```
library(repr)
options(repr.plot.width=12, repr.plot.height=10)
```

# AP_Entropy

```
library(tidyverse)
library(gridExtra)
library(patchwork)
library(stringr)
library(reshape2)
library(rstan)
library(bridgesampling)
library(loo)
library(ape)
library(bayesplot)
library(shinystan)
library(coda)
options(mc.cores = parallel::detectCores())
rstan_options("auto_write" = TRUE)
```

```
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.1.4 ✔ readr 2.1.5
✔ forcats 1.0.0 ✔ stringr 1.5.1
✔ ggplot2 3.5.0 ✔ tibble 3.2.1
✔ lubridate 1.9.3 ✔ tidyr 1.3.1
✔ purrr 1.0.2
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
Attaching package: ‘gridExtra’
The following object is masked from ‘package:dplyr’:
combine
Attaching package: ‘reshape2’
The following object is masked from ‘package:tidyr’:
smiths
Loading required package: StanHeaders
rstan version 2.32.6 (Stan version 2.32.2)
For execution on a local, multicore CPU with excess RAM we recommend calling
options(mc.cores = parallel::detectCores()).
To avoid recompilation of unchanged Stan programs, we recommend calling
rstan_options(auto_write = TRUE)
For within-chain threading using `reduce_sum()` or `map_rect()` Stan functions,
change `threads_per_chain` option:
rstan_options(threads_per_chain = 1)
Attaching package: ‘rstan’
The following object is masked from ‘package:tidyr’:
extract
This is loo version 2.6.0
- Online documentation and vignettes at mc-stan.org/loo
- As of v2.0.0 loo defaults to 1 core but we recommend using as many as possible. Use the 'cores' argument or set options(mc.cores = NUM_CORES) for an entire session.
Attaching package: ‘loo’
The following object is masked from ‘package:rstan’:
loo
Attaching package: ‘ape’
The following object is masked from ‘package:dplyr’:
where
This is bayesplot version 1.11.1
- Online documentation and vignettes at mc-stan.org/bayesplot
- bayesplot theme set to bayesplot::theme_default()
* Does _not_ affect other ggplot2 plots
* See ?bayesplot_theme_set for details on theme setting
Loading required package: shiny
This is shinystan version 2.6.0
Attaching package: ‘coda’
The following object is masked from ‘package:rstan’:
traceplot
```

```
<- rgb(206/255, 143/255, 137/255)
karminrot <- rgb(217/255, 205/255, 177/255)
gold <- rgb(143/255, 143/255, 149/255) anthrazit
```

Next, the data are loaded into a tibble called `d`

.

```
<- read_csv("../data/data_with_counts.csv")
d $Glottocode[d$Glottocode == "kose1239"] <- "awiy1238"
d<- d$Glottocode
taxa head(d)
```

```
Rows: 827 Columns: 13
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (5): ISO, Glottocode, Language, Family, Area
dbl (8): N_Speakers, Case, AP_Entropy, VerbFinal, VerbMiddle, v1, vm, vf
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
```

ISO | Glottocode | Language | Family | Area | N_Speakers | Case | AP_Entropy | VerbFinal | VerbMiddle | v1 | vm | vf |
---|---|---|---|---|---|---|---|---|---|---|---|---|

<chr> | <chr> | <chr> | <chr> | <chr> | <dbl> | <dbl> | <dbl> | <dbl> | <dbl> | <dbl> | <dbl> | <dbl> |

aai | arif1239 | Arifama-Miniafia | Austronesian | Oceania | 3470 | 0 | 0.7510324 | 0.8172043 | 0.15053763 | 3 | 14 | 76 |

aak | anka1246 | Ankave | Angan | Oceania | 1500 | NA | 0.9549905 | 0.6121212 | 0.30303030 | 14 | 50 | 101 |

aau | abau1245 | Abau | Sepik | Oceania | 7500 | 1 | 0.6152539 | 0.9565217 | 0.02173913 | 1 | 1 | 44 |

abt | ambu1247 | Ambulas | Ndu | Oceania | 33000 | 1 | 0.8478617 | 0.6666667 | 0.29411765 | 2 | 15 | 34 |

aby | anem1248 | Aneme Wake | Yareban | Oceania | 650 | 1 | 0.8426579 | 0.6562500 | 0.22916667 | 11 | 22 | 63 |

acd | giky1238 | Gikyode | Atlantic-Congo | Africa | 10400 | 0 | 0.5916728 | 0.1904762 | 0.73809524 | 3 | 31 | 8 |

In the next step, the *maximum clade credibility trees* are loaded as `phylo`

objects.

```
<- list.files("../data/mcc_trees", pattern = ".tre", full.names = TRUE)
mcc_trees
<- purrr::map(mcc_trees, read.tree)
trees
# get a vector of tip labels of all trees in trees
<- purrr::map(trees, function(x) x$tip.label) %>% unlist %>% unique tips
```

A brief sanity check to ensure that all tips of the trees correspond to a Glottocode in `d`

.

```
if (length(setdiff(tips, taxa)) == 0 ) {
print("All tips in trees are in taxa")
else {
} print("Not all tips in trees are in taxa")
}
```

`[1] "All tips in trees are in taxa"`

All Glottocodes in `d`

that do not occur as a tip are *isolates* (for the purpose of this study).

