Target-Aware Sequential Inference: Pooled versus Stratified Anytime-Valid Designs

Abstract: Sequential studies with heterogeneous strata often target a weighted population mean while collecting data under a different, possibly adaptive allocation. This creates two distinct design choices: how observations are allocated and whether inference is performed directly for the target or by aggregating simultaneous stratum-level confidence sequences. We compare these architectures under anytime-valid inference. For a single prespecified target, direct target sampling yields one bounded pooled process. When simultaneous stratum-level reporting, post-hoc reweighting, or robustness over several targets is required, a stratified construction aggregates local confidence sequences. For local half-widths with power-law rate n raised to minus beta, we derive the asymptotically width-optimal allocation; its exponent is 1/(1+beta), and root-n variance-adaptive boundaries yield the 2/3 rule. We establish validity under predictable adaptive sampling, show oracle tracking under a vanishing exploration floor, extend the design to uncertain target distributions and familywise best-system identification, and quantify the first-order cost of unnecessary local multiplicity. Simulations and a public benchmark replay show two robust patterns: variance adaptation can matter more than fine allocation tuning, and pooled target-specific inference can reduce stopping cost dramatically when local simultaneous guarantees are not needed. The framework connects stratified sampling, confidence sequences, adaptive allocation, and ranking and selection through a common target-aware sequential design problem.