.0341   0.0391 0.0152
#>     emotion       plan   283       0.0362 0.0013   0.0337   0.0387 0.0150
#>  coregulate    emotion   339       0.0358 0.0011   0.0337   0.0378 0.0149
#>    cohesion  consensus   844       0.0334 0.0008   0.0318   0.0351 0.0139
#>        plan   cohesion   155       0.0328 0.0019   0.0292   0.0367 0.0136
#>  coregulate  consensus   265       0.0319 0.0011   0.0298   0.0342 0.0132
#>     discuss   cohesion   188       0.0317 0.0016   0.0284   0.0349 0.0132
#>     emotion    emotion   218       0.0310 0.0015   0.0282   0.0338 0.0129
#>    cohesion       plan   239       0.0265 0.0009   0.0247   0.0283 0.0110
#>     monitor    discuss   538       0.0259 0.0007   0.0245   0.0272 0.0107
#>  coregulate    monitor   170       0.0250 0.0012   0.0227   0.0274 0.0104
#>        plan coregulate   106       0.0248 0.0018   0.0212   0.0285 0.0103
#>    cohesion coregulate   202       0.0244 0.0010   0.0224   0.0263 0.0101
#>    cohesion    emotion   196       0.0240 0.0010   0.0218   0.0258 0.0099
#>     monitor       plan   309       0.0233 0.0007   0.0219   0.0247 0.0097
#>   consensus   cohesion    94       0.0225 0.0018   0.0191   0.0260 0.0093
#>     monitor  consensus   228       0.0206 0.0008   0.0190   0.0221 0.0085
#>     emotion    monitor   103       0.0190 0.0013   0.0163   0.0216 0.0079
#>     discuss    monitor    88       0.0186 0.0015   0.0159   0.0215 0.0077
#>     emotion coregulate    97       0.0182 0.0013   0.0157   0.0205 0.0075
#>    cohesion    discuss   101       0.0162 0.0011   0.0142   0.0184 0.0067
#>     monitor    emotion   130       0.0153 0.0008   0.0136   0.0169 0.0063
#>  coregulate   cohesion    71       0.0143 0.0012   0.0120   0.0167 0.0059
#>   synthesis  consensus   304       0.0137 0.0005   0.0127   0.0148 0.0057
#>   consensus  synthesis    48       0.0134 0.0015   0.0105   0.0164 0.0055
#>   synthesis      adapt   153       0.0131 0.0006   0.0120   0.0142 0.0054
#>     monitor coregulate    83       0.0117 0.0009   0.0101   0.0133 0.0048
#>     monitor   cohesion    80       0.0113 0.0009   0.0097   0.0131 0.0047
#>     discuss       plan    46       0.0113 0.0013   0.0090   0.0139 0.0047
#>    cohesion    monitor    56       0.0109 0.0010   0.0089   0.0130 0.0045
#>       adapt   cohesion   139       0.0107 0.0005   0.0098   0.0116 0.0044
#>       adapt  consensus   243       0.0107 0.0005   0.0098   0.0116 0.0044
#>  coregulate coregulate    46       0.0104 0.0012   0.0082   0.0129 0.0043
#>    cohesion   cohesion    46       0.0094 0.0010   0.0075   0.0113 0.0039
#>   consensus      adapt    30       0.0092 0.0014   0.0068   0.0120 0.0038
#>  coregulate  synthesis    37       0.0089 0.0011   0.0070   0.0111 0.0037
#>  coregulate      adapt    32       0.0080 0.0011   0.0060   0.0101 0.0033
#>       adapt    emotion    61       0.0077 0.0006   0.0065   0.0088 0.0032
#>   synthesis       plan    49       0.0075 0.0007   0.0063   0.0088 0.0031
#>   synthesis    emotion    46       0.0072 0.0007   0.0058   0.0085 0.0030
#>   synthesis    discuss    41       0.0067 0.0007   0.0055   0.0083 0.0028
#>   synthesis coregulate    29       0.0053 0.0007   0.0041   0.0067 0.0022
#>     monitor    monitor    26       0.0052 0.0007   0.0038   0.0067 0.0021
#>       adapt    discuss    30       0.0050 0.0006   0.0039   0.0062 0.0021
#>     monitor  synthesis    23       0.0047 0.0007   0.0033   0.0062 0.0020
#>   synthesis   cohesion    22       0.0045 0.0007   0.0033   0.0058 0.0018
#>        plan  synthesis    11       0.0041 0.0010   0.0023   0.0063 0.0017
#>     monitor      adapt    16       0.0036 0.0007   0.0024   0.0050 0.0015
