Take this dataframe (df) representing an ABCD test on conversions.

When I take the descriptive means or fit a `brms`

model to df with default uninformed priors I see that recipe A (the Intercept) has a mean of 49.3%.

```
library(brms)
prob_to_lor <- function(prob){log(prob / (1-prob))}
df <- structure(list(
recipe = c("A", "B", "C", "D"),
events = c(4926L, 4968L, 5051L, 5136L),
trials = c(10000, 10000, 10000, 10000),
rate = c(0.4926, 0.4968, 0.5051, 0.5136)),
row.names = c(NA, -4L), class = c("tbl_df", "tbl", "data.frame"))
fit1 <- brm(
data = df,
family = binomial,
formula = "events | trials(trials) ~ 0 + Intercept + recipe",
cores = 7,
seed = 42
)
fixef(fit1)['Intercept','Estimate'] |> plogis()
```

[1] 0.492667

I am perplexed that including a prior for the Intercept with a mean of 0.49 has the effect of increasing the estimate for the Intercept upward to 0.50.

```
# Convert prior probabilities to log-odds ratios for logistic regression
PRIOR_MU_INTERCEPT <- .49
PRIOR_SD_INTERCEPT <- .025
PRIOR_SD_RECIPE <- .005
PRIOR_MU_INTERCEPT_LOR <- prob_to_lor(PRIOR_MU_INTERCEPT)
PRIOR_SD_INTERCEPT_LOR <- prob_to_lor(PRIOR_MU_INTERCEPT + PRIOR_SD_INTERCEPT) - prob_to_lor(PRIOR_MU_INTERCEPT)
PRIOR_SD_RECIPE_LOR <- prob_to_lor(PRIOR_MU_INTERCEPT + PRIOR_SD_RECIPE) - prob_to_lor(PRIOR_MU_INTERCEPT)
my_stanvars <- c(
stanvar(PRIOR_MU_INTERCEPT_LOR, name = "PRIOR_MU_INTERCEPT_LOR"),
stanvar(PRIOR_SD_INTERCEPT_LOR, name = "PRIOR_SD_INTERCEPT_LOR"),
stanvar(PRIOR_SD_RECIPE_LOR, name = "PRIOR_SD_RECIPE_LOR")
)
my_priors <-
prior(normal(PRIOR_MU_INTERCEPT_LOR, PRIOR_SD_INTERCEPT_LOR),
class = b, coef = Intercept) +
prior(normal(0, PRIOR_SD_RECIPE_LOR), class = b, coef = recipeB) +
prior(normal(0, PRIOR_SD_RECIPE_LOR), class = b, coef = recipeC) +
prior(normal(0, PRIOR_SD_RECIPE_LOR), class = b, coef = recipeD)
fit2 <- brm(
data = df,
family = binomial,
formula = my_formula,
prior = my_priors,
stanvars = my_stanvars,
cores = 7,
seed = 42
)
fixef(fit2)['Intercept','Estimate'] |> plogis()
```

[1] 0.4999328

In my mental model, if the Intercept in the raw data has a mean of 0.493, then a prior centered at 0.49 should only be able to bias the posterior estimate toward 0.49, but clearly in this case the posterior estimate is moving away from the prior for the Intercept, and I am struggling to understand how and why.