Choose your reading route
eduKateSG · Why Mathematics? · Sequential Probability Ratio Tests for Reliability Innovations: Log-Likelihood Evidence, Error Bounds, Stopping Rules and Detection Delay
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–24
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes concrete when a reliability system must decide as evidence arrives rather than wait for a fixed batch. The sequential probability ratio test (SPRT), developed by Abraham Wald, compares the cumulative likelihood of observations under two stated hypotheses. At every step it can favour the null hypothesis, favour the alternative, or continue sampling.
For a Bernoulli teaching example, let H0 have event probability p0=0.1 and H1 have p1=0.3. A success adds log(0.3/0.1)=1.0986 to the log-likelihood ratio; a failure adds log(0.7/0.9)=−0.2513. For observations 1,0,0,1, cumulative values are 1.0986, 0.8473, 0.5960 and 1.6946. With target error probabilities α=0.05 and β=0.10, Wald's approximate boundaries are log((1−β)/α)=log 18≈2.890 and log(β/(1−α))≈−2.251. The four observations cross neither boundary, so the correct action is to continue.
Did You Know? A sequential test includes “continue” as a formal decision. Waiting for more evidence can be the mathematically correct action rather than indecision.
This article answers one intent: The mechanism accumulates log-likelihood evidence and stops only when that evidence crosses a boundary tied to stated error goals.
Sequential testing makes the stopping rule part of the mathematics. The likelihood ratio, error targets and decision boundaries must be designed together. The worked Bernoulli path shows why “continue” is a real conclusion, then later chapters add Gaussian innovations, variance alternatives, dependence, truncation and operational decisions. Students transfer logarithms, probability and inequalities; practitioners transfer disciplined calibration and full evidence logging.
Full chapter index
Chapters 1–5 · Understand the question
Chapters 6–10 · Reproduce the example
Chapters 11–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–24 · Study and verify
Begin with the object being measured, because a detector name is less useful than the statistic it changes. Reliability evidence often arrives one measurement, inspection or event at a time. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
A monitor updates after every standardized innovation. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Repeated ordinary fixed-sample tests inflate error rates. Use a valid sequential design. Transfer the idea by changing one input at a time and explaining the direction before computing.
A useful audit begins by writing the claim in plain language, then identifying the observation, reference model and comparison rule. In this chapter, the central claim is: Reliability evidence often arrives one measurement, inspection or event at a time. Test it first on a short sequence where every intermediate value can be inspected. Then change one condition and explain why the conclusion does or does not survive.
The mathematical value appears when a pattern is connected to one reproducible calculation. Classical SPRT compares two simple hypotheses with fully specified data distributions. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
H0 sets p=0.1 and H1 sets p=0.3. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Composite or drifting alternatives need extensions. State models and parameter values. Keep raw evidence beside the transformed statistic so diagnosis remains possible.
For transfer, sketch a before-and-after case and label what is measured, what is predicted and what is inferred. The concrete case here is: H0 sets p=0.1 and H1 sets p=0.3. That example is illustrative rather than evidence about a particular school, learner or machine. A sound explanation keeps calculation, interpretation and action in separate sentences.
Separate the mechanism from the interpretation so a number does not become unsupported certainty. Multiply the probability of observed data under H1 and divide by its probability under H0. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
A success contributes 0.3/0.1=3. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Ratios can underflow across long runs. Work in logarithms. Compare with a simple baseline and an alternative cause before deciding.
Reliability work needs a counterexample as well as a successful example. Here the important boundary is: Ratios can underflow across long runs. Ask what other mechanism could create the same observed pattern, what additional measurement would separate the explanations, and how uncertainty should be recorded. This turns a formula into a defensible decision process.
Keep timing, units and assumptions visible before applying a threshold. Logarithms turn the product of evidence into an additive sequence. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Success adds 1.0986 and failure subtracts 0.2513. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
An increment is conditional on the assumed distribution. Derive it before coding. Explain the key equation aloud and point to the term carrying each meaning.
A student can practise this idea without specialised equipment. Start with a tiny table, reproduce the stated operation, and predict how the output changes when one entry is doubled, delayed or removed. The practical next step is: Derive it before coding. Explain the result aloud, including the assumption that would most change the answer.
