CausalSmith Working Papers

For fixed interaction order $\beta\ge1$ and fixed assignment probability $p\in(0,1)$, this paper establishes a finite-population design-based minimax mean squared error frontier, up to constants depending on $(\beta,p)$, for estimating the all-treated-versus-all-control effect under known bounded-degree low-order polynomial interference and common-probability Bernoulli assignment. The constants depend only on $(\beta,p)$ and are uniform in $n$, $d$, and $B$; they may deteriorate as $p$ approaches $0$, $1$, or a zero of the Bernoulli contrast. For radius $B$, population size $n$, and degree bound $d$, the paper's coefficient-mass and uniformly bounded-outcome minimax risks share the scale $$B^2\min\!\left\{1,\frac{dA_d}{n}\right\},$$ where $A_d$, the complete-block score energy, is the design second moment of the complete-block score used by the score-weighted neighborhood inverse-probability estimator (SNIPE, the shifted-neighbourhood inverse-probability estimator introduced by \citet{CortezRodriguezEichhornYu2023}). Equivalently, $$dA_d\asymp_{\beta,p} d\binom d{k_\star(d,\beta,p)},$$ where $k_\star(d,\beta,p)$, the largest interaction order with nonzero all-one-versus-all-zero Bernoulli contrast, is the exposed order in that contrast. A clipped SNIPE estimator attains the minimax rate on both bounded classes, while unprojected SNIPE attains the linear-regime variance scale. Complete directed $d$-blocks, with $\lfloor n/d\rfloor$ active blocks in the block construction of Definition 15, supply the matching lower-bound construction. When $d\mid n$, the same blocks give the exact worst-case risk $B^2A_d/m=B^2dA_d/n$, where $m=n/d$ is the number of complete blocks, for unprojected SNIPE. The complete-block local linear benchmark is solved exactly for block-local design-unbiased weights on fixed complete directed blocks under the coefficient-mass schedule class, with minimax risk $B^2A_d/m$ for a population of $m$ complete $d$-blocks. For the fair-coin design, even-order Bernoulli contrasts cancel, $A_d=4\sum_{1\le r\le \min\{\beta,d\},\,r\text{ odd}}\binom dr$, and first-order interference yields the frontier $B^2\min\{1,d^2/n\}$.

This paper establishes finite-sample minimax bounds for a scalar average treatment effect in a real-outcome observed-data model with a finite discrete adjustment variable. The model imposes consistency, conditional exchangeability, and fixed overlap parameter $\epsilon$; allows arbitrary cell masses; fixes the known outcome scale $M$, with conditional means in an interval of radius $M/2$; bounds conditional second central moments by $M^2$; and restricts the maximal cell-effect deviation to at most $\sigma M$, where $\sigma$ is the heterogeneity radius. For sample size $n$ and alphabet size $d$, the number of adjustment cells, the paper constructs two clipped estimators and a clipped three-branch selector: a heavy--light signed Chebyshev factorial estimator for rare cells, an occupancy-weighted treated-control estimator for empirically crossed cells, and a constant-zero branch. The known-radius selector satisfies $$\sup_{P\in\mathcal P_{d,\epsilon,M,\sigma}} E\{(\widehat\tau_n^\star-\tau(P))^2\} \le C_\epsilon M^2 \left[ \frac1n+ \min\left\{ 1,\frac{d^2}{n^2\log^2(en)},\sigma^2+\frac d{n^2} \right\} \right].$$ The same model class admits the lower benchmark $$c_\epsilon M^2 \left[ \frac1n+\min\left(1,\frac d{n^2}\right) +\sigma^2\min\left\{1,\frac{d^2}{n^2\log^2(en)}\right\} \right].$$ These two bounds match in order at exact homogeneity, at the unrestricted-radius endpoint, for every fixed positive radius, in the saturated alphabet regime, and at the parametric-dominance elbows. The results give a radius-indexed minimax bracket for point-estimation mean-squared error under fixed overlap, known outcome scale $M$, and known heterogeneity radius $\sigma$. The selector attaining the upper bound is a known-radius procedure: it uses $\sigma$ as an input to choose deterministically between its polynomial, collision, and zero branches, and the regime theorem isolates the residual shrinking-radius region generated by the displayed upper and lower benchmarks.

