Fit the conditional law of each variable given all others and use its predictor support to estimate its neighbours. In a Gaussian graphical model, this is linear regression with coefficients given by conditional regression coefficients from a Gaussian precision matrix. The Nodewise Lasso supplies sparse fits; the OR rule for nodewise graph selection and AND rule for nodewise graph selection combine possibly disagreeing directional supports into an undirected graph.
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.
The intersection rule retains an edge only when both directions of nodewise regression select it. It is more conservative than the OR rule for nodewise graph selection; the distinction matters because separate finite-sample fits can disagree.
The union rule retains an edge whenever either of the two nodewise regression fits selects the other variable. It is more inclusive than the AND rule for nodewise graph selection and makes an undirected edge decision from potentially asymmetric estimated neighbourhoods.
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