Under standard graphical assumptions, the Markov boundary of a target variable is the smallest set of features that renders every other feature redundant. Once the boundary is observed, the target is conditionally independent of the rest of the table. This is a tempting object for tabular prediction, since it names exactly the columns a model should need. Yet modern regressors are still trained on the full feature set. We ask whether the Markov boundary is genuinely useful for prediction on SCM3K, a 3,450-task synthetic SCM benchmark with feature counts from 40 to 1000 and six SCM families, evaluated with six regressors. The answer is more nuanced than the theory suggests. Restricting a regressor to the oracle boundary often improves prediction substantially, and the improvement grows as the feature space becomes larger and sparser. But the natural pipeline of recovering the boundary with causal discovery and training on the recovered mask does not deliver. Existing estimators exhaust the compute budget before reaching the regime where the boundary helps most, and even where they run they rarely beat the full feature set. We trace this to three causes. Discovery optimizes structural recovery rather than prediction. False negatives and false positives carry sharply asymmetric predictive cost. The exact boundary is only one of many feature sets that beat all features. We then develop what these facts imply for prediction-aligned feature selection and for tabular models that learn to use causal structure.

To gain intuition for how p-locality functions, we will introduce another notion of locality, called Markov locality, which will use the language of Markov blankets. We will prove that under relatively relaxed conditions p-locality and Markov locality are equivalent. This will allow us to relate the notion of locality to various graph structures commonly used to represent probability distributions, and will be a key step in proving Properties 2.1 and 2.2. We start by defining the Markov boundary, M(X,S), of a random variable X contained in a set of random variables S, as a minimal set such that p(X|S) = p(X|M(X,S)). The Markov boundary defines a minimal set of variables such that, conditioned on these variables, conditioning on no additional random variables in S changes the probability of X [39]. Similarly, we define the Markov blanket, M(X,S) for X in S as any set of variables such that conditioning on M(X,S), makes X conditionally independent from all other variables [39]. In this way, the Markov boundary is a Markov blanket but not all blankets are boundaries. Markov locality: Given probability distribution p(Z) and function f: RNX+NΘ RNΘ, the update function f(Z) is Markov-local with respect to the distribution p over Z if and only if k: Z Ωs.t. AMarkov boundary can be thought of as the set of variables that'locally' communicate with the parameter Θk, thus providing a natural measure of locality. Importantly, for Markov-locality to be of use, we would like the Markov boundaries of random variables in the model of interest to be unique.

Lasso performance, with the added benefit of providing multiple solutions.