This paper studies honest confidence sets for a transported complier effect in a two-sample encouragement design. Outcomes, receipt, encouragement, and covariates are observed in a source population, while covariates are observed in a target population. Under the transported complier-effect model class with binary receipt, bounded outcomes, instrument overlap, monotonicity, and target-to-source covariate domination, the target complier effect $\theta_T$, the target complier-conditional outcome contrast, is identified as a transported reduced-form ratio and lies in the compact causal range $\Theta=[-1,1]$. The expected-length frontier is governed by the effective strength $t_n=n\mu_n^2/\kappa_n$, where $\mu_n$ is the transported first-stage mean and $\kappa_n$ is the Kish second moment of the transport weights. For every fixed threshold $t_0>0$, oracle honest confidence sets have minimax expected length of order $\min\{1,t_0^{-1/2}\}$, both globally and conditional on each admissible deterministic source-covariate, transport-weight, and propensity geometry. The lower bound is matched by an oracle Anderson--Rubin/Fieller score inversion. In uniform finite-cell designs, empirical target cell frequencies yield a score inversion that is sample-only in the uniform class with balanced assignment, while the regular nonuniform extension additionally uses known source cell probabilities and a known cell-varying propensity, the covariate being observed as a finite-cell label and the cell count obeying $k_n=o(\sqrt n)$, attaining the same order; the same conclusion extends to regular nonuniform source cells with known cell probabilities and known cell-varying propensity.

This paper establishes two new unconditional logarithmic minimax lower bounds for distance-compressed boundary regression. In the unsigned experiment, a rule observes outcomes and scalar Euclidean distances from each boundary query point. Over the compact nonparametric law class $\mathcal P_{\mathrm{NP}}(L,q)$ in Definition 9, with $q\geq1$ and $L\geq4$, both the CTY common-map risk in Definition 12 and the larger point-indexed outer risk in Definition 14 have minimax expected boundary sup-loss bounded below on the scale $$a_n=\left({\log n\over n}\right)^{1/4}.$$ The support-boundary hypercube constructs separated boundary perturbations with matching scalar-distance information at the queried point, yielding the same lower-bound scale even when the rule may use point-indexed Borel sections. Economically, this experiment isolates the risk cost of summarizing proximity by scalar distance after angular location and treatment-side information have been removed: a distance-only rule cannot tell which boundary arc or treatment side generated nearby observations, so recovering the entire boundary curve incurs a logarithmic testing penalty. The second unconditional lower bound holds in a signed-distance experiment with known treatment geometry, already on a fixed rectangular subexperiment. Conditional on the three CTY-style analytic inputs collected in Definition 33, the stabilized local-polynomial estimator and that lower bound characterize the signed-distance expected outer-risk rate on the same scale for every fixed polynomial order $p$, moment exponent $\nu\ge2$, and envelope $L\ge L_0(p)$. Thus the unsigned contribution is lower-bound sharpness, while the signed two-sided frontier is explicitly conditional on $\mathsf{AI}_{p,\nu,L}$.

This paper studies estimation of the partially linear coefficient $\theta_0$, the treatment coefficient, when the treatment innovation is independent of covariates, sub-Gaussian, and separated from Gaussianity by a fixed nonzero cumulant. The identifying resource is a zero of the treatment-innovation moment-generating function: the associated polynomial-exponential weight annihilates real shifts induced by treatment-code error, and a contour average of observable residual transforms identifies $\theta_0$ under the stated boundary zero-freeness, nuisance zero-freeness, and positive-count conditions. The statistical theorem is a fixed-code result. For fixed primitive constants, i.i.d. product sampling, the partially linear conditional-mean restrictions, bounded coefficient and regression ranges, sub-Gaussian treatment and outcome-noise envelopes, a nonempty fixed-code class, and the displayed current $L^1(P_X)$ treatment-code radius gate, where $P_X$ is the covariate marginal, a finite translated-dyadic contour bank and a total Borel statistic attain matched $c/n$ and $C/n$ mean-squared-error bounds on the non-Gaussian spectral class. For $\gamma\in(1/2,1)$, where $\gamma$ is the probability level, the same statistic controls the generalized lower $(1-\gamma)$-quantile of absolute error by order $(\gamma n)^{-1/2}$. The same fixed-separation rate holds on the aligned Jin--Mackey--Syrgkanis ACE comparison class. The ACE comparison records the published finite-order ACE upper guarantee and compares it with the contour upper guarantee on the common clipped-code class. The paper also gives a bounded-outcome Gaussian diagnostic and explicit sine-ratio reductions for mixture benchmarks. The represented-data construction is a conditional reproducibility layer for the same ordinary Borel statistic. When a compiled bounded spectral adapter satisfies the full canonical build-and-compilation specification for the parameter record, transported fixed records, and base treatment-code sequence, its represented-data execution realizes the statistic used in the statistical risk statements.

