Contribution
It names the sequential probability ratio test and gives its theory: when it stops, how its boundaries relate to the error probabilities, its operating characteristic, and the distribution and mean of its sample size. Against any other test of the same strength, it is optimal up to the overshoot. Before it, sequential procedures were double or multiple sampling schemes (Dodge and Romig, Bartky) with no optimality theory. After it, a test of a simple hypothesis against a simple alternative needs, on average, about half the observations of the Neyman–Pearson test with the same α and β.
Key insight
Accumulate the log likelihood ratio Z_m = Σ log f₁(xᵢ)/f₀(xᵢ) and stop when it leaves (log B, log A). The boundaries are set by the error rates alone, because on every sample path that accepts H₁ the ratio is at least A, so P₁(accept H₁) ≥ A·P₀(accept H₁). No sampling distribution has to be computed. “Another surprising feature of the sequential probability ratio test is that the test can be carried out without determining any probability distributions whatsoever” (p. 119). The motivation is Bayesian (§3.1). With a prior, a rule that stops when either posterior passes a level d > ½ is a pair of likelihood-ratio thresholds that do not depend on m. Dropping the prior gives the SPRT.
Assumptions
- Two simple hypotheses, densities (or probability functions) f₀ and f₁ of one random variable X (§1). Part II reduces one-sided and composite problems to this case, or to mixtures over a weight function.
- For the definition and the error inequalities (3.18)–(3.19): only that the test terminates with probability one (3.11). “The independence of the observations is, however, not necessary for the validity of (3.18) and (3.19)” (p. 130), with an autoregressive example.
- For the operating characteristic, the sample size and the optimality theorem: independent, identically distributed observations, so that zᵢ = log f₁(xᵢ)/f₀(xᵢ) are i.i.d. with nonzero variance, and A > 1 > B (α + β < 1).
- The practical formulas neglect the excess of Z_n over the boundary at stopping: “If both | Ez | and the standard deviation of z are small”, which “will be the case when f₁(x) is near f₀(x)” (pp. 135, 143).
Key results
- Definition (3.8)–(3.10). Accept H₁ if p₁ₘ/p₀ₘ ≥ A, accept H₀ if p₁ₘ/p₀ₘ ≤ B, take another observation if B < p₁ₘ/p₀ₘ < A, with 0 < B < A. The test “has been justified here merely on an intuitive basis” (p. 127), from the posterior thresholds (3.1)–(3.7).
- Termination (3.11). It stops with probability one under either hypothesis, by Lemma 1 of Wald’s “On cumulative sums of random variables” (1944), cited and not reproved.
- The fundamental inequalities (3.16)–(3.19). 1 − β ≥ Aα and β ≤ B(1 − α), that is α/(1 − β) ≤ 1/A and β/(1 − α) ≤ B. Hence α ≤ 1/A, β ≤ B (3.20)–(3.21).
- Setting the boundaries (3.24)–(3.30). The exact boundaries satisfy A(α, β) ≤ (1 − β)/α and B(α, β) ≥ β/(1 − α). With A = (1 − β)/α and B = β/(1 − α), the realized errors satisfy α′ ≤ α/(1 − β), β′ ≤ β/(1 − α) and α′ + β′ ≤ α + β, so “at most one of the probabilities α and β may be increased” (p. 132). The gap is “merely on account of the fact that the number m of observations is discontinuous”. “If fractional observations were possible … A(α, β) and B(α, β) would be exactly equal to (1 − β)/α and β/(1 − α) respectively” (p. 132).
- Operating characteristic (3.32)–(3.35). Let h ≠ 0 solve E e^{hz} = 1. Then E e^{hZ_n} = 1 (3.32), the probability γ of accepting H₁ is (1 − E*)/(E** − E*) exactly (3.34), and γ ≈ (1 − B^h)/(A^h − B^h) (3.35). “It is easy to verify that h = 1 if H = H₀, and h = −1 if H = H₁.” The paper gives bounds (3.42), (3.44), and an exact solution when z is a lattice variable (3.45)–(3.48).
- Expected sample size (4.4)–(4.14). EZ_n = En·Ez (4.4), so En = [(1 − γ)E*Z_n + γE**Z_n]/Ez (4.7) and En ≈ [(1 − γ) log B + γ log A]/Ez (4.8), with bounds through the maximal expected overshoot ξ, ξ′ (4.9)–(4.14).