For each isolate I create a `list`

object with the fields `"tip.lable"`

and `"Nnode"`

, to make them compatible with the `phylo`

objects for language families. These list objects are added to the list `trees`

.

```
<- setdiff(taxa, tips)
isolates
# convert isolates to one-node trees and add to trees
for (i in isolates) {
<- list()
tr "tip.label"]] <- i
tr[["Nnode"]] <- 0
tr[[length(trees) + 1]] <- tr
trees[[ }
```

Just to make sure, I test whether the tips of this forest are exactly the Glottocodes in the data.

```
<- c()
tips for (tr in trees) {
<- c(tips, tr$tip.label)
tips
}
if (all(sort(tips) == sort(taxa))) {
print("taxa and tips are identical")
else {
} print("taxa and tips are not identical")
}
```

`[1] "taxa and tips are identical"`

Now I re-order the rows in `d`

to make sure they are in the same order as `tips`

.

`<- d[match(tips, d$Glottocode), ] d `

The trees in `trees`

form a forest, i.e., a disconnected directed acyclic graph. I want to construct a representation of this graph which I can pass to Stan, consisting of

- the number of nodes
`N`

, - a Nx2-matrix
`edges`

, and - a length-N vector of edge lengths,
`edge_lengths`

. - a vector of root nodes
- a vector of tip nodes

First I create an Nx4 matrix where each row represents one node. The columns are

- node number
- tree number
- tree-internal node number
- label

For internal nodes, the label is the empty string.

```
<- c()
node_matrix <- 1
node_number for (tn in 1:length(trees)) {
<- trees[[tn]]
tr <- length(tr$tip.label) + tr$Nnode
tr_nodes for (i in 1:tr_nodes) {
<- ""
tl if (i <= length(tr$tip.label)) {
<- tr$tip.label[i]
tl
}<- rbind(node_matrix, c(node_number, tn, i, tl))
node_matrix <- node_number+1
node_number
} }
```

I convert the matrix to a tibble.

```
<- tibble(
nodes node_number = as.numeric(node_matrix[,1]),
tree_number = as.numeric(node_matrix[,2]),
internal_number = as.numeric(node_matrix[,3]),
label = node_matrix[,4]
)head(nodes)
tail(nodes)
```

node_number | tree_number | internal_number | label |
---|---|---|---|

<dbl> | <dbl> | <dbl> | <chr> |

1 | 1 | 1 | aman1265 |

2 | 1 | 2 | wari1266 |

3 | 1 | 3 | |

4 | 2 | 1 | bora1263 |

5 | 2 | 2 | muin1242 |

6 | 2 | 3 |

node_number | tree_number | internal_number | label |
---|---|---|---|

<dbl> | <dbl> | <dbl> | <chr> |

1549 | 95 | 1 | uduk1239 |

1550 | 96 | 1 | wiru1244 |

1551 | 97 | 1 | kamu1260 |

1552 | 98 | 1 | yagu1244 |

1553 | 99 | 1 | kark1258 |

1554 | 100 | 1 | nucl1454 |

Next I define a function that maps a tree number and an internal node number to the global node number.

`<- function(tree_number, local_id) which((nodes$tree_number == tree_number) & (nodes$internal_number == local_id)) get_global_id `

Now I create a matrix of edges and, concomittantly, a vector of edge lengths.

```
<- c()
edges <- c()
edge_lengths for (tn in 1:length(trees)) {
<- trees[[tn]]
tr if (length(tr$edge) > 0 ) {
for (en in 1:nrow(tr$edge)) {
<- tr$edge[en, 1]
mother <- tr$edge[en, 2]
daughter <- get_global_id(tn, mother)
mother_id <- get_global_id(tn, daughter)
daughter_id <- tr$edge.length[en]
el <- rbind(edges, c(mother_id, daughter_id))
edges <- c(edge_lengths, el)
edge_lengths
}
} }
```

Finally I identify the root nodes and tip nodes.

```
<- c(sort(setdiff(edges[,1], edges[,2])), match(isolates, nodes$label))
roots <- which(nodes$label != "") tips
```