Make the example auditable by predicting direction, calculating and checking a nearby case. Add each new increment to the running log ratio. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
The sequence reaches 1.6946 after four observations. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Correlation makes nominal increments miscalibrated. Check temporal assumptions. Report observation, assumption, calculation and authorised action as separate layers.
A useful audit begins by writing the claim in plain language, then identifying the observation, reference model and comparison rule. In this chapter, the central claim is: Add each new increment to the running log ratio. Test it first on a short sequence where every intermediate value can be inspected. Then change one condition and explain why the conclusion does or does not survive.
Begin with the object being measured, because a detector name is less useful than the statistic it changes. Calculate every step so direction and arithmetic can be audited. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Values 1.0986,0.8473,0.5960,1.6946 remain between boundaries. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Short paths can be highly variable. Report the whole path. Transfer the idea by changing one input at a time and explaining the direction before computing.
For transfer, sketch a before-and-after case and label what is measured, what is predicted and what is inferred. The concrete case here is: Values 1.0986,0.8473,0.5960,1.6946 remain between boundaries. That example is illustrative rather than evidence about a particular school, learner or machine. A sound explanation keeps calculation, interpretation and action in separate sentences.
| Step | Observation | Log increment | Cumulative log ratio |
|---|---|---|---|
| 1 | 1 | 1.0986 | 1.0986 |
| 2 | 0 | −0.2513 | 0.8473 |
| 3 | 0 | −0.2513 | 0.5960 |
| 4 | 1 | 1.0986 | 1.6946 |
The mathematical value appears when a pattern is connected to one reproducible calculation. Approximate Wald boundaries use chosen Type I and Type II error targets. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
With α=.05,β=.10 the limits are about 2.890 and −2.251. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Actual errors can differ because of overshoot and model mismatch. Validate operating characteristics. Keep raw evidence beside the transformed statistic so diagnosis remains possible.
Reliability work needs a counterexample as well as a successful example. Here the important boundary is: Actual errors can differ because of overshoot and model mismatch. Ask what other mechanism could create the same observed pattern, what additional measurement would separate the explanations, and how uncertainty should be recorded. This turns a formula into a defensible decision process.
Separate the mechanism from the interpretation so a number does not become unsupported certainty. Cross the upper boundary for H1, the lower for H0, otherwise continue. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
After observation four the action is continue. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Continuing indefinitely may be operationally impossible. Add a justified truncation rule. Compare with a simple baseline and an alternative cause before deciding.
A student can practise this idea without specialised equipment. Start with a tiny table, reproduce the stated operation, and predict how the output changes when one entry is doubled, delayed or removed. The practical next step is: Add a justified truncation rule. Explain the result aloud, including the assumption that would most change the answer.
Keep timing, units and assumptions visible before applying a threshold. Alpha concerns favouring H1 under H0; beta concerns favouring H0 under H1. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Different maintenance consequences can justify asymmetric targets. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Statistical errors are not identical to safety risk. Map them to decision costs. Explain the key equation aloud and point to the term carrying each meaning.
A useful audit begins by writing the claim in plain language, then identifying the observation, reference model and comparison rule. In this chapter, the central claim is: Alpha concerns favouring H1 under H0; beta concerns favouring H0 under H1. Test it first on a short sequence where every intermediate value can be inspected. Then change one condition and explain why the conclusion does or does not survive.
Make the example auditable by predicting direction, calculating and checking a nearby case. Sequential designs are judged partly by how many observations they require under each state. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
A strong shift can stop faster than a weak one. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Average sample size hides long tails. Report distributions and quantiles. Report observation, assumption, calculation and authorised action as separate layers.
For transfer, sketch a before-and-after case and label what is measured, what is predicted and what is inferred. The concrete case here is: A strong shift can stop faster than a weak one. That example is illustrative rather than evidence about a particular school, learner or machine. A sound explanation keeps calculation, interpretation and action in separate sentences.
Begin with the object being measured, because a detector name is less useful than the statistic it changes. The cumulative statistic usually passes beyond a boundary rather than landing exactly on it. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
A large sensor innovation can jump well above the upper limit. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Simple boundary formulas ignore exact overshoot. Use simulation or corrected approximations. Transfer the idea by changing one input at a time and explaining the direction before computing.