- Savings for a normal mean (§4.3, Table 1). With E₁(z) = E₀(−z) = ½(θ₀ − θ₁)², the ratios E_i(n)/n(α, β) do not depend on θ₀, θ₁, and for α, β from .01 to .05 the saving is 47–63%: “at least 47 per cent” (p. 148).
- Distribution of n (§4.4). The fundamental identity E{e^{Z_n t}[φ(t)]^{−n}} = 1 (4.35) gives the characteristic function of n. For normal z with one boundary, m = (Ez)²n/2σ_z² has the density c/(2Γ(½)m^{3/2}) e^{−c²/4m − m + c} with c = (Ez) log A/σ_z² (4.51). The paper notes that a density for a discrete n is an artefact of neglecting the excess.
- Termination and truncation (§§4.5–4.6, Tables 2–3). Lower bounds on the probability of stopping by n₀. Upper bounds on the errors when the test is cut off at n₀ and decided by the sign of Z_{n₀}.
- Optimality (§4.7). Theorem: for any sequential test S of strength (α, β) that terminates with probability one, E₀(n|S) ≥ [(1 − α) log(β/(1 − α)) + α log((1 − β)/α)]/E₀(z) (4.80) and E₁(n|S) ≥ [β log(β/(1 − α)) + (1 − β) log((1 − β)/α)]/E₁(z) (4.81). The SPRT attains both when the excess is zero, so it is then “exactly an optimum test”, and otherwise “very nearly” one (p. 157). Footnote 12: “The author conjectures that the sequential probability ratio test is exactly an optimum test even if the excess of Z_n over the boundaries is not zero. However, he did not succeed in proving this.”
- Part II. For one-sided alternatives, test θ₀ against a chosen θ₁ and rely on a monotone operating characteristic (§5.1). The binomial plan has acceptance and rejection numbers linear in m (5.9)–(5.10). Double dichotomies are tested exactly through the unlike pairs (0, 1) and (1, 0), whose conditional law depends only on the odds ratio u (5.17)– (5.24), and the test stays valid when success rates drift from trial to trial alike in both series. The normal mean with known σ has h = (θ₁ + θ₀ − 2θ)/(θ₁ − θ₀) (5.37). §6 replaces the alternative by a weight-function mixture (6.1), so the SPRT bounds a weighted average of the second-kind errors (6.5). With spherical weights it gives a test on Σx̄ⱼ² (6.10)–(6.14), and for a normal mean with unknown variance a ratio that is homogeneous of degree zero in the observations, so free of scale (6.22)–(6.26). “There are several unsolved problems in this case” (p. 119).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | The SPRT stops with probability one | strong | cited from Wald 1944, Lemma 1 |
| C2 | α/(1 − β) ≤ 1/A and β/(1 − α) ≤ B for any terminating SPRT, independence not required | strong | (3.12)–(3.19), a two-line proof; re-derived in THEORY-952 |
| C3 | Boundaries (1 − β)/α and β/(1 − α) give α′ + β′ ≤ α + β, and neither error can rise much | strong for the sum and for α′ ≤ α/(1 − β); “for all practical purposes” for the rest | (3.26)–(3.30); re-derived and simulated |
| C4 | Neglecting the excess, P(accept H₀) = (A^h − 1)/(A^h − B^h) and En = [(1 − γ) log B + γ log A]/Ez | strong as approximations; exact without excess | (3.32)–(3.35), (4.4)–(4.8), from Wald 1944; re-derived in THEORY-951 |
| C5 | No sequential test of the same strength has smaller E₀(n) or E₁(n) than (4.80)–(4.81), and the SPRT attains them without excess | strong | §4.7 theorem, proved here; re-derived |
| C6 | The SPRT is exactly optimal even with excess | weak (conjecture) | footnote 12 |
| C7 | For a normal mean it saves 47–63% of observations for α, β in .01–.05 | strong within the approximation | Table 1, from (4.30)–(4.34); reproduced |
| C8 | The procedure extends to one-sided and composite hypotheses through monotone OC curves and weight functions | moderate | Part II; the minimax choice of weights is left as a calculus-of-variations problem |
Concepts
- Sequential test: a partition of each m-dimensional sample space into accept-H₀, accept-H₁ and continue regions R⁰ₘ, R¹ₘ, Rₘ (§2.1).