We study independent, heterogeneous Bernoulli assignment in bipartite experiments, where intervention units are assigned treatment and outcome units may depend on assignments in known intervention neighborhoods. For the finite-population contrast between all-treated and all-control neighborhood outcomes, we analyze an exposure-weighted Hájek estimator under bipartite neighborhood interference. With bounded potential outcomes, we derive a graph-and-design-dependent envelope that upper-bounds the design-based variance scale of its linearization and can be minimized subject to a positivity-constrained expected-treatment budget. Under the full assumptions of Theorem 4 and Theorem 5, the envelope-optimal design supports asymptotic normality and a graph-only Wald scale yields asymptotically conservative coverage. We also give conditions under which heterogeneous probabilities strictly improve the envelope relative to homogeneous assignment and study a separable overlap-based surrogate under an admissible budget and bounded outcome degree. A complementary unbounded-degree construction shows that degree dispersion and comparable surrogate weights alone do not control this approximation. Together, the results provide an outcome-model-free design criterion with convex, inferential, and separation guarantees, and identify approximation questions for surrogate criteria.

This paper studies offline policy learning for deterministic treatment rules when overlap can deteriorate near either propensity boundary in the same region where the treatment contrast is small. The target is observed-law welfare regret, defined directly from the conditional treatment contrast. The law class imposes bounded outcomes, positivity, a margin condition for small treatment contrasts, a zero-effect convention, and a joint overlap-decay restriction that ties weak treatment-arm information to the small-contrast region. Under the margin-window normalization and the auxiliary calibrations used by the two-point construction, the paper proves an observed-law minimax lower bound $$M_n(\alpha,\gamma)\ge c\,n^{-r_\star(\alpha,\gamma)}, \qquad r_\star(\alpha,\gamma) = \frac{1+\alpha}{2+\alpha+\beta_{\alpha,\gamma}}, \qquad \beta_{\alpha,\gamma} = \begin{cases} 0,&\gamma=0,\\ \alpha\gamma/(\alpha+1),&\gamma>0. \end{cases}$$ The exponent is obtained by balancing margin mass, local contrast size, and the probability of observing the informative treatment arm. The paper also gives a conditional analysis of a specified clipped cross-fitted AIPW empirical welfare rule with supplied nuisance estimates satisfying explicit rate, boundedness, cross-fitting, and localized empirical-process conditions. In the nonbinding nuisance-and-clipping regime, this conditional upper exponent matches the lower-bound exponent up to logarithmic factors; in the binding regime, the analysis gives the rule's nuisance-limited exponent.

This paper studies how to place measurement rounds in a finite-population rollout experiment when the target is the full-adoption contrast but the rollout budget stops at a treated fraction $q<1$. Under static rollout consistency, a degree-$\beta$ polynomial restriction on the rollout mean curve, and a common round-mean variance envelope---restrictions that low-order interference motivates but that we impose rather than derive from a microfounded interference model---unbiased linear estimation reduces to a polynomial extrapolation problem from $[0,q]$ to $1$. For a fixed schedule, the worst-case variance over the positive-semidefinite covariance class meeting this diagonal envelope equals the variance scale times the squared $\ell^1$ (total-variation) norm of the polynomial-exact weights. The resulting diagonal-envelope amplification criterion, defined over linear unbiased estimators, attains its minimax value up to multiplicative constants at shifted Chebyshev-Lobatto measurement fractions when $q\le q_{\max}<1$ and the number of rollout intervals satisfies $k\ge c\beta$ for some $c>1$. The minimax amplification is of order $$\left(\frac{(1+\sqrt{1-q})^2}{q}\right)^{2\beta},$$ up to constants depending only on $q_{\max}$ and the oversampling ratio. For the exact nested rollout covariance problem, the same Chebyshev schedule is rate-feasible through the envelope upper bound, and exact optimality under the true rollout covariance structure is posed as a separate covariance-specific design question.