Next I define a function computing the post-order.

```
<- function(roots, edges) {
get_postorder <- roots
input <- c()
output while (length(input) > 0) {
<- input[1]
pivot <- setdiff(edges[edges[,1] == pivot, 2], output)
daughters if (length(daughters) == 0) {
<- input[-1]
input <- c(output, pivot)
output else {
} <- c(daughters, input)
input
}
}return(output)
}
```

```
<- c()
undefined <- c()
po for (i in get_postorder(roots, edges)) {
if ((i %in% edges[,2]) & !(i %in% undefined)) {
<- c(po, i)
po
}
}<- match(po, edges[,2]) edge_po
```

```
<- function(prior_pc) {
predictive_plot <- prior_pc %>%
p1 mutate(
x_jittered = 1 + rnorm(n(), mean = 0, sd = 0.01),
y_jittered = tv + rnorm(n(), mean = 0, sd = sd(prior_pc$tv)/10)
%>%
) ggplot(aes(x = x_jittered, y = tv)) +
geom_boxplot(aes(x = factor(1), y = tv), width = 0.5, fill=karminrot) +
geom_point(alpha = 0.5, size = 0.1, position = position_identity()) +
theme_grey(base_size = 16) +
theme(
axis.text.x = element_blank(),
axis.ticks.x = element_blank(),
axis.title.x = element_blank(),
axis.title.y = element_text(size = 18),
plot.title = element_text(size = 20, hjust = 0.5)
+
) labs(title = "total variance", y = "") +
geom_point(aes(x=1, y=empirical$tv, color="empirical"), size=5, fill='red') +
guides(color = "none")
<- prior_pc %>%
p2 mutate(
x_jittered = 1 + rnorm(n(), mean = 0, sd = 0.01),
y_jittered = lv + rnorm(n(), mean = 0, sd = sd(prior_pc$lv)/10)
%>%
) ggplot(aes(x = x_jittered, y = lv)) +
geom_boxplot(aes(x = factor(1), y = lv), width = 0.5, fill=gold) +
geom_point(alpha = 0.5, size = 0.1, position = position_identity()) +
theme_grey(base_size = 16) +
theme(
axis.text.x = element_blank(),
axis.ticks.x = element_blank(),
axis.title.x = element_blank(),
axis.title.y = element_text(size = 18),
plot.title = element_text(size = 20, hjust = 0.5)
+
) labs(title = "lineage-wise variance", y = "") +
geom_point(aes(x=1, y=empirical$lv, color="empirical"), size=5, fill='red') +
guides(color = "none")
<- prior_pc %>%
p3 mutate(
x_jittered = 1 + rnorm(n(), mean = 0, sd = 0.01),
y_jittered = cv + rnorm(n(), mean = 0, sd = sd(prior_pc$cv)/10)
%>%
) ggplot(aes(x = x_jittered, y = cv)) +
geom_boxplot(aes(x = factor(1), y = cv), width = 0.5, fill=anthrazit) +
geom_point(alpha = 0.5, size = 0.1, position = position_identity()) +
theme_grey(base_size = 16) +
theme(
axis.text.x = element_blank(),
axis.ticks.x = element_blank(),
axis.title.x = element_blank(),
axis.title.y = element_text(size = 18),
plot.title = element_text(size = 20, hjust = 0.5)
+
) labs(title = "cross-lineage variance", y = "") +
geom_point(aes(x=1, y=empirical$cv, color="empirical"), size=5, fill='red') +
labs(color = "") +
guides(color = "none")
return(grid.arrange(p1, p2, p3, ncol=3))
}
```

This is a single continuous trait, so I will use a continuous trait model.

The entropy of a binary variable has a theoretical lower and upper bound. Therefore it makes sense to model it as a beta-distributed variable.

In the baseline model, the trait values at the tips are drawn from the same beta distribution.

\[ \begin{aligned} y_i &\sim \text{Beta}(\lambda_i\phi, \lambda_i(1 - \phi))\\ \text{logistic}(\phi) &\sim \mathcal N(0,2)\\ \log \lambda_i &\sim \mathcal N(0,1)\\ \end{aligned} \]

`<- d$AP_Entropy Y `

```
<- list()
apentropy_data $Y <- Y
apentropy_data$Ndata <- length(Y) apentropy_data
```

```
<- "
pooling_code data {
int<lower=1> Ndata;
vector[Ndata] Y;
}
transformed data {
real epsilon = 1e-5;
vector<lower=0, upper=1>[Ndata] Y_adj;
for (n in 1:Ndata) {
if (Y[n] == 0) {
Y_adj[n] = epsilon;
} else if (Y[n] == 1) {
Y_adj[n] = 1 - epsilon;
} else {
Y_adj[n] = Y[n];
}
}
}
parameters {
real z;
real mu_lambda;
real<lower=0> sigma_lambda;
vector<lower=0>[Ndata] lambda;
}
model {
z ~ normal(0, 1);
mu_lambda ~ lognormal(0, 1);
sigma_lambda ~ exponential(10);
lambda ~ lognormal(mu_lambda, sigma_lambda);
for (i in 1:Ndata) {
real phi = inv_logit(z);
real a = lambda[i] * phi;
real b = lambda[i] * (1-phi);
Y_adj[i] ~ beta(a, b);
}
}
generated quantities {
real z_prior = normal_rng(0, 1);
real mu_lambda_prior = lognormal_rng(0, 1);
real sigma_lambda_prior = exponential_rng(1);
vector[Ndata] lambda_prior;
vector<lower=0,upper=1>[Ndata] Y_prior;
vector<lower=0,upper=1>[Ndata] Y_posterior;
vector[Ndata] log_lik;
for (i in 1:Ndata) {
lambda_prior[i] = lognormal_rng(mu_lambda_prior, sigma_lambda_prior);
}
for (i in 1:Ndata) {
real phi_prior = inv_logit(z_prior);
real a_prior = lambda_prior[i] * phi_prior;
real b_prior = lambda_prior[i] * (1-phi_prior);
real phi = inv_logit(z);
real a = lambda[i] * phi;
real b = lambda[i] * (1-phi);
Y_prior[i] = beta_rng(a_prior, b_prior);
Y_posterior[i] = beta_rng(a, b);
log_lik[i] = beta_lpdf(Y_adj[i] | a, b);
}
}
"
```

`<- stan_model(model_code = pooling_code) model_apentropy_pooling `

```
# fit_apentropy_pooling <- rstan::sampling(
# model_apentropy_pooling,
# data = apentropy_data,
# chains = 4,
# iter = 10000,
# thin = 1
# )
```