#>       adapt    monitor    17       0.0035 0.0006   0.0024   0.0046 0.0014
#>     emotion  synthesis     8       0.0027 0.0007   0.0014   0.0042 0.0011
#>       adapt coregulate    11       0.0025 0.0006   0.0015   0.0037 0.0011
#>        plan      adapt     6       0.0025 0.0008   0.0011   0.0043 0.0010
#>     emotion      adapt     7       0.0024 0.0007   0.0012   0.0040 0.0010
#>   synthesis    monitor     8       0.0022 0.0006   0.0011   0.0035 0.0009
#>    cohesion  synthesis     6       0.0020 0.0007   0.0009   0.0034 0.0008
#>       adapt       plan     8       0.0020 0.0005   0.0010   0.0032 0.0008
#>    cohesion      adapt     5       0.0018 0.0006   0.0007   0.0031 0.0007
#>  share_lower share_upper credible
#>       0.0537      0.0557     TRUE
#>       0.0523      0.0554     TRUE
#>       0.0511      0.0541     TRUE
#>       0.0456      0.0481     TRUE
#>       0.0457      0.0481     TRUE
#>       0.0397      0.0427     TRUE
#>       0.0321      0.0340     TRUE
#>       0.0288      0.0323     TRUE
#>       0.0275      0.0301     TRUE
#>       0.0270      0.0302     TRUE
#>       0.0268      0.0300     TRUE
#>       0.0252      0.0283     TRUE
#>       0.0240      0.0262     TRUE
#>       0.0230      0.0246     TRUE
#>       0.0230      0.0246     TRUE
#>       0.0204      0.0227     TRUE
#>       0.0197      0.0230     TRUE
#>       0.0176      0.0201     TRUE
#>       0.0166      0.0181     TRUE
#>       0.0159      0.0184     TRUE
#>       0.0160      0.0175     TRUE
#>       0.0141      0.0162     TRUE
#>       0.0140      0.0161     TRUE
#>       0.0140      0.0157     TRUE
#>       0.0132      0.0145     TRUE
#>       0.0121      0.0152     TRUE
#>       0.0124      0.0142     TRUE
#>       0.0118      0.0145     TRUE
#>       0.0117      0.0140     TRUE
#>       0.0102      0.0117     TRUE
#>       0.0102      0.0112     TRUE
#>       0.0094      0.0114    FALSE
#>       0.0088      0.0118    FALSE
#>       0.0093      0.0109    FALSE
#>       0.0091      0.0107    FALSE
#>       0.0091      0.0102    FALSE
#>       0.0079      0.0108    FALSE
#>       0.0079      0.0092    FALSE
#>       0.0068      0.0090    FALSE
#>       0.0066      0.0089    FALSE
#>       0.0065      0.0085    FALSE
#>       0.0059      0.0076    FALSE
#>       0.0057      0.0070    FALSE
#>       0.0050      0.0069    FALSE
#>       0.0053      0.0061    FALSE
#>       0.0044      0.0068    FALSE
#>       0.0050      0.0059    FALSE
#>       0.0042      0.0055    FALSE
#>       0.0040      0.0054    FALSE
#>       0.0037      0.0058    FALSE
#>       0.0037      0.0054    FALSE
#>       0.0040      0.0048    FALSE
#>       0.0041      0.0048    FALSE
#>       0.0034      0.0053    FALSE
#>       0.0031      0.0047    FALSE
#>       0.0028      0.0050    FALSE
#>       0.0029      0.0046    FALSE
#>       0.0025      0.0042    FALSE
#>       0.0027      0.0036    FALSE
#>       0.0026      0.0036    FALSE
#>       0.0024      0.0035    FALSE
#>       0.0023      0.0034    FALSE
#>       0.0017      0.0028    FALSE
#>       0.0016      0.0028    FALSE
#>       0.0016      0.0026    FALSE
#>       0.0014      0.0026    FALSE
#>       0.0014      0.0024    FALSE
#>       0.0009      0.0026    FALSE
#>       0.0010      0.0021    FALSE
#>       0.0010      0.0019    FALSE
#>       0.0006      0.0018    FALSE
#>       0.0006      0.0015    FALSE
#>       0.0004      0.0018    FALSE
#>       0.0005      0.0016    FALSE
#>       0.0005      0.0015    FALSE
#>       0.0004      0.0014    FALSE
#>       0.0004      0.0013    FALSE
#>       0.0003      0.0013    FALSE
plot(eb)