Reliability work needs a counterexample as well as a successful example. Here the important boundary is: Simple boundary formulas ignore exact overshoot. Ask what other mechanism could create the same observed pattern, what additional measurement would separate the explanations, and how uncertainty should be recorded. This turns a formula into a defensible decision process.
The mathematical value appears when a pattern is connected to one reproducible calculation. Set a maximum sample number and define a terminal decision rule. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
An inspection window may end after 100 observations. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Truncation changes error and delay properties. Recalibrate the full design. Keep raw evidence beside the transformed statistic so diagnosis remains possible.
A student can practise this idea without specialised equipment. Start with a tiny table, reproduce the stated operation, and predict how the output changes when one entry is doubled, delayed or removed. The practical next step is: Recalibrate the full design. Explain the result aloud, including the assumption that would most change the answer.
Separate the mechanism from the interpretation so a number does not become unsupported certainty. For Gaussian innovations, each increment follows from competing means or variances. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
A mean-shift alternative yields a linear function of the innovation. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Unknown shift size is not a simple hypothesis. Use a mixture or generalized method. Compare with a simple baseline and an alternative cause before deciding.
A useful audit begins by writing the claim in plain language, then identifying the observation, reference model and comparison rule. In this chapter, the central claim is: For Gaussian innovations, each increment follows from competing means or variances. Test it first on a short sequence where every intermediate value can be inspected. Then change one condition and explain why the conclusion does or does not survive.
Keep timing, units and assumptions visible before applying a threshold. Likelihood evidence can target a change in innovation variance. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Large squared standardized residuals favour a high-variance model. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Outliers can imitate sustained variance change. Combine persistence and robust checks. Explain the key equation aloud and point to the term carrying each meaning.
For transfer, sketch a before-and-after case and label what is measured, what is predicted and what is inferred. The concrete case here is: Large squared standardized residuals favour a high-variance model. That example is illustrative rather than evidence about a particular school, learner or machine. A sound explanation keeps calculation, interpretation and action in separate sentences.
Make the example auditable by predicting direction, calculating and checking a nearby case. Classical derivations rely on a specified sequential data model. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Whitened innovations better support independent-increment calculations. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Adaptive filters can change the distribution while monitoring. Freeze or model adaptation. Report observation, assumption, calculation and authorised action as separate layers.
Reliability work needs a counterexample as well as a successful example. Here the important boundary is: Adaptive filters can change the distribution while monitoring. Ask what other mechanism could create the same observed pattern, what additional measurement would separate the explanations, and how uncertainty should be recorded. This turns a formula into a defensible decision process.
Begin with the object being measured, because a detector name is less useful than the statistic it changes. Update only when a valid observation and likelihood model are available. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
A missing sample contributes no evidence, not a zero residual. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Elapsed time still affects fault consequences. Record time and sample index. Transfer the idea by changing one input at a time and explaining the direction before computing.
A student can practise this idea without specialised equipment. Start with a tiny table, reproduce the stated operation, and predict how the output changes when one entry is doubled, delayed or removed. The practical next step is: Record time and sample index. Explain the result aloud, including the assumption that would most change the answer.
The mathematical value appears when a pattern is connected to one reproducible calculation. A bank of tests can represent several fault types or magnitudes. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Tests target positive bias, negative bias and variance increase. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Multiple monitoring increases false-alarm opportunity. Calibrate the family. Keep raw evidence beside the transformed statistic so diagnosis remains possible.
A useful audit begins by writing the claim in plain language, then identifying the observation, reference model and comparison rule. In this chapter, the central claim is: A bank of tests can represent several fault types or magnitudes. Test it first on a short sequence where every intermediate value can be inspected. Then change one condition and explain why the conclusion does or does not survive.
Separate the mechanism from the interpretation so a number does not become unsupported certainty. Measure time from a known change to the stopping decision. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Small shifts usually require more evidence. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Delay depends on sampling rate and model separation. Report delay by scenario. Compare with a simple baseline and an alternative cause before deciding.