`# saveRDS(fit_apentropy_pooling, "models/fit_apentropy_pooling.rds")`

`<- readRDS("models/fit_apentropy_pooling.rds") fit_apentropy_pooling `

```
print(
fit_apentropy_pooling,pars = c("z", "mu_lambda", "sigma_lambda", "lp__")
)
```

```
Inference for Stan model: pooling_apentropy.
4 chains, each with iter=10000; warmup=5000; thin=1;
post-warmup draws per chain=5000, total post-warmup draws=20000.
mean se_mean sd 2.5% 25% 50% 75% 97.5%
z 0.67 0.00 0.03 0.61 0.65 0.67 0.69 0.73
mu_lambda 1.57 0.00 0.06 1.45 1.53 1.57 1.61 1.69
sigma_lambda 0.73 0.00 0.06 0.61 0.69 0.73 0.78 0.86
lp__ -577.29 1.93 61.09 -690.71 -618.56 -580.68 -537.98 -448.53
n_eff Rhat
z 10641 1.00
mu_lambda 2408 1.00
sigma_lambda 1068 1.01
lp__ 1002 1.01
Samples were drawn using NUTS(diag_e) at Thu Mar 7 07:56:16 2024.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
```

```
<- extract_log_lik(
log_lik
fit_apentropy_pooling,parameter_name = "log_lik",
merge_chains = FALSE
)<- relative_eff(exp(log_lik))
reff <- loo(log_lik, r_eff=reff)) (loo_apentropy_pooling
```

```
Warning message:
“Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.
”
```

```
Computed from 20000 by 827 log-likelihood matrix
Estimate SE
elpd_loo 252.3 15.2
p_loo 145.7 9.0
looic -504.6 30.5
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 602 72.8% 3437
(0.5, 0.7] (ok) 170 20.6% 294
(0.7, 1] (bad) 52 6.3% 29
(1, Inf) (very bad) 3 0.4% 5
See help('pareto-k-diagnostic') for details.
```

```
<- stan(model_code = pooling_code, data = apentropy_data, chains = 0)
new_mod <- bridge_sampler(
(marginal_apentropy_pooling
fit_apentropy_pooling,
new_mod,maxiter = 100000,
cores = 4
))
```

```
the number of chains is less than 1; sampling not done
```

```
Iteration: 1
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Iteration: 67
```

```
Bridge sampling estimate of the log marginal likelihood: 220.7127
Estimate obtained in 67 iteration(s) via method "normal".
```

### predictive checks

I will use the same three statistics, but computed on relative instead of absolute frequencies.

`<- function(Y) return(var(Y)) total_variance `

```
<- function(Y) {
lineage_wise_variance <- tibble(y = Y, family = d$Family) %>%
r group_by(family) %>%
summarize(v = var(y)) %>%
filter(complete.cases(.)) %>%
pull(v) %>%
mean()
return(r)
}
```

```
<- function(Y) {
crossfamily_variance <- tibble(y = Y, family = d$Family) %>%
r group_by(family) %>%
summarize(m = mean(y)) %>%
pull(m) %>% var
return(r)
}
```

```
<- tibble(
empirical tv = total_variance(Y),
lv = lineage_wise_variance(Y),
cv = crossfamily_variance(Y)
)
```

