Statistical Constraints

1 INTRODUCTION

• Statistical constraints use inference to determine which assignments satisfy a statistical property at a prescribed significance level.They can identify parameter values consistent with samples or sample sets compatible with hypothetical distributions.Source: §1 · p. 1
• The paper introduces statistical constraints embedding the t-test and Kolmogorov-Smirnov test, with bound-consistency filtering algorithms for some constraints.Source: §1 · p. 1
• Applications range from standard statistical problems to inspection scheduling for plans with desirable statistical properties.Source: §1 · p. 1

2.1 Statistical inference

• A statistical model represents possible CDFs, and inference seeks which model element likely generated observed data.The background distinguishes parametric models ⟨Θ, g, Ω⟩ from non-parametric models ⟨D, Ω⟩.Source: §2.1 · p. 1
• The background excludes semi-parametric models for brevity, and surveys the Student’s t-test and Kolmogorov-Smirnov test.Source: §2.1 · p. 1
• Hypothesis testing compares a null hypothesis with observed data using a test statistic and p-value against significance level α.A p-value below α leads to rejection; at or above α, the evidence is insufficient to reject the null hypothesis.Source: §2.1 · p. 1

2.1.1 Student’s t-test

• The one-sample t-test compares a sample mean with a specified mean, while the two-sample version compares means assuming equal variances.The two-sample case permits different sample sizes; other statistics may apply under different assumptions, such as unequal variances.Source: §2.1.1 · p. 2

2.1.2 Kolmogorov-Smirnov test

• The one-sample KS test compares a sample with a reference CDF on continuous support; its statistic measures the greatest distance between empirical and target CDFs.The background also describes two-sample testing and one-tailed tests for stochastic dominance.Source: §2.1.2 · p. 2

2.2 Constraint programming

• Constraint programming represents a problem as variables, domains, and constraints; filtering prunes infeasible values, while search explores assignments.Propagation repeats filtering until no more values can be pruned.Source: §2.2 · p. 2

3 STATISTICAL CONSTRAINTS

• A statistical constraint embeds a parametric or non-parametric model and significance-level test; an assignment satisfies it when the test does not reject its null hypothesis.Source: §3 · p. 3
• Unlike classical testing, the approach models sample variates as decision variables while leaving the sample-generating random variables unmodelled.The paper says this modelling strategy opens the way to novel applications.Source: §3 · p. 3

3.1.1 Student’s t test constraint

• The Student’s t-test constraint tests hypotheses about a sample mean or compares means across two samples, while effective filtering strategies remain future work.Source: §3.1.1 · p. 3

3.1.2 Parametric Kolmogorov-Smirnov constraint

• The parametric KS constraint tests sample variates against a parameterized distribution, with this section focusing on the exponential distribution and monotonic bound propagation.Its test parameter w selects the hypothesis, and the constraint holds when the one-sample KS test does not reject that null hypothesis.Source: §3.1.2 · p. 3

3.2 Non-parametric statistical constraint

• The paper introduces a non-parametric version of the Kolmogorov-Smirnov constraint for statistical modelling.Source: §3.2 · p. 3

3.2.1 Non-parametric Kolmogorov-Smirnov constraint

• Non-parametric KS uses two decision-variable samples; w selects a dominance or same-distribution test, and the constraint supports bound propagation.It is satisfied when the two-sample KS test does not reject the null hypothesis selected by w.Source: §3.2.1 · pp. 3–4

4.1 Classical problems in statistics

• The section presents a parametric t-test example and a nonparametric KS-test example as applications of statistical constraints.The examples address estimating plausible means and finding sample sets compatible with a reference distribution, respectively.Source: §4.1 · pp. 4–5
• At α = 0.05, propagating the t-test constraint reduces the candidate mean domain to {8, 9, 10, 11}.The example uses the sample {8, 14, 6, 12, 12, 9, 10, 9, 10, 5}; the decision variables are ground in this instance.Source: §4.1 · p. 4, Figure 1 · p. 4
• For the two-sample KS example, 365 of 6561 candidate sets are rejected at α = 0.05.The candidate sets come from eight variables with three values each; the figure contrasts an infeasible and a feasible set against the reference empirical CDF’s confidence band.Source: §4.1 · pp. 4–5, Figure 3 · p. 5

4.2 Inspection scheduling

• The inspection model combines a cumulative constraint with per-unit KS tests of whether inspection intervals follow exponential(λ).Decision variables represent inspection start times, end times, durations, inspector requirements, and intervals between consecutive inspections.Source: §4.2 · p. 5
• The feasible inspection plan’s unit-1 interval CDF lies fully within the exponential(λ) confidence bands at significance level α.The plan covers 10 units over 365 days; side constraints cap consecutive-inspection intervals at M days and require the final inspection in the last month.Source: §4.2 · p. 5, Figure 6 · p. 6

4.3 Further application areas

• The paper identifies regression analysis, distribution fitting, experimental design, quality management, and supply-chain coordination as further application areas.One proposed supply-chain use is to find a production schedule whose ordering process mimics a specified stochastic process while meeting other technical constraints.Source: §4.3 · p. 6

5 RELATED WORKS

• Unlike stochastic constraint programming, statistical constraints handle distributions that are partially specified or unspecified and model random variates as decision variables; unlike SPREAD, they include a significance level.The paper relates its approach to, stochastic constraint programming, and SPREAD.Source: §5 · p. 6

• Statistical constraints link statistical inference and constraint programming, a bridge described as new to the literature.Source: §6 · p. 6
• Constraint programming’s declarative nature can identify assignments that feature specific statistical properties.Source: §6 · p. 6
• The work presents filtering algorithms enforcing bound consistency and applications to statistical problems and inspection scheduling.The scheduling application seeks inspection plans with desirable statistical properties.Source: §6 · p. 6