Finite-Sample Inference under Interference
Under interference, the variance of an estimator can involve outcomes that are never observed together. One way to obtain a confidence bound is to avoid estimating that variance and use known outcome bounds instead. A second question is how to identify a policy mean from observational assignments. (Tchetgen-Tchetgen and VanderWeele 2012) addresses both questions through the assignment design: concentration gives conservative finite-sample bounds, while joint inverse-probability weights identify allocation-averaged outcomes under additional assumptions.
1. A finite-sample bound
Start with independent groups and bounded group contributions. This case makes the concentration argument transparent. The paper also treats the two-stage design without assuming stratified interference.
Proposition 1. Suppose \(V_g\) are independent unbiased group contributions to a target \(G^{-1}\sum_g\mathbb{E}[V_g]\), with known bounds \(a_g\le V_g\le b_g\). Then
Solving for \(t\) gives a confidence radius \(\sqrt{\log(2/\alpha)\sum_g(b_g-a_g)^2/(2G^2)}\). Within-group interference is unrestricted because it is already contained in each bounded contribution.
The independence assumption is specific to this example. Selecting a fixed number of groups creates dependence between their assignment indicators, so that design requires the paper’s corresponding finite-sample argument.
2. Weight the whole allocation
Let \(f_g(\mathbf a_g\mid L_g)\) be the observed joint assignment probability, and let \(\pi_s(\mathbf a_{g,-j})\) be a target distribution for peers’ assignments. Under consistency, conditional exchangeability of the entire allocation vector given \(L_g\), and positivity on the target support,
has expectation equal to the group mean potential outcome with own treatment fixed at \(a\) and peers assigned according to \(\pi_s\).
Expand the expectation as a sum over \(\mathbf A_g\). For every allocation in the target support, its observed probability cancels the same factor in the denominator. The remaining sum averages potential outcomes with the target probabilities. Taking expectation over covariates then gives the stated mean.
3. Why an individual propensity is generally insufficient
Weighting only by \(\Pr(A_{gj}=a\mid L_{gj})\) leaves the conditional distribution of peers’ assignments in the expectation. Under interference, that remaining distribution affects outcomes and need not equal the policy distribution of interest.
This identity requires probabilities for joint allocations, which can be difficult to estimate in large groups. An assignment model or a sufficient exposure mapping can reduce the dimension. Either reduction must be justified, and neither identifies outcomes at target allocations that the observed design never produces.
4. Deriving the concentration radius
For a centered variable \(V_g-\mathbb{E}V_g\) lying in an interval of length \(b_g-a_g\), Hoeffding’s lemma bounds its moment-generating function by
The exponential of a sum is a product. Independence allows its expectation to factor, so the one-group bounds multiply. Applying Markov’s inequality gives
Minimizing the exponent at \(\lambda=4Gt/\sum_g(b_g-a_g)^2\) yields the upper-tail exponent in the proposition. Apply the same calculation to \(-V_g\) and add the two tail probabilities. Setting the resulting bound equal to \(\alpha\) gives the displayed confidence radius.
For contributions in \([0,1]\), the radius is \(\sqrt{\log(2/\alpha)/(2G)}\). Only the number of independent groups appears. Under unrestricted interference, everyone within a group may move together, so adding people need not add independent information. A bound that improves with group size needs restrictions that rule out this possibility.
Known deterministic weights can also be accommodated by applying the lemma to \(w_gV_g\), whose range has length \(|w_g|(b_g-a_g)\). The squared weights then enter the exponent. This is useful when the target weights groups by population size rather than equally.
5. The allocation-weighting identity without omitted steps
Fix a group, a focal individual \(j\), and covariates \(L_g=l\). Write \(\mathbf a=(a,\mathbf a_{-j})\). Conditional exchangeability and consistency imply
The first equality uses exchangeability for the complete assignment vector. The second is algebraic cancellation on the target support. If the target peer distribution depends on covariates, replace \(\pi_s\) by \(\pi_s(\cdot\mid l)\) throughout. Averaging this identity over individuals and \(L_g\) gives the group target.
6. A two-person example of the wrong weight
Let two treatments be independent Bernoulli\((p)\) and let person one’s outcome be \(Y_1(a_1,a_2)=a_1+2a_2\). The target sets own treatment to \(a\) and assigns the peer independently with probability \(q\). Its mean is \(a+2q\).
Weighting only by own treatment gives
The peer assignment still follows the observed design. The full allocation weight adds
which changes the peer mean from \(p\) to \(q\) and yields the desired \(a+2q\). When \(q=p\), the extra factor is one. The individual propensity is sufficient in that special case because the target preserves the observed conditional peer law, not because randomization eliminates spillovers.
If the design instead treats exactly one of the two people, the allocations \((0,0)\) and \((1,1)\) have zero probability. A policy requiring either of them is not identified by this design without further outcome restrictions. No large inverse weight can compensate for an allocation that never occurs.
7. Positivity also controls precision
For bounded outcomes \(0\leq Y_{gj}\leq1\), the weighted contribution can still be large when the joint assignment probability is small. If the target-to-observed weight is bounded by \(H\), then the group contribution lies in \([0,H]\) and the concentration radius scales with \(H\).
Small full-allocation probabilities alone do not determine weight stability: the numerator may be equally small. The relevant quantity is the ratio between target and observed probabilities. The next calculation shows how a persistent difference between the two assignment laws accumulates across peers.
There is an exact second-moment calculation behind this concern. Suppose a focal unit has \(k\) peers, all assignments are independent Bernoulli\((p)\), and the target sets own treatment to \(a\) while assigning peers independently with probability \(q\). Let \(p_a=p^a(1-p)^{1-a}\) and denote the full weight by
Each peer’s likelihood ratio has mean one. Squaring it and averaging over a single Bernoulli draw gives
Independence therefore yields
For fixed assignment probabilities with a nonzero gap \(q-p\), the second moment grows exponentially in the number of peers. Since the weight has mean one, subtracting one gives its variance; for \(Y=1\), this is also the variance of the weighted outcome. Thus positive support everywhere can coexist with very poor precision.
If instead \(q-p\) is of order \(k^{-1/2}\), and \(p\) remains bounded away from zero and one, the peer factor stays bounded. The same observed network can therefore support precise evaluation of nearby policies while giving much less information about policies that change many peer assignments substantially.
A sufficient exposure mapping reduces the number of assignment configurations that must be distinguished. One must then compute probabilities for those exposure cells and their pairs. Sufficiency defines the outcome model; the resulting probabilities determine precision.
The bounded-variable exponential lemma
The concentration calculation can be completed without taking its one-variable bound on faith. For \(V\in[a,b]\), set \(K(\lambda)=\log\mathbb E e^{\lambda(V-\mathbb EV)}\). Boundedness permits differentiation under the expectation. Then \(K(0)=K'(0)=0\), and \(K''(\lambda)\) is the variance of \(V\) under the probability law tilted by \(e^{\lambda V}\). Under any law supported on \([a,b]\),
Integrating this second-derivative bound twice yields \(K(\lambda)\le\lambda^2(b-a)^2/8\) for either sign of \(\lambda\). Independence across groups and the optimization already shown then prove the full two-sided inequality. If all ranges have length zero, the estimator is constant and the probability bound is replaced by the exact statement that its error is zero.
For a fixed selected set of groups, the same proof applies conditionally when their within-group randomizations are independent. It covers the average effect of those selected groups. Passing to the population average additionally requires the between-group selection term; conditional concentration alone does not remove that term.