Dear Experts,

I am a Bayesian newbie. I have a question about the difference between add_fitted_draws() and add_predicted_draws() when applied to brms binomial (Bernoulli) models. Specifically, using a cut-off of 50%, I compared predictions from add_fitted_draws() with those from add_predicted_draws. I expected the 2 sets of values to be similar - but they differed. Below, I provide a minimal example.

Thank you in advance for your kind guidance and advice.

```
rm(list=ls())
library(rms)
library(tidyverse)
library(brms)
library(tidybayes)
library(janitor)
pr_flrm = c(
prior(normal(0, 0.857), class = "b", coef = "vs1"),
prior(normal(0, 0.00857), class = "b", coef = "hp"))
flrm <- brm(am ~ hp + vs,
data=mtcars,
family='bernoulli',
prior=pr_flrm,
sample_prior = TRUE, ## so that we can visualize the prior distribution
save_pars = save_pars(all = TRUE),
seed=1202)
## predictions from add_fitted_draws
mtcars %>%
dplyr::select(hp, vs) %>%
add_fitted_draws(flrm, seed = 1234, n=1000) %>%
clean_names %>%
mutate(prediction = (value>0.50)*1) %>% ## using a cut-off of 0.50
group_by(row) %>% ## each subject
summarise(across(c(prediction), ~ list(smean.sd(.x)))) %>%
unnest_wider(prediction)
# A tibble: 32 x 3
row Mean SD
<int> <dbl> <dbl>
1 1 0.342 0.475
2 2 0.342 0.475
3 3 0.479 0.500
4 4 0.382 0.486
5 5 0.073 0.260
6 6 0.414 0.493
7 7 0.042 0.201
8 8 0.631 0.483
9 9 0.468 0.499
10 10 0.328 0.470
# ... with 22 more rows
## predictions from add_predicted_draws
mtcars %>%
dplyr::select(hp, vs) %>%
add_predicted_draws(flrm, seed = 1234, n=1000) %>%
clean_names %>%
group_by(row) %>% ## each subject
summarise(across(c(prediction), ~ list(smean.sd(.x)))) %>% ## smean.sd() from Hmsic
unnest_wider(prediction)
# A tibble: 32 x 3
row Mean SD
<int> <dbl> <dbl>
1 1 0.461 0.499
2 2 0.45 0.498
3 3 0.491 0.500
4 4 0.467 0.499
5 5 0.322 0.467
6 6 0.495 0.500
7 7 0.295 0.456
8 8 0.528 0.499
9 9 0.481 0.500
10 10 0.416 0.493
# ... with 22 more rows
```

.