I have taken up this project for the purposes of learning a bit more about Bayesian statistics and how to program with Stan. I've bulit up some simple classes and idioms to hopefully make it fairly straightforward to write a stan model and then interface with it via the sklearn API -- this is quite convenient for experimenting with different datasets.

The first model I implemented is simply a linear regression model with T-distributed output

  y0 ~ cauchy(0, 1);
  nu ~ cauchy(0, 1);
  sigma ~ normal(0, 1);  // half-normal
  lam ~ exponential(1);
  theta ~ normal(0, lam);
  y ~ student_t(nu, y0 + Q * theta, sigma);

the input data needs to be scaled to unit variance before feeding it in. I obviously do not claim that the following examples are exemplars of remarkable performance -- they are illustrations.

Here's some examples on the Diabetes dataset. Parameter (marginal) posteriors:

Out of sample predictions with uncertainty quantification (Linear Regression)

Another interesting model is VAR(p) models. So far, I have developed an AR(p) model, with a parameterization in terms of the reflection coefficients (see /RJTK/levinson-durbin-recursion or any book on signal processing). This guarantees the stability of every posterior sample.