Random Forests as Statistical Procedures: Design, Variance, and Dependence

• URL: http://arxiv.org/abs/2602.13104v2
• Date: Tue, 17 Feb 2026 17:50:58 GMT
• Title: Random Forests as Statistical Procedures: Design, Variance, and Dependence
• Authors: Nathaniel S. O'Connell,
• Abstract summary: We develop a finite-sample, design-based formulation of random forests in which each tree is an explicit randomized conditional regression function.<n>This perspective yields an exact variance identity for the forest predictor that separates finite-aggregation variability from a structural dependence term.<n>The resulting framework clarifies how resampling, feature-level randomization, and split selection govern resolution, tree variability, and dependence.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Random forests are widely used prediction procedures, yet are typically described algorithmically rather than as statistical designs acting on a fixed set of covariates. We develop a finite-sample, design-based formulation of random forests in which each tree is an explicit randomized conditional regression function. This perspective yields an exact variance identity for the forest predictor that separates finite-aggregation variability from a structural dependence term that persists even under infinite aggregation. We further decompose both single-tree dispersion and inter-tree covariance using the laws of total variance and covariance, isolating two fundamental design mechanisms-reuse of training observations and alignment of data-adaptive partitions. These mechanisms induce a strict covariance floor, demonstrating that predictive variability cannot be eliminated by increasing the number of trees alone. The resulting framework clarifies how resampling, feature-level randomization, and split selection govern resolution, tree variability, and dependence, and establishes random forests as explicit finite-sample statistical designs whose behavior is determined by their underlying randomized construction.

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