`<- extract(fit_apentropy_pooling) generated_quantities `

```
<- sample(1:dim(generated_quantities$Y_prior)[1], 1000)
spl <- tibble(
prior_pc tv = apply(generated_quantities$Y_prior[spl, ], 1, total_variance),
lv = apply(generated_quantities$Y_prior[spl, ], 1, lineage_wise_variance),
cv = apply(generated_quantities$Y_prior[spl, ], 1, crossfamily_variance)
)predictive_plot(prior_pc)
```

```
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
```

```
<- sample(1:dim(generated_quantities$Y_posterior)[1], 1000)
spl <- tibble(
prior_pc tv = apply(generated_quantities$Y_posterior[spl, ], 1, total_variance),
lv = apply(generated_quantities$Y_posterior[spl, ], 1, lineage_wise_variance),
cv = apply(generated_quantities$Y_posterior[spl, ], 1, crossfamily_variance)
)predictive_plot(prior_pc)
```

```
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
```

```
<- list()
apentropy_data $Y <- Y
apentropy_data$Ndata <- length(Y)
apentropy_data$Ntrees <- length(trees)
apentropy_data$Nnodes <- nrow(nodes)
apentropy_data$Nedges <- length(edge_po)
apentropy_data$observations <- match(d$Glottocode, nodes$label)
apentropy_data$roots <- roots
apentropy_data$edges <- edges[edge_po,]
apentropy_data$edge_lengths <- edge_lengths[edge_po] apentropy_data
```

```
<- "
brown_code data {
int<lower=1> Ndata;
vector[Ndata] Y;
int<lower=1> Ntrees;
int<lower=1> Nnodes;
int<lower=1> Nedges;
int<lower=1,upper=Nnodes> observations[Ndata];
int<lower=1, upper=Nnodes> roots[Ntrees];
int<lower=1, upper=Nnodes> edges[Nedges, 2];
vector[Nedges] edge_lengths;
}
transformed data {
real epsilon = 1e-5;
vector<lower=0, upper=1>[Ndata] Y_adj;
for (n in 1:Ndata) {
if (Y[n] == 0) {
Y_adj[n] = epsilon;
} else if (Y[n] == 1) {
Y_adj[n] = 1 - epsilon;
} else {
Y_adj[n] = Y[n];
}
}
}
parameters {
vector[Nnodes] z;
vector<lower=0>[Ndata] lambda;
real mu_root;
real<lower=0> sigma;
real<lower=0> sigma_root;
real mu_lambda;
real<lower=0> sigma_lambda;
}
model {
mu_root ~ normal(0, 1);
sigma ~ exponential(10);
sigma_root ~ exponential(1);
mu_lambda ~ normal(0,1);
sigma_lambda ~ exponential(10);
lambda ~ lognormal(mu_lambda, sigma_lambda);
for (i in 1:Ntrees) {
int idx = roots[i];
z[idx] ~ normal(mu_root, sigma_root);
}
for (i in 1:Nedges) {
int j = Nedges - i + 1;
int mother_node = edges[j, 1];
int daughter_node = edges[j, 2];
real local_mu = z[mother_node];
real local_sigma = sigma * sqrt(edge_lengths[j]);
z[daughter_node] ~ normal(local_mu, local_sigma);
}
for (i in 1:Ndata) {
real phi = inv_logit(z[observations[i]]);
real a = lambda[i] * phi;
real b = lambda[i] * (1 - phi);
Y_adj[i] ~ beta(a, b);
}
}
generated quantities {
real mu_root_prior = normal_rng(0, 1);
real sigma_prior = exponential_rng(1);
real sigma_root_prior = exponential_rng(10);
real mu_lambda_prior = normal_rng(0, 10);
real sigma_lambda_prior = exponential_rng(10);
vector[Ndata] lambda_prior;
vector[Nnodes] z_prior;
vector<lower=0,upper=1>[Ndata] Y_prior;
vector<lower=0,upper=1>[Ndata] Y_posterior;
vector[Ndata] log_lik;
for (i in 1:Ndata) {
lambda_prior[i] = lognormal_rng(mu_lambda_prior, sigma_lambda_prior);
}
for (i in 1:Ntrees) {
int idx = roots[i];
z_prior[idx] = normal_rng(mu_root_prior, sigma_root_prior);
}
for (i in 1:Nedges) {
int j = Nedges - i + 1;
int mother_node = edges[j, 1];
int daughter_node = edges[j, 2];
real local_mu_prior = z_prior[mother_node];
real local_sigma_prior = sigma_prior * sqrt(edge_lengths[j]);
z_prior[daughter_node] = normal_rng(local_mu_prior, local_sigma_prior);
}
for (i in 1:Ndata) {
real phi_prior = inv_logit(z_prior[observations[i]]);
real a_prior = lambda_prior[i] * phi_prior;
real b_prior = lambda_prior[i] * (1 - phi_prior);
real phi = inv_logit(z[observations[i]]);
real a = lambda[i] * phi;
real b = lambda[i] * (1 - phi);
Y_prior[i] = beta_rng(a_prior, b_prior);
Y_posterior[i] = beta_rng(a, b);
log_lik[i] = beta_lpdf(Y_adj[i] | a, b);
}
}
"
```

`<- stan_model(model_code = brown_code) model_apentropy_brown `

```
# fit_apentropy_brown <- rstan::sampling(
# model_apentropy_brown,
# data = apentropy_data,
# chains = 4,
# iter = 100000,
# thin = 1,
# control = list(adapt_delta = 0.99, max_treedepth = 15)
# )
# saveRDS(fit_apentropy_brown, "models/fit_apentropy_brown.rds")
```

`<- readRDS("models/fit_apentropy_brown.rds") fit_apentropy_brown `

```
print(
fit_apentropy_brown,pars = c("mu_root", "sigma", "sigma_root", "mu_lambda", "sigma_lambda", "lp__")
)
```

```
Inference for Stan model: brown_apentropy.
4 chains, each with iter=1e+05; warmup=50000; thin=1;
post-warmup draws per chain=50000, total post-warmup draws=2e+05.
mean se_mean sd 2.5% 25% 50% 75% 97.5%
mu_root 0.79 0.00 0.08 0.63 0.74 0.79 0.85 0.96
sigma 0.63 0.00 0.12 0.39 0.55 0.63 0.70 0.85
sigma_root 0.61 0.00 0.08 0.47 0.56 0.61 0.67 0.78
mu_lambda 2.19 0.00 0.11 1.99 2.12 2.19 2.26 2.41
sigma_lambda 0.85 0.00 0.09 0.69 0.79 0.85 0.91 1.03
lp__ 1786.38 8.84 289.71 1304.74 1588.19 1756.15 1951.15 2451.46
n_eff Rhat
mu_root 93512 1
sigma 1133 1
sigma_root 35128 1
mu_lambda 2466 1
sigma_lambda 5144 1
lp__ 1075 1
Samples were drawn using NUTS(diag_e) at Thu Mar 7 23:51:05 2024.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
```

```
<- extract_log_lik(
log_lik
fit_apentropy_brown,parameter_name = "log_lik",
merge_chains = FALSE
)
```

```
<- c()
good_indices for (i in 1:dim(log_lik)[1]) {
if (sum(is.na(log_lik[i,,])) == 0) {
<- c(good_indices, i)
good_indices
} }
```

