Bayesian Lasso standardized data application

Question

I am attempting to duplicate Belmonte et al. 2014. The paper applies a Bayesian Lasso to state space models to get better out of sample prediction. Bitto et al 2019 expands on this model. Their expansion includes a simulation study showing that the Bayesian Lasso shrinkage estimator recovered the true coefficients. Both papers, along with Park and Casella 2008, assume that the covariates are standardized mean zero and variance one. However, in most cases data we have is not standardized. How is it possible for data to first be standardized, then analyzed with a Bayesian Lasso and still recover the true DGP in a simulation study? Why is it when I standardize the variables the Bayesian Lasso returns the wrong coefficients and when I don't standardize it returns the right coefficients?

References:

Bitto et al 2019: https://www.sciencedirect.com/science/article/pii/S0304407618302070

Belmonte et al. 2014: https://onlinelibrary.wiley.com/doi/full/10.1002/for.2276

Park and Casella 2008: https://www.tandfonline.com/doi/abs/10.1198/016214508000000337

Answer 1

In a regression problem, you can have predictors with the most different scales. for example, you may have a predictor that ranges between 0 and 1 alongside another that ranges in the thousands, but still, the first one may be an index with a very strong influence on your outcome variable $y$, much more than the second one. You don't know in principle.

But, let's say they have an equal share of the predicted value $\hat y$. The variance of $\hat y$ can be decomposed (following this fundamental property of variance) in the quadratic form of the parameters $\hat \beta$ and the variance of the predictors.

This means that if we want our two predictors to contribute equally to the variability of $\hat y$, their parameters must be inversely proportional to the respective predictor st. deviation. If they only had equal variance, their parameter would have been the same value.

So, if you scale the predictors to have equal variance, their parameter will be proportional to the influence they have on $\hat y$ (more or less, collinearity also plays a role, but LASSO generally leaves out collinear predictors). But why is this so important for LASSO method?

Because LASSO applies a regularization on all the parameters of the model, which means from a bayesian perspective, that it applies a prior with the same scale on all the parameters. So for the model to make a sensible selection of the predictors, and a good shrinkage of the parameters, the predictors must have a scale proportional to their importance.

For all these reasons, if you don't have prior knowledge of which predictors have more or less effect on $\hat y$, you just scale all them to have equal variance, otherwise you may get a model dangerously biased towards higher-scale predictors.