with the design matrix X=(\boldsymbol{1}, \boldsymbol{x}_1, ..., \boldsymbol{x}_k), the corresponding regression coefficients \boldsymbol{b}=(b_0, b_1, ..., b_k)^T and the residual variance-variance matrix \Sigma. The k predictors, vectors \boldsymbol{x}_1, ..., \boldsymbol{x}_k, have the same length of n as the response variable \boldsymbol{y}.

Can the model be implemented in brms if I want to impose a prior distribution on the regression coefficients (except b_0): b_i \sim N(b, \tau^2) (i=1,2,...,k)? Also, how to specify the autoregressive structure of the residuals with a band matrix \Sigma?

Although my understanding of time series is limited, the covariance structure you want looks very similar to the one provided by cor_ar or is there something I am missing? Other than that it should be simple.

Yes, the variance-covariance structure for the residuals can be modeled through cor_ar or cor_arma in brms. But how to impose the constraint b_i \sim N(b, \tau^2) on the regressors coefficients?

Oh, I see - you want there to be a single \tau shared across all b_i? Then you will probably be best served by horseshoe: https://rdrr.io/cran/brms/man/horseshoe.html, although this is slightly different than your model, it should IMHO serve the same purpose. Or am I missing something?

The horseshoe is indeed an interesting idea. However, the horseshoe prior would not work in my case because I just realized there are a couple of issues I didn’t state correctly in my original post.

(1) Actually it is not that I want to impose a prior on some of the population effects, but rather I’d like to assume that b_i follows a Gaussian distribution N(b, \tau^2), where both b and \tau^2 are unknown.

(2) Sometimes the Gaussian assumption is placed only on some, not all, of the population effects b_i.

So you can easily get away with zero-centered effects as long as you add a new fixed effect for sum of the predictors (please double check my math though)

So it is up to whether you are willing to use a slightly different shape than normal for the prior on \hat{b}_i.

You should IMHO be able to handle 2) in this framework as well.

But you are getting close to the limits of brms :-)