`<- log_lik[good_indices,,] log_lik `

```
<- relative_eff(exp(log_lik))
reff <- loo(log_lik, r_eff=reff)) (loo_apentropy_brown
```

```
Warning message:
“Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.
”
```

```
Computed from 199980 by 827 log-likelihood matrix
Estimate SE
elpd_loo 401.3 21.2
p_loo 301.5 9.7
looic -802.5 42.3
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 189 22.9% 24488
(0.5, 0.7] (ok) 466 56.3% 573
(0.7, 1] (bad) 170 20.6% 12
(1, Inf) (very bad) 2 0.2% 4
See help('pareto-k-diagnostic') for details.
```

```
<- stan(model_code = brown_code, data = apentropy_data, chains = 0)
new_mod <- bridge_sampler(
(marginal_apentropy_brown
fit_apentropy_brown,
new_mod,cores = 4,
maxiter = 10000
))
```

```
the number of chains is less than 1; sampling not done
```

```
Iteration: 1
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Iteration: 19
```

```
Bridge sampling estimate of the log marginal likelihood: 1749.126
Estimate obtained in 19 iteration(s) via method "normal".
```

`<- extract(fit_apentropy_brown) generated_quantities `

```
<- sample(1:dim(generated_quantities$Y_prior)[1], 1000)
spl <- tibble(
prior_pc tv = apply(generated_quantities$Y_prior[spl, ], 1, total_variance),
lv = apply(generated_quantities$Y_prior[spl, ], 1, lineage_wise_variance),
cv = apply(generated_quantities$Y_prior[spl, ], 1, crossfamily_variance)
)predictive_plot(prior_pc)
```

```
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
```

```
<- sample(1:dim(generated_quantities$Y_posterior)[1], 1000)
spl <- tibble(
prior_pc tv = apply(generated_quantities$Y_posterior[spl, ], 1, total_variance),
lv = apply(generated_quantities$Y_posterior[spl, ], 1, lineage_wise_variance),
cv = apply(generated_quantities$Y_posterior[spl, ], 1, crossfamily_variance)
)predictive_plot(prior_pc)
```

```
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
```

## OU model

```
<- "
ou_code data {
int<lower=1> Ndata;
vector[Ndata] Y;
int<lower=1> Ntrees;
int<lower=1> Nnodes;
int<lower=1> Nedges;
int<lower=1,upper=Nnodes> observations[Ndata];
int<lower=1, upper=Nnodes> roots[Ntrees];
int<lower=1, upper=Nnodes> edges[Nedges, 2];
vector[Nedges] edge_lengths;
}
transformed data {
real epsilon = 1e-5;
vector<lower=0, upper=1>[Ndata] Y_adj;
for (n in 1:Ndata) {
if (Y[n] == 0) {
Y_adj[n] = epsilon;
} else if (Y[n] == 1) {
Y_adj[n] = 1 - epsilon;
} else {
Y_adj[n] = Y[n];
}
}
}
parameters {
vector[Nnodes] z;
vector<lower=0>[Ndata] rho;
real mu;
real<lower=0> sigma;
real<lower=0> lambda;
real mu_rho;
real<lower=0> sigma_rho;
}
model {
mu ~ normal(0, 10);
sigma ~ exponential(1);
lambda ~ cauchy(0, 2.5);
mu_rho ~ normal(0,1);
sigma_rho ~ exponential(10);
rho ~ lognormal(mu_rho, sigma_rho);
for (i in 1:Ntrees) {
int idx = roots[i];
z[idx] ~ normal(mu, sigma * sqrt(2 * lambda));
}
for (i in 1:Nedges) {
int j = Nedges - i + 1;
int mother_node = edges[j, 1];
int daughter_node = edges[j, 2];
real local_mu = mu + (z[mother_node] - mu) * exp(-lambda * edge_lengths[j]);
real local_sigma = sigma * sqrt((1 - exp(-2 * lambda * edge_lengths[j])) / (2 * lambda));
z[daughter_node] ~ normal(local_mu, local_sigma);
}
for (i in 1:Ndata) {
real phi = inv_logit(z[observations[i]]);
real a = rho[i] * phi;
real b = rho[i] * (1 - phi);
Y_adj[i] ~ beta(a, b);
}
}
generated quantities {
real mu_prior = normal_rng(0, 10);
real sigma_prior = exponential_rng(1);
real lambda_prior = fabs(cauchy_rng(0, 2.5));
real mu_rho_prior = normal_rng(0,1);
real sigma_rho_prior = exponential_rng(10);
vector[Ndata] rho_prior;
vector[Nnodes] z_prior;
vector<lower=0,upper=1>[Ndata] Y_prior;
vector<lower=0,upper=1>[Ndata] Y_posterior;
vector[Ndata] log_lik;
mu_prior = normal_rng(0, 10);
sigma_prior = exponential_rng(1);
lambda_prior = fabs(cauchy_rng(0, 2.5));
mu_rho_prior = normal_rng(0,1);
sigma_rho_prior = exponential_rng(10);
for (i in 1:Ndata) {
rho_prior[i] = lognormal_rng(mu_rho_prior, sigma_rho_prior);
}
for (i in 1:Ntrees) {
int idx = roots[i];
z_prior[idx] = normal_rng(mu_prior, sigma_prior * sqrt(2 * lambda_prior));
}
for (i in 1:Nedges) {
int j = Nedges - i + 1;
int mother_node = edges[j, 1];
int daughter_node = edges[j, 2];
real local_mu_prior = mu_prior + (z_prior[mother_node] - mu_prior) * exp(-lambda_prior * edge_lengths[j]);
real local_sigma_prior = sigma_prior * sqrt((1 - exp(-2 * lambda_prior * edge_lengths[j])) / (2 * lambda_prior));
z_prior[daughter_node] = normal_rng(local_mu_prior, local_sigma_prior);
}
for (i in 1:Ndata) {
real phi_prior = inv_logit(z_prior[observations[i]]);
real a_prior = rho_prior[i] * phi_prior;
real b_prior = rho_prior[i] * (1 - phi_prior);
real phi = inv_logit(z[observations[i]]);
real a = rho[i] * phi;
real b = rho[i] * (1 - phi);
Y_prior[i] = beta_rng(a_prior, b_prior);
Y_posterior[i] = beta_rng(a, b);
log_lik[i] = beta_lpdf(Y_adj[i] | a, b);
}
}
"
```

