9. Stochastic Search Variable Selection

SSVS is a variable selection technique for linear models and GLMs. Consider the linear equation for \(\mu\)

\[\begin{split} \begin{align*} \mu & = \beta_0 + \beta_1 x_1 + ... + \beta_k x_k \\ \beta_i & = \delta_i \alpha_i \\ \alpha_i & \sim N(0,\tau) \\ \delta_i & \sim Bern(p_i) \end{align*} \end{split}\]

where each \(\beta_i\) coefficient consists of a Normally-distributed \(\alpha_i\) multiplied by a \(\delta_i\) indicatator that is Bernoulli-distributed. If the indicator equals 1, then its \(\beta_i\) equals \(\alpha_i\), otherwise the coefficient is zero and the variable is not selected.

A posteriori, we can analyze the number of times each variable is selected in the MCMC samples, and then choose the model (combination of \(\delta\)’s) visited most.