ShrinkagePriors

Shrinkage priors are a class of Bayesian priors used in statistical modeling, particularly in situations where a large number of parameters are being estimated. The primary goal of shrinkage priors is to induce sparsity in the parameter estimates, meaning that many parameters are shrunk towards zero, effectively setting them to be exactly zero or very close to it. This is especially useful in high-dimensional problems where the number of predictors is large relative to the number of observations, as it can help to prevent overfitting and improve the predictive performance of the model.

These priors work by assigning a higher probability density to values of parameters close to zero compared to other regions of the parameter space. This "pulls" the estimates of many parameters towards zero, effectively performing a form of regularization. Common examples of shrinkage priors include the Lasso prior (which is closely related to the L1 penalty in frequentist methods), the Horseshoe prior, and the Dirichlet-Laplace prior. Each of these priors has different properties regarding the degree and nature of shrinkage they induce.

Shrinkage priors are often contrasted with standard non-informative priors or diffuse priors, which do not impose such a strong belief in sparsity. By shrinking less important parameters to zero, shrinkage priors can help to identify and retain only the most relevant predictors in a model, leading to simpler and more interpretable results. They are a powerful tool in modern statistical inference and machine learning.