Profiling intercepts in nodewise regression (${\hat{a}}_k={\overline{X}}_k-{\overline{X}}_{-k}^{\mathsf{T}}{\theta }_{-k,k}$)

Minimizing the squared error over each free regression intercept gives the displayed formula. Substitution replaces all data columns by centred columns. The regression intercept is generally not the marginal mean of the response when predictor means are nonzero; the distinction matters in conditional regression coefficients from a Gaussian precision matrix.