Intersection Supermartingale Products

• Intersection supermartingale products are rigorous probabilistic constructions that combine single-stratum nonnegative supermartingales to test composite hypotheses under controlled Type I error.
• They leverage adaptive sampling and predictable stratum selection to yield anytime-valid P-values, substantially increasing efficiency in risk-limiting audits.
• Empirical results show that these methods can significantly reduce sample size and risk, outperforming traditional approaches like SUITE and SHANGRLA.

Intersection supermartingale products are a rigorous probabilistic apparatus combining single-stratum nonnegative supermartingale (NNSM) processes to enable composite union-intersection hypothesis tests, particularly relevant in stratified risk-limiting audits (RLAs). This construction controls Type I error guarantees nonparametrically, applies under any predictable sampling strategy, and supports adaptive allocation of sampling resources across strata. Intersection supermartingale products yield anytime-valid $P$-values and can substantially increase efficiency in practical RLA settings, offering improvements over earlier approaches such as SUITE and SHANGRLA by exploiting the generality and flexibility of NNSMs (Spertus et al., 2022).

1. Single-Stratum Nonnegative Supermartingales

The fundamental component is the single-stratum NNSM, defined for a stratum $s$ of size $N_s$ with bounded nonnegative assorter values $X_{1s},\dots,X_{N_s s}\in[0,u_s]$. To test the one-sided null hypothesis $H_{0s}: \mu_s \leq \theta_s$ for some $\theta_s\in[0,u_s]$, samples are drawn sequentially. At each sample, a predictable, typically upward-biased estimator $\tau_{i-1,s}$ of the mean $\mu_s$ is used to define a nonnegative factor $Y_{is}(\theta_s)$ satisfying $$\mathbb{E}[Y_{is}(\theta_s)\mid \text{history}]\leq 1$$ under $s$0.

The single-stratum NNSM process is then

$s$1

Ville’s inequality ensures $s$2 under $s$3 for any $s$4, granting finite-sample error control and anytime validity.

2. Intersection Supermartingale Construction

For testing the global null hypothesis $s$5 for all strata $s$6, intersection supermartingale products combine the single-stratum NNSMs into a global process. At each sampling epoch $s$7, a (predictable) stratum selector $s$8 identifies the stratum to be sampled, measurable with respect to the filtration $s$9 generated by the previous samples and stratum selections. Let $N_s$0 be the total draws from stratum $N_s$1 up to time $N_s$2.

The update at time $N_s$3 is:

$N_s$4

where $N_s$5 is the new sample from stratum $N_s$6. The global intersection supermartingale is

$N_s$7

By construction, $N_s$8 under $N_s$9, and Ville's inequality immediately applies:

$X_{1s},\dots,X_{N_s s}\in[0,u_s]$0

Therefore, the first crossing time $X_{1s},\dots,X_{N_s s}\in[0,u_s]$1 such that $X_{1s},\dots,X_{N_s s}\in[0,u_s]$2 achieves strong control of the familywise Type I error for the intersection hypothesis.

3. Adaptive Stratum Selection and Efficiency

The validity of the global process for any predictable sampling rule enables adaptive allocation. Two canonical strategies are:

• Proportional Allocation: Sampling strata in proportion to their sizes ($X_{1s},\dots,X_{N_s s}\in[0,u_s]$3), a nonadaptive but valid rule.
• Lower-Sided Screening (Hard Stop): In parallel, a one-sided test is run for each stratum for the alternative $X_{1s},\dots,X_{N_s s}\in[0,u_s]$4, using reversed bets. When this lower-sided supermartingale exceeds $X_{1s},\dots,X_{N_s s}\in[0,u_s]$5, sampling for that stratum ceases; subsequent $X_{1s},\dots,X_{N_s s}\in[0,u_s]$6 selections are restricted to remaining strata.

The lower-sided screening approach can reduce expected sample size by up to approximately $X_{1s},\dots,X_{N_s s}\in[0,u_s]$7 in illustrative two-stratum cases. Once a stratum ceases to provide evidence for $X_{1s},\dots,X_{N_s s}\in[0,u_s]$8, it is efficiently excluded from further sampling (Spertus et al., 2022).

