Experimentation.SuperPopulation
Super-population network inference: locally dependent fields, network-sum and mean CLTs, HAC variance estimation, and consistency bounds.
Basic 5 core · 0 supporting The design-based experimentation substrate (Experimentation/DesignBased/) fixes the potential outcomes and puts all randomness in the assignment, so its probability space is a finite product measure over the treatment ve ★ mem_nbhd_iff★ self_mem_nbhd
Super-population locally-dependent network field
The design-based experimentation substrate (Experimentation/DesignBased/) fixes the potential
outcomes and puts all randomness in the assignment, so its probability space is a finite product
measure over the treatment vector. The super-population / model-based route is the opposite:
the units (and their outcomes) are themselves drawn from a population, the randomness is the
sampling / population draw, and the dependence between units is governed by a network rather
than by an assignment mechanism. This file provides the base object for that route.
A NetworkDependence is the model-based sibling of Causalean.Stat.IIDSample: instead of mutual
independence (iIndepFun) it carries a bounded-range / m-dependent network — node-level
random summands X i : Ω → ℝ on a common ambient space (Ω, μ), together with a reflexive,
symmetric adjacency relation such that index sets with no edge between them carry independent
summand tuples. This is exactly the data of a dependency graph
(Causalean.SteinMethod.DepGraph), so toDepGraph exposes the field to the proved Stein
dependency-graph CLT (stein_cdf_clt_of_depGraph), which the CLT file specializes.
This is the m-dependent (exact-independence-beyond-the-network) layer. Decaying-dependence (ψ- or mixing) models, where far-apart nodes are only approximately independent, are a different super-population abstraction and are not part of this module.
A super-population locally-dependent network field: bundles node-level random summands on a common ambient probability space together with a network relation between units that is reflexive and symmetric, requires every summand to be measurable, and requires any two collections of units joined by no edge to carry independent summand tuples (exact m-dependence beyond the network). This is the model-based counterpart of the finite design — the randomness is the population draw, not the assignment — and the m-dependence sibling of an i.i.d. sample.
Definition (Lean source)
The network field, viewed as a Stein dependency graph for its summands — a pure field rename, so the proved dependency-graph CLT applies verbatim.
Definition (Lean source)
The closed network neighborhood N i = {j | adj i j} of a unit.
Definition (Lean source)
Unit j lies in unit i's network neighborhood if and only if i and j are adjacent in the underlying interference network.
Formal statement
Proof (Lean source)
Every unit lies in its own network neighborhood.
Formal statement
Proof (Lean source)
HAC 2 core · 0 supporting Inference for a super-population network field needs a variance estimator that is robust to the network dependence — the analog of a HAC / cluster-robust estimator. ★ netHACVarEst★ netHACVarEst_integral_eq_variance
Network-HAC variance estimator
Inference for a super-population network field needs a variance estimator that is robust to the network dependence — the analog of a HAC / cluster-robust estimator. Because far-apart units are independent (m-dependence beyond the network), the variance of the network sum collapses to a sum of within-neighborhood cross-products, so the natural estimator is
V̂ = ∑ᵢ ∑_{j ∈ N i} Xᵢ Xⱼ,
i.e. the empirical sum of products over network-adjacent pairs. This file defines that estimator
and records its unbiasedness anchor: under mean-zero summands its expectation is exactly the
variance of the network sum. Sequence-level consistency of V̂ for the true variance along a
growing network is developed in the HACConsistency modules.
The network-HAC variance estimator for a network field: the empirical sum of summand products over network-adjacent pairs, V̂(ω) = ∑ᵢ ∑_{j ∈ N i} Xᵢ(ω)·Xⱼ(ω). It targets the variance of the network sum, whose cross-terms vanish outside the network neighborhoods.
Definition (Lean source)
Unbiasedness of the network-HAC estimator. Under a probability measure with square- integrable summands that are mean zero, the expectation of the network-HAC estimator equals the variance of the network sum: E[V̂] = Var(∑ᵢ Xᵢ).
Formal statement
Proof (Lean source)
CLT 1 core · 0 supporting For a sequence of super-population network fields (one per population size n) with bounded degree m, uniformly bounded summands |X i| ≤ Bₙ shrinking with Bₙ → 0 and N·Bₙ³ → 0, mean-zero summands, and unit total variance, ★ networkSum_clt
m-dependent CLT for a super-population network field
For a sequence of super-population network fields (one per population size n) with bounded
degree m, uniformly bounded summands |X i| ≤ Bₙ shrinking with Bₙ → 0 and N·Bₙ³ → 0,
mean-zero summands, and unit total variance, the network sum ∑ᵢ Xᵢ converges in distribution
to the standard normal. This is the model-based (super-population, network-dependence) analog of
the design-based prodDesign_clt, and is obtained by viewing the field as a Stein dependency
graph (NetworkDependence.toDepGraph) and invoking the proved bounded-degree dependency-graph CLT
stein_cdf_clt_of_depGraph — the m-dependence (exact independence beyond the network) is exactly
its leave-out independence hypothesis.
m-dependent network CLT (super-population). Consider a sequence of super-population network fields F n, each on a probability space with measure μ n, whose dependency graph has degree at most m, whose summands are uniformly bounded in absolute value by a sequence B n tending to zero fast enough that card(Vₙ)·(Bₙ)³ → 0, and whose summands are mean zero with the network sum having unit total variance for every n. Then the network sum's cumulative distribution function converges pointwise to the standard normal CDF: P[∑ᵢ (F n).X i ≤ s] → Φ(s).