`<- stan_model(model_code = ou_code) model_apentropy_ou `

```
# fit_apentropy_ou <- rstan::sampling(
# model_apentropy_ou,
# data = apentropy_data,
# chains = 4,
# iter = 50000,
# thin = 1,
# control = list(adapt_delta = 0.99, max_treedepth = 15)
# )
# saveRDS(fit_apentropy_ou, "models/fit_apentropy_ou.rds")
```

`<- readRDS("models/fit_apentropy_ou.rds") fit_apentropy_ou `

```
print(
fit_apentropy_ou,pars = c("mu", "sigma", "lambda", "mu_rho", "sigma_rho", "lp__")
)
```

```
Inference for Stan model: ou_apentropy.
4 chains, each with iter=50000; warmup=25000; thin=1;
post-warmup draws per chain=25000, total post-warmup draws=1e+05.
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff
mu 0.79 0.00 0.09 0.62 0.73 0.79 0.85 0.96 57780
sigma 0.80 0.00 0.09 0.63 0.74 0.80 0.86 0.99 1394
lambda 0.43 0.00 0.15 0.20 0.32 0.41 0.51 0.77 3946
mu_rho 2.27 0.00 0.11 2.07 2.20 2.27 2.34 2.49 2650
sigma_rho 0.88 0.00 0.09 0.70 0.81 0.87 0.94 1.06 3351
lp__ 1448.49 4.92 181.84 1103.95 1324.23 1443.50 1568.29 1813.03 1366
Rhat
mu 1
sigma 1
lambda 1
mu_rho 1
sigma_rho 1
lp__ 1
Samples were drawn using NUTS(diag_e) at Sat Mar 9 08:22:31 2024.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).
```

```
<- extract_log_lik(
log_lik
fit_apentropy_ou,parameter_name = "log_lik",
merge_chains = FALSE
)
```

```
<- c()
good_indices for (i in 1:dim(log_lik)[1]) {
if (sum(is.na(log_lik[i,,])) == 0) {
<- c(good_indices, i)
good_indices
} }
```

`<- log_lik[good_indices,,] log_lik `

```
<- relative_eff(exp(log_lik))
reff <- loo(log_lik, r_eff=reff)) (loo_apentropy_ou
```

```
Warning message:
“Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.
”
```

```
Computed from 88304 by 827 log-likelihood matrix
Estimate SE
elpd_loo 412.6 21.4
p_loo 335.2 9.6
looic -825.1 42.7
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 146 17.7% 16889
(0.5, 0.7] (ok) 457 55.3% 768
(0.7, 1] (bad) 222 26.8% 32
(1, Inf) (very bad) 2 0.2% 6
See help('pareto-k-diagnostic') for details.
```

```
<- stan(model_code = ou_code, data = apentropy_data, chains = 0)
new_mod <- bridge_sampler(
(marginal_apentropy_ou
fit_apentropy_ou,
new_mod,cores = 4,
maxiter = 10000
))
```

```
the number of chains is less than 1; sampling not done
Warning message:
“logml could not be estimated within maxiter, rerunning with adjusted starting value.
Estimate might be more variable than usual.”
```

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Iteration: 10000
Iteration: 1
```

```
Bridge sampling estimate of the log marginal likelihood: 1757.244
Estimate obtained in 10001 iteration(s) via method "normal".
```

`<- extract(fit_apentropy_ou) generated_quantities `

```
<- sample(1:dim(generated_quantities$Y_prior)[1], 1000)
spl <- tibble(
prior_pc tv = apply(generated_quantities$Y_prior[spl, ], 1, total_variance),
lv = apply(generated_quantities$Y_prior[spl, ], 1, lineage_wise_variance),
cv = apply(generated_quantities$Y_prior[spl, ], 1, crossfamily_variance)
)predictive_plot(prior_pc)
```

```
Warning message:
“There was 1 warning in `mutate()`.
ℹ In argument: `y_jittered = tv + rnorm(n(), mean = 0, sd =
sd(prior_pc$tv)/10)`.
Caused by warning in `rnorm()`:
! NAs produced”
Warning message:
“There was 1 warning in `mutate()`.
ℹ In argument: `y_jittered = lv + rnorm(n(), mean = 0, sd =
sd(prior_pc$lv)/10)`.
Caused by warning in `rnorm()`:
! NAs produced”
Warning message:
“There was 1 warning in `mutate()`.
ℹ In argument: `y_jittered = cv + rnorm(n(), mean = 0, sd =
sd(prior_pc$cv)/10)`.
Caused by warning in `rnorm()`:
! NAs produced”
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message:
“Removed 30 rows containing non-finite outside the scale range
(`stat_boxplot()`).”
Warning message:
“Removed 30 rows containing missing values or values outside the scale range
(`geom_point()`).”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message:
“Removed 30 rows containing non-finite outside the scale range
(`stat_boxplot()`).”
Warning message:
“Removed 30 rows containing missing values or values outside the scale range
(`geom_point()`).”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message:
“Removed 30 rows containing non-finite outside the scale range
(`stat_boxplot()`).”
Warning message:
“Removed 30 rows containing missing values or values outside the scale range
(`geom_point()`).”
```