We study a population identification question for a bivariate latent linear non-Gaussian model in which the number $m$ of middle source slots and an axis normalization are maintained inputs. The representation-level target asks whether a truncated joint-cumulant vector through order $L$ can also arise from the opposite axis-normalized arrow convention with the same $m$. At $L=2m+2$, the unordered finite loading-slope support is generically recovered and the opposite cumulant-map fiber is empty, while the same-arrow parametrization retains ambiguity beyond middle-slot relabelling. The closure of the opposite-arrow compatibility locus has codimension exactly one in each arrow image variety. For $m\ge3$, generic real opposite-fiber exclusion already holds at order $2m+1$; loading-support recovery uses the apolar order $2m+2$. There exist Euclidean-open parameter neighborhoods whose intersections with the feasible regions are nonempty, sharpening the population geometry of the maintained representation class.

This paper studies the population treatment coefficient from pooling a staggered-adoption panel in a unit-and-time fixed-effect Poisson pseudo-likelihood. We show that this projection coefficient can be negative even when every cohort-time proportional treatment effect is positive. The failure reflects heterogeneous effects interacting with the fixed-effect PPML score: a four-cohort design with equal shares and flat untreated means provides an explicit sign reversal, while a positive-weight proportional treatment-on-the-treated target remains positive. The result characterizes the interpretation of the deterministic population projection targeted by the pooled criterion.

This paper studies when a covariance relaxation for interference-aware randomized design is exactly implementable by a finite $\{\pm1\}$ assignment law. The setting is a design-based finite-population model with two equal homophilous communities, sign-symmetric assignment, and a block-weighted graph. The design objective combines a graph Laplacian term, a Laplacian-pseudoinverse term, a Schatten--2 robustness penalty, and an aggregate balance penalty. A symmetry reduction shows that the relaxed problem is exactly the optimization over a two-parameter block elliptope, while the implementable problem is the same objective restricted to covariances induced by block-exchangeable sign designs. The relaxation is exact in a strict cut region, where the unique relaxed optimum is generated by a two-point randomized design. Independent fair assignment is implementable throughout the model and is a finite-robustness relaxed optimum precisely on the affine locus $a+3b=2m$ and $r=2b(a+b)$; elsewhere it emerges as the asymptotic target as the robustness weight diverges. For odd community size, parity creates an open parameter region with a uniquely optimal relaxed covariance and strictly positive implementability loss. Finally, an exact finite active-set formula computes the loss for all admissible $m,a,b,r,\kappa$, with zero loss for even community size.

This paper studies lower bounds for estimating an interior continuous-treatment dose-response partial mean. The target is $$\theta_P(t_0)=\int \mu_P(t_0,x)p_{X,P}(x)\,dx,$$ interpretable as a causal dose-response mean under consistency, no unmeasured confounding, and local positivity. Over the same Hölder dose class $\mathcal P_{\alpha,\beta,s}(M,c_0,\varepsilon_0,t_0)$, under iid sampling, bounded outcomes, an interior evaluation dose, the stated anisotropic Hölder restrictions, and a strict-slack baseline condition, we prove the same-class minimax lower bound $$R_n\{\mathcal P_{\alpha,\beta,s}(M,c_0,\varepsilon_0,t_0),t_0\} \ge c\,n^{-2\alpha/(2\alpha+1)}$$ for all sufficiently large $n$. The lower-bound exponent is the one-dimensional treatment-regression exponent and holds for every fixed $\beta>0$, although the constant and slack-baseline feasibility may depend on $\beta$. We compare this lower floor with the published higher-order influence-function benchmark $$\rho_n= n^{-2\alpha/(2\alpha+1)} \vee n^{-2/(1+d/(4s)+1/\alpha)}.$$ When $d\le 4s$, the lower-bound exponent equals the exponent of this published comparator, whose upper theorem concerns a distinct localized-regularity class. When $4s<d$, the benchmark is governed by the covariate-smoothness term and has a strictly smaller exponent than the same-class lower-floor exponent. The paper establishes a same-class lower bound and an exact algebraic comparison with the external HOIF benchmark; a matching same-class upper analysis would complete the minimax characterization.