- Strength (α, β): the two error probabilities. Tests of equal strength are compared by E₀(n) and E₁(n).
- Optimum test: admissible, and minimizing max[E₀(n), E₁(n)] among tests of equal strength (§2.2). Efficiency is the ratio to that minimum.
- Cylindric point of type i: a finite sample path on which the test first accepts Hᵢ (§3.2).
- Operating characteristic L: the probability of accepting H₀ as a function of the true distribution (§§3.4, 5.2.4).
- Current test procedure: the fixed-sample most powerful test.
Connections
Its predecessors, by the Historical Note, are Dodge and Romig’s double sampling (1929), Bartky’s multiple sampling for a binomial mean (1943) and Hotelling’s chain experiments. Friedman and Wallis posed the problem in March 1943, and Wald devised the test in April 1943. Barnard found a sequential test of double dichotomies independently, and it “differs from the one obtained by applying the sequential probability ratio test” (p. 121). The test was first issued in a restricted report (September 1943), and the OC and sample-size theory rest on Wald’s “On cumulative sums of random variables” (1944). In the record, Bogacz et al. (LIT-569, NOTE-461) restate it as the test “of Barnard and Wald” that is fastest for a given error rate, with the drift-diffusion model as its continuum limit, and THEORY-068 reads that limit.
Bearing on the record
- THEORY-952 (the boundaries and error inequalities) and THEORY-951 (OC, sample size and optimality, and their coincidence with Brownian motion with drift) are filed from this reading.
- THEORY-068 and LIT-569. “Fastest for a given error rate” is, in this paper, a theorem only up to the overshoot (§4.7), with exact optimality a conjecture (footnote 12). The record’s statement through LIT-569 should be read that way. The continuum limit is not in Wald. What is in Wald is that the error bounds become equalities “if fractional observations were possible” (p. 132), and that the overshoot-free OC, sample size and sample-size density, (3.35), (4.8) and (4.51), are exactly those of Brownian motion with drift (re-derived in THEORY-951). That is the sense in which the drift-diffusion model is the SPRT’s continuum limit.
- CASE-155, note of 2026-10-25, and CLAIM-453. “Wald’s sequential probability ratio walk” holds as Wald defines it, with two provisos. (1) The walk Z_m = Σ log f₁/f₀ is the SPRT’s statistic. The SPRT is the walk with fixed boundaries, so the identification needs commitment as a bound on the log odds, which CLAIM-453 already states as a condition. (2) The case’s log odds start at the prior odds, not at zero. Wald’s §3.1 covers that exactly: the posterior log odds are log(g₁/g₀) + Z_m, and posterior thresholds are likelihood-ratio thresholds shifted by the prior. The case’s “drift is half the variance” for Gaussian evidence is Wald’s h = −1 under H₁ (p. 135). For normal z, h = −2Ez/σ_z² (4.39), so Ez = σ_z²/2.
- CLAIM-262 (the owner’s continuous statement) and CASE-160 need nothing from Wald beyond the DDM limit, which they take from THEORY-068.
The proposed edits to those entries are written for the consolidator, not made here. No instruction for machine-learning practice, so nothing for the anthology.
Limitations
- Optimality is proved only without overshoot. It holds exactly when z takes only the values ±d and log A, log B are multiples of d, and approximately when f₁ is near f₀. The general case is conjectured (footnote 12).
- The working formulas are approximations. The OC and the sample size are exact only through the conditional expectations E*, E** and E*Z_n, E**Z_n, which need the overshoot distribution. Bounds are given, and for the normal case the paper’s own bounds may be loose.
- The truncation bounds are loose by the author’s own account: “It would be desirable to find closer limits” (p. 154).
- Simple hypotheses. Composite problems are reduced to simple ones with a choice of θ₁ or of weight functions. The minimax choice of weights is left open except in symmetric cases.
- Key lemmas are elsewhere. Termination, the fundamental identity and the bounds on E*, E** are cited from the 1944 cumulative-sums paper.
Open questions
- Is the SPRT exactly optimal with overshoot? Footnote 12 conjectures so. It was later proved by Wald and Wolfowitz (1948), a work the record does not hold.
- How should weight functions be chosen for composite alternatives in general (§6)?
- Tighter bounds on the errors of the truncated test (§4.6).