2.4. Region selection for the Bayesian probit model

The central task in our problem is to select a small set of voxels that can effectively classify different conditions. Thus we would like to fit a sparse model such that at most locations the regression coefficient β₁(s_j) is zero or nearly zero. To this end, we modified the covariance matrix in the prior distribution in Eq. (3) as

where D_α = diag[ã₁τ₁, …, ã_nτ_n], ã_j = 1 if α_j = 0 and ã_j = c_j if α_j = 1, α = (α₁, …, α_n) is an indicator vector for variable selection. We set τ_j (> 0) small and c_j (always > 1) large to make those β₁(s_j) with α_j = 0 clustered around 0 with variance τ_j, whereas those β₁(s_j) with α_j = 1 more dispersed with variance c_jτ_j. Under this setting, the prior of β₁(s_j) corresponds to a mixture of two normal distributions, and is more effective to differentiate the two groups of β₁(s_j)’s [33]. In addition, we assume that α₁, …, α_n are i.i.d. observations from the Bernoulli distribution P(α_j = 1) = 1 − P(α_j = 0) = p_j = 0.5. The prior distribution for β₀ is specified as π(β₀) = N(0, c₃), where c₃ = 10⁵. The prior of γ depends on the model of the correlation function H(γ). In the Appendix, we consider an example with a Matérn correlation function with a decay parameter γ >0, for which we use π(γ) = N(a₂, b₂) truncated to be positive. We will discuss the choice of a₂ and b₂ using the empirical semivariogram in the Results Section. All the priors are assumed to be independent. A graphical representation of our BPSV model that depicts the dependency relationship between the variables is presented in Figure 2.

Graphical representation of our proposed model with m observations and n predictors.