The Sequential Probability Ratio Test (SPRT), developed by Abraham Wald during World War II and published in 1947, is the mathematical ancestor of accumulator models of decision making. It provides an optimal solution to the problem of choosing between two hypotheses by sequentially sampling evidence until the accumulated log-likelihood ratio crosses one of two boundaries.
The SPRT Procedure
If Λₙ ≥ A: accept H₁ (upper boundary)
If Λₙ ≤ B: accept H₀ (lower boundary)
Otherwise: sample another observation
Boundaries: A ≈ log[(1−β)/α], B ≈ log[β/(1−α)]
Optimality and Connection to DDM
Wald and Wolfowitz proved that among all sequential tests with the same error rates α and β, the SPRT requires the fewest observations on average. This optimality result makes the SPRT the gold standard for sequential decision procedures. The Drift Diffusion Model is the continuous-time analog of the SPRT: the drift rate corresponds to the expected log-likelihood ratio per unit time, and the boundaries correspond to the SPRT thresholds A and B.
The connection between SPRT and the DDM provides a normative justification for accumulator models: under certain assumptions, the brain's decision mechanism implements an approximately optimal sequential test, trading off speed against accuracy by adjusting boundary separation.