A statistician has access to multiple data streams and must decide at each step which one to sample from. The goal is to test a global null hypothesis—all sources are null—against the alternative that at least one is non-null, without knowing in advance which stream carries the signal. Traditional sequential testing struggles here because optimal allocation requires knowing where the evidence is, but discovering where the evidence is requires sequential testing.
The betting framework solves this by tracking wealth rather than p-values. At each step, you wager on outcomes from the chosen data source. Winnings compound across selections. The test rejects when accumulated wealth crosses a threshold. The remarkable result is that this approach matches oracle performance—it's log-optimal and matches expected rejection time bounds even without knowing which arm provides the strongest evidence. The wealth process integrates evidence across sources automatically.
The technical innovation involves a modified upper-confidence-bound algorithm applied to rewards that aren't directly observable but can be estimated well enough to guide allocation. You don't need to see the reward to bet on it; you need to estimate it well enough to know where to look next. New concentration inequalities for Kelly optimal growth rates make this formal.
This reframes sequential testing as portfolio management. Hypotheses become investments, data sources become assets, evidence becomes return. The unified metric—wealth growth—handles heterogeneous information naturally because compounding doesn't care about the source. This matters beyond statistics: any problem involving selective attention across information channels with unknown value has this structure. Where to monitor, what to measure, which signal to track—wealth accumulation provides a single currency for evidence that doesn't require knowing the answer in advance.