I think I found the reason for the seemingly diverging inference when I compute bayes factors from the two models. It is the prior for the effect of interest that if kept constant can produces different outcomes for the two models. For example

```
priors_ = c(
set_prior('normal(log(.4), 0.05)', class='b', coef='Intercept'),
set_prior('normal(0, 0.05)', class='b', coef='MotionCongruencyincongruent'),
set_prior('normal(.3, 0.15)', class='sigma'),
set_prior('normal(0.3, .2)', class='sd')
)
mf_ <- bf(
RT ~ 0 + Intercept + MotionCongruency + (MotionCongruency || Run)#, sigma ~ MotionCongruency + (MotionCongruency | Run)
)
hyp_string <- 'MotionCongruencyincongruent = 0'
brms_mm = brm(
mf_,
data = rt_dat,
prior = priors_,
sample_prior = TRUE,
family = shifted_lognormal(link = "identity"),
save_pars = save_pars(all=TRUE),
control = list(adapt_delta = 0.9, max_treedepth = 15))
```

preduces a BF_{01} = 0.44

while the same prios, just scaled to ms, in the exgaussian model:

```
mf_ <- bf(
RT ~ 0 + Intercept + MotionCongruency + (MotionCongruency || Run)
)
priors_ = c(
set_prior('normal(400, 50)', class='b', coef='Intercept'),
set_prior('normal(0, 50)', class='b', coef='MotionCongruencyincongruent'),
set_prior('normal(300, 150)', class='sigma'),
set_prior('normal(300, 200)', class='sd')
)
hyp_string <- 'MotionCongruencyincongruent = 0'
rt_dat_ms <- rt_dat
rt_dat_ms$RT <- rt_dat_ms$RT * 1000
brms_mm = brm(
mf_,
data = rt_dat_ms,
prior = priors_,
sample_prior = TRUE,
family = exgaussian(link = "identity"),
save_pars = save_pars(all=TRUE),
control = list(adapt_delta = 0.9, max_treedepth = 15)
)
```

preduces a BF_{01} = 5.58.

Both versions, I think, are reasonable regarding my expectations, or do I miss something?

It seems that for both models I can adjust the prior so that it mimics the respective other model. For example setting the prior of the shifted_lognormal model to `set_prior('normal(0, 1.5)', class='b', coef='MotionCongruencyincongruent')`

yields BF_{01} = 7.58 (however, then the prior predicitions are far off again).

Do I missinterpret the sigma values of the priors?