```
<- sample(1:dim(generated_quantities$Y_posterior)[1], 1000)
spl <- tibble(
prior_pc tv = apply(generated_quantities$Y_posterior[spl, ], 1, total_variance),
lv = apply(generated_quantities$Y_posterior[spl, ], 1, lineage_wise_variance),
cv = apply(generated_quantities$Y_posterior[spl, ], 1, crossfamily_variance)
)predictive_plot(prior_pc)
```

```
Warning message:
“There was 1 warning in `mutate()`.
ℹ In argument: `y_jittered = tv + rnorm(n(), mean = 0, sd =
sd(prior_pc$tv)/10)`.
Caused by warning in `rnorm()`:
! NAs produced”
Warning message:
“There was 1 warning in `mutate()`.
ℹ In argument: `y_jittered = lv + rnorm(n(), mean = 0, sd =
sd(prior_pc$lv)/10)`.
Caused by warning in `rnorm()`:
! NAs produced”
Warning message:
“There was 1 warning in `mutate()`.
ℹ In argument: `y_jittered = cv + rnorm(n(), mean = 0, sd =
sd(prior_pc$cv)/10)`.
Caused by warning in `rnorm()`:
! NAs produced”
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message:
“Removed 30 rows containing non-finite outside the scale range
(`stat_boxplot()`).”
Warning message:
“Removed 30 rows containing missing values or values outside the scale range
(`geom_point()`).”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message:
“Removed 30 rows containing non-finite outside the scale range
(`stat_boxplot()`).”
Warning message:
“Removed 30 rows containing missing values or values outside the scale range
(`geom_point()`).”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message:
“Removed 30 rows containing non-finite outside the scale range
(`stat_boxplot()`).”
Warning message:
“Removed 30 rows containing missing values or values outside the scale range
(`geom_point()`).”
```

## model comparison

### Bayes factor

```
tibble(model = c("pooling", "Brownian", "OU"),
bf = c(
$logml,
marginal_apentropy_pooling$logml,
marginal_apentropy_brown$logml
marginal_apentropy_ou
)%>%
) mutate(delta_bf = round(bf-max(bf), 3)) %>%
arrange(desc(delta_bf))
```

model | bf | delta_bf |
---|---|---|

<chr> | <dbl> | <dbl> |

OU | 1757.2442 | 0.000 |

Brownian | 1749.1261 | -8.118 |

pooling | 220.7127 | -1536.532 |

### PSIS-LOO

```
tibble(model = c("pooling", "Brownian", "OU"),
looic = c(
$estimates[3,1],
loo_apentropy_pooling$estimates[3,1],
loo_apentropy_brown$estimates[3,1]
loo_apentropy_ou
)%>%
) mutate(delta_looic = round(looic-min(looic), 3)) %>%
arrange(delta_looic)
```

model | looic | delta_looic |
---|---|---|

<chr> | <dbl> | <dbl> |

OU | -825.1421 | 0.000 |

Brownian | -802.5082 | 22.634 |

pooling | -504.5621 | 320.580 |

`<- generated_quantities$z z `

`<- generated_quantities$mu mu `

`<- sample(1:dim(z)[1], 1000) spl `

`<- mu[spl] mu_spl `

`<- t(z[spl,]) z_spl `

`<- generated_quantities$sigma[spl] sigma_spl `

`<- generated_quantities$lambda[spl] lambda_spl `

`<- function (i,j) (sigma_spl[i] * sqrt((1 - exp(-2 * lambda_spl[i] * apentropy_data$edge_lengths[j])) / (2 * lambda_spl[i]))) f `

`<- t(outer(1:1000, 1:length(apentropy_data$edge_lengths), Vectorize(f))) local_sigma `

`<- sigma_spl * sqrt(2 * lambda_spl) global_sigma `

```
<- matrix(0, nrow=apentropy_data$Nnodes, ncol=1000)
z_normal for (i in 1:apentropy_data$Ntrees) {
<- apentropy_data$roots[i]
idx <- (z_spl[idx,] - mu_spl) / global_sigma
z_normal[idx,]
}for (i in 1:apentropy_data$Nedges) {
<- apentropy_data$edges[i,1]
mother <- apentropy_data$edges[i,2]
daugther <- mu_spl + (z_spl[mother,] - mu_spl) * exp(-lambda_spl * apentropy_data$edge_lengths[i])
local_mu <- (z_spl[daugther,] - local_mu) / local_sigma[i,]
z_normal[daugther,] }
```

`write_csv(as_tibble(z_normal), "../data/ap_entropy_ou.csv")`

```
Warning message:
“The `x` argument of `as_tibble.matrix()` must have unique column names if
`.name_repair` is omitted as of tibble 2.0.0.
ℹ Using compatibility `.name_repair`.”
```