sigma;
}
transformed parameters {
vector[len_a] a = a_tilde * sigma_s;
vector[n] mu = X * a;
}
model {
target+= - quad_form(S, a_tilde);
target+= normal_lpdf(y | mu, sigma);
}
```
I’m including the penalisation via the prior: p(a|\sigma\_s) = exp(- \frac{a^T S a}{2 \sigma\_s^2}). The a\_tilde term is to avoid divergent transitions by using a non-centred parametrisation.
I get the model matrix, X, and the penalisation matrix, S, using mgcv’s function smoothCon in R:
```r
sm_spec <- s(x, bs = "bs", k = 10)
sm_obj <- smoothCon(sm_spec, data = df, absorb.cons = TRUE)[[1]]
X <- sm_obj$X
S <- sm_obj$S[[1]]
```
Fitting this stan model, again the posterior looks reasonable and now I only get 1 to 2 divergent transitions in 4000 which I am more comfortable with getting rid of by increasing adapt\_delta.
I know of two big differences between my parametrisation and brms’s. First, I have flat priors on all parameters, whereas brms has a set of default priors (I’m not sure how to replicate equivalent priors to the brms ones in mine).
Second, I understand (from this blog by Tristan Mahr: [Random effects and penalized splines are the same thing - Higher Order Functions](https://www.tjmahr.com/random-effects-penalized-splines-same-thing/) ), that under the hood brms uses mgcv to convert the penalized spline problem to a mixed effects model.
My questions are as follows:
1. How should I go about trying to reduce divergent transitions for brms in this example?
2. brms have chosen a sensible parametrisation for computational reasons, if I used my stan code for more complex problems in the future, would I run into problems?
Fit using my stan code:
Fit using brms:
Fit using mgcv (REML):
---
### Author:  [@PaulVBell](https://discourse.mc-stan.org/u/PaulVBell)
#### Post date: [August 14, 2026, 1:03pm UTC](https://discourse.mc-stan.org/t/divergent-transitions-fitting-smooths-in-brms/41517/2 "2026-08-14T13:03:38Z")
Quick update to this, I forgot to include an intercept term, new model is:
```stan
transformed parameters {
vector[len_a] a = a_tilde * sigma_s;
vector[n] mu = a_0 + X * a;
}
model {
target+= cauchy_lpdf(sigma_s |0, 2.5);
target+= - quad_form(S, a_tilde);
target+= normal_lpdf(y | mu, sigma);
}
```
I had to include a prior on sigma\_s to avoid divergences but otherwise this didn’t change the previous comparison.
---
### Author:  [@js592](https://discourse.mc-stan.org/u/js592)
#### Post date: [August 14, 2026, 1:38pm UTC](https://discourse.mc-stan.org/t/divergent-transitions-fitting-smooths-in-brms/41517/3 "2026-08-14T13:38:03Z")
I’ve had similar issues implementing various types of smooths – a few divergent transitions but the overall fit looks fine. I suspect it has something to do with either potential collinearity between the basis columns and/or scaling of the design matrix. Can you post a pairs plot of your posterior draws? It might be also worth looking into doing a QR decomposition of X, but in that case I’m not sure how the penalty matrix S would be derived.
---
### Author:  [@PaulVBell](https://discourse.mc-stan.org/u/PaulVBell)
#### Post date: [August 14, 2026, 3:11pm UTC](https://discourse.mc-stan.org/t/divergent-transitions-fitting-smooths-in-brms/41517/4 "2026-08-14T15:11:42Z")
Thanks for the reply. Here is the pairs plot for a selection of brms parameters. It looks OK to me.
> [@js592](#):
>
> It might be also worth looking into doing a QR decomposition of X 𝑋, but in that case I’m not sure how the penalty matrix S 𝑆 would be derived.
Do you mean for the version I wrote or brms?
---
### Author:  [@js592](https://discourse.mc-stan.org/u/js592)
#### Post date: [August 14, 2026, 4:19pm UTC](https://discourse.mc-stan.org/t/divergent-transitions-fitting-smooths-in-brms/41517/5 "2026-08-14T16:19:09Z")
I’m not sure what `brms` does under the hood. I meant for the Stan implementation – if you’re doing an unpenalized spline then you can just swap the design matrix into something like [Regression Models](https://mc-stan.org/docs/stan-users-guide/regression.html#QR-reparameterization.section), which could help if collinearity between the basis functions is an issue. But since you’re then doing inference on the transformed parameters, I don’t know if the penalty matrix formulation of the likelihood would still behave as intended.
---
### Author:  [@avehtari](https://discourse.mc-stan.org/u/avehtari)
#### Post date: [August 15, 2026, 3:41pm UTC](https://discourse.mc-stan.org/t/divergent-transitions-fitting-smooths-in-brms/41517/6 "2026-08-15T15:41:10Z")
Smooths tend to have the problem that you get funnels with both centered and non-centered parameterizations as illustrated by @Niko in his blog post [**Divergent transitions in Hilbert Space Gaussian process posteriors and how to avoid them**](https://www.generable.com/post/hsgp-reparam). Although the blog post uses HSGP as the example, the same problem happens with the usual spline implementations. Sometimes increasing adapt\_delta a little bit can help, but sometimes it may require switching to adaptive parameterization, but as it is not automatically used by brms, it may require some effort.
---
### Author:  [@PaulVBell](https://discourse.mc-stan.org/u/PaulVBell)
#### Post date: [August 18, 2026, 8:48am UTC](https://discourse.mc-stan.org/t/divergent-transitions-fitting-smooths-in-brms/41517/7 "2026-08-18T08:48:47Z")
Thanks for the answer and the link to the blog post, that’s very helpful for my understanding.