I have been using the excellent `projpred`

package to conduct variable selection on fitted models from `brms`

.

I have also read a few papers on projection predictive inference .

Assuming I fit a simple linear regression model with 4 variables (our reference model):

`brms(y ~ x1 + x2 + x3 + x4, data=data, family=gaussian())`

And we would like to run Projection Predictive Inference (PPI) on this.

From what I have read, PPI (forward search) starts with an intercept-only model, and the reference model is projected onto this. It then proceeds to build all 4 submodels of size 1, and picks the best submodel of size one, which has the minimum KL divergence wrt. the posterior predictive distribution of the reference model. Say it was found to be x_2, we then use the one variable model `y ~ x2`

, and we build all possible size 2 models where x_2 has to be present and so on.

In this model building phase (for all possible size one models, all possible size two models, …), how exactly is it done internally ?

- Is a brms object refitted all this times ?
- Is the simple lm() formula used in fitting sub models ?
- Is it that, we use the reference model, and when fitting sub models of size one using say variable x_2, we set the coefficients of the rest of the variables to 0.

No. 1 & 2 was due to section 4.1 of *Robust and efficient projection predictive inference (2023)*

[…] Following this,

we fit all size-two modelsincluding the intercept and 𝑥(1) (“size-two” does not count the intercept here), and once more select the one closest to the reference model in terms of KL divergence of their posterior predictive distributions. Denote this second predictor to be selected 𝑥 (2) . This is repeated until either all predictors are selected, or some pre-defined limit on the model size is reached.

No. 3 was due to the paper I read on page 2 of : *Projection Predictive Inference for Generalized Linear and Additive Multilevel Models (2020)*

In the context of variable selection, one typically constrains the projection to a smaller subset of variables where the

excluded variables have their coefficients fixed at zero. Then, the projection procedure sequentially projects the posterior onto an incremental subspace, until all the variables have entered the projection.

Thanks,