4. Empirical Results and Case Studies

Application to both real-world and simulated audits demonstrates marked efficiency gains. In the 2018 Whitmer–Schuette race pilot audit in Kalamazoo, MI (two strata: CVR $X_{1s},\dots,X_{N_s s}\in[0,u_s]$9, no-CVR $H_{0s}: \mu_s \leq \theta_s$0), the SUITE approach yielded a $H_{0s}: \mu_s \leq \theta_s$1-value of $H_{0s}: \mu_s \leq \theta_s$2. Re-analysis with intersection supermartingales reduced this to approximately $H_{0s}: \mu_s \leq \theta_s$3, representing a more than order-of-magnitude reduction in measured risk, even after conditioning on observed samples.

In a large-scale simulation across 58 California counties (total $H_{0s}: \mu_s \leq \theta_s$4), employing countywise Empirical-Bernstein supermartingales combined by the intersection product, sampling proportional to county turnout and stopping at $H_{0s}: \mu_s \leq \theta_s$5 with $H_{0s}: \mu_s \leq \theta_s$6, $H_{0s}: \mu_s \leq \theta_s$7 of $H_{0s}: \mu_s \leq \theta_s$8 replicates completed after $H_{0s}: \mu_s \leq \theta_s$9 draws. The global $\theta_s\in[0,u_s]$0-value computation, requiring a small LP (since the EB log-$\theta_s\in[0,u_s]$1 is linear in county means), took approximately $\theta_s\in[0,u_s]$2 seconds per instance on commodity hardware (Spertus et al., 2022).

5. Practical Algorithm and Implementation

The intersection supermartingale audit can be expressed concisely as:

$\tau_{i-1,s}$1

The function compute_factor is required to ensure $\theta_s\in[0,u_s]$3 under $\theta_s\in[0,u_s]$4, guaranteeing the supermartingale property. For numerical safety, truncation of $\theta_s\in[0,u_s]$5-factors is allowed provided it only decreases the value, preserving the error control.

A valid omnibus $\theta_s\in[0,u_s]$6-value at time $\theta_s\in[0,u_s]$7 is $\theta_s\in[0,u_s]$8, forming an anytime-valid $\theta_s\in[0,u_s]$9-process.

6. Theoretical Guarantees and Connections

Intersection supermartingale products deliver Type I error control for global intersection nulls, generalizing classical sequential tests. The supermartingale property is robust to arbitrary predictable sampling rules, encompassing both non-adaptive and adaptive stratum selection. This universal validity supports optimal resource allocation in practical RLAs and ensures computational tractability even in high-dimensional stratification (e.g., many counties).

The intersection product structure provides a rigorous foundation for union–intersection testing in stratified designs and generalizes to any family of one-sided hypotheses with stratumwise NNSMs. The compatibility with Ville’s inequality and anytime $\tau_{i-1,s}$0-value construction supports efficient, sequential evidence accumulation. A plausible implication is that this approach could transfer to other multi-group sequential analysis tasks in statistics or machine learning involving stratified or dependent data (Spertus et al., 2022).

7. Related Methodologies and Generalizations

Intersection supermartingale products substantially synergize techniques from SHANGRLA, SUITE, and ALPHA. They inherit the computational tractability and modularity of SHANGRLA’s reduction to mean testing, the union-intersection logic of SUITE, and the nonparametric flexibility of ALPHA’s NNSM-based construction. Key references include Stark (2022) on intersection supermartingales for union–intersection tests, Waudby‐Smith et al. (2021) and Stark (2020) on betting martingales and the SHANGRLA framework.

The underlying principles are not reliant on specifics of ballot sampling and are extensible to other contexts—anywhere nonnegative supermartingale tests for intersection nulls, stratified adaptivity, or anytime error control are vital. This suggests broad applicability wherever composite hypotheses are tested under sequential, adaptive, or stratified sample collection regimes.

References:

• "Sweeter than SUITE: Supermartingale Stratified Union-Intersection Tests of Elections" (Spertus et al., 2022)
• Stark (2022), “Intersection supermartingales for union–intersection tests,” Stat Sci.
• Waudby-Smith et al. (2021), “Betting martingales for RLAs.”
• Stark (2020), “SHANGRLA: A framework for RLAs.”