For transfer, sketch a before-and-after case and label what is measured, what is predicted and what is inferred. The concrete case here is: Small shifts usually require more evidence. That example is illustrative rather than evidence about a particular school, learner or machine. A sound explanation keeps calculation, interpretation and action in separate sentences.
Keep timing, units and assumptions visible before applying a threshold. A boundary crossing can trigger confirmation, inspection or safe-mode review. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Strong evidence for a bias model prioritises a sensor check. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
The test does not prove physical cause. Seek independent evidence. Explain the key equation aloud and point to the term carrying each meaning.
Reliability work needs a counterexample as well as a successful example. Here the important boundary is: The test does not prove physical cause. Ask what other mechanism could create the same observed pattern, what additional measurement would separate the explanations, and how uncertainty should be recorded. This turns a formula into a defensible decision process.
Make the example auditable by predicting direction, calculating and checking a nearby case. The topic connects probability, logarithms, recurrence and inequalities. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
A learner can reproduce the four cumulative values by hand. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Fast stopping is not guaranteed. Practise both directions. Report observation, assumption, calculation and authorised action as separate layers.
A student can practise this idea without specialised equipment. Start with a tiny table, reproduce the stated operation, and predict how the output changes when one entry is doubled, delayed or removed. The practical next step is: Practise both directions. Explain the result aloud, including the assumption that would most change the answer.
Begin with the object being measured, because a detector name is less useful than the statistic it changes. Specify hypotheses, increments, boundaries, dependence assumptions, maximum length and actions. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Unit tests include all-success, all-failure and alternating sequences. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Silent defaults make results unauditable. Archive every version. Transfer the idea by changing one input at a time and explaining the direction before computing.
A useful audit begins by writing the claim in plain language, then identifying the observation, reference model and comparison rule. In this chapter, the central claim is: Specify hypotheses, increments, boundaries, dependence assumptions, maximum length and actions. Test it first on a short sequence where every intermediate value can be inspected. Then change one condition and explain why the conclusion does or does not survive.
The mathematical value appears when a pattern is connected to one reproducible calculation. SPRT compares two stated likelihoods; CUSUM accumulates targeted shifts; GLR estimates unknown parameters. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
All three use sequential evidence differently. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Names are not interchangeable. Choose by uncertainty structure. Keep raw evidence beside the transformed statistic so diagnosis remains possible.
For transfer, sketch a before-and-after case and label what is measured, what is predicted and what is inferred. The concrete case here is: All three use sequential evidence differently. That example is illustrative rather than evidence about a particular school, learner or machine. A sound explanation keeps calculation, interpretation and action in separate sentences.
Separate the mechanism from the interpretation so a number does not become unsupported certainty. Optional stopping is valid only because the test and boundaries were designed for it. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Recompute the path after changing p1 to .2. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
Checking an ordinary p-value repeatedly is not SPRT. Explain the stopping rule first. Compare with a simple baseline and an alternative cause before deciding.
Reliability work needs a counterexample as well as a successful example. Here the important boundary is: Checking an ordinary p-value repeatedly is not SPRT. Ask what other mechanism could create the same observed pattern, what additional measurement would separate the explanations, and how uncertainty should be recorded. This turns a formula into a defensible decision process.
Keep timing, units and assumptions visible before applying a threshold. Wald gives the foundational sequential test; industrial-statistics work develops applications; official laboratory reports show monitoring use. The importance of mathematics here is that the statement becomes a quantity or logical test that another reader can reproduce.
Recent eduKate articles cover CUSUM, GLR and innovation diagnostics. Follow the order of operations, keep the time index and units, and check whether the direction makes sense before trusting the number.
No reference supplies one universal reliability threshold. Validate for the real process. Explain the key equation aloud and point to the term carrying each meaning.
A student can practise this idea without specialised equipment. Start with a tiny table, reproduce the stated operation, and predict how the output changes when one entry is doubled, delayed or removed. The practical next step is: Validate for the real process. Explain the result aloud, including the assumption that would most change the answer.
The nearest published companions are innovation whiteness tests, CUSUM monitoring, generalized likelihood-ratio fault detection and multiple-model adaptive estimation. The eduKateSