Sampled Priors for Intercept don't match the setting

Interfaces brms
1

Hello!

I try to run a quite simple multiple regression model for educational purposes. Before running the model I defined priors and wanted to conduct prior predictive checking by running the model with the option “sample_prior = “only””

bf.1 <- bf(bernote ~ wital + witsch + witbo + mtp) 

get_prior(bf.1,
          family = gaussian(), 
          data = dat_z)

prior.1 <- c(brms::prior(normal(0, 0.5), class = b, coef = witsch),
             brms::prior(normal(0, 0.5), class = b, coef = wital),
             brms::prior(normal(0, 0.5), class = b, coef = witbo),
             brms::prior(normal(0, 0.5), class = b, coef = mtp),
             brms::prior(normal(3.5, 2.5), class = Intercept),
             brms::prior(normal(0, 2.5), class = sigma, lb = 0))

validate_prior(prior.1, bf.1,
               family =gaussian(),
               data = dat_z)

As can be seen, I wanted a normal(3.5, 2.5) prior for the Intercept. The validate_prior() command confirmed that it was set up correctly. Therefore I ran the prior predictive model:

mod.1_ppc <- brm(data = dat_z,
                 family = gaussian(),
                 formula = bf.1,
                 prior = prior.1,
                 sample_prior = "only")

But when I looked at the visual pp_check and the output of the model, the sampled priors for the Intercept were completely off the limits, where the Intercept had a mean of 5.19 and an SD of 104.48.


> print(mod.1_ppc)
 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: bernote ~ wital + witsch + witbo + mtp 
   Data: dat_z (Number of observations: 31) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept     5.19    104.58  -202.51   217.19 1.00     4061     2700
wital        -0.01      0.50    -0.98     0.96 1.00     4368     2857
witsch        0.00      0.52    -1.00     1.03 1.00     4149     2566
witbo        -0.02      0.50    -0.99     0.97 1.00     4326     2897
mtp           0.01      0.50    -0.98     0.99 1.00     4242     2784

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     1.99      1.52     0.07     5.55 1.00     2488     1573

Strangely enough, if I run the actual model and sample the priors with samp_prior = TRUE, the priors look fine:

mod.1 <- brm(data = dat_z,
             family = gaussian(),
             formula = bf.1,
             prior = prior.1,
             sample_prior = TRUE,
             warmup = 2000, iter = 5000)

prior_draws(mod.1) |> describe()

    vars          n   mean   sd median trimmed  mad   min   max range  skew kurtosis   se
Intercept    1 12000  3.48 2.49   3.49    3.49 2.49 -5.09 13.19 18.27  0.00    -0.03 0.02
b_wital      2 12000 -0.01 0.50   0.00   -0.01 0.50 -1.69  1.84  3.54  0.01    -0.07 0.00
b_witsch     3 12000  0.01 0.50   0.01    0.01 0.50 -1.83  1.84  3.67 -0.04    -0.02 0.00
b_witbo      4 12000  0.00 0.50   0.00    0.00 0.50 -2.12  2.09  4.21 -0.01    -0.02 0.00
b_mtp        5 12000  0.00 0.50  -0.01   -0.01 0.51 -1.86  2.00  3.86  0.02    -0.02 0.00
sigma        6 12000  1.99 1.50   1.70    1.83 1.49  0.00 10.46 10.46  0.97     0.76 0.01

Can someone explain this to me and how to fix this? Did I specify something wrong?

Many thanks for taking your time and help!
Rainer

2

Brms by default puts priors on the centered intercept. You can try suppressing the intercept and put a prior on a manually defined intercept. See here for a very similar question and the solution: Use of `sample_prior='only'` in brms - #8 by paul.buerkner

1 Like
3

Many thanks for the quick reply, that solved the problem. But I am still puzzled why brms behaves like this, since I put a prior on the default Intercept and the other priors are normal around 0, i.e. their mean is 0. I dont get why the Intercept prior is changing in that case?

And why does it deliver the correct prior samples in the actual model?

4

You can see from your summary that the error on the intercept is really large, just like in the other issue. Thus you can’t really trust the mean estimate. In the other thread @paul.buerkner points out that this can happen when your predictors are on a very wide scale, causing troubles with the sampling. Unfortunately I don’t know more details than that. Probably if you scale and center your predictors that would also help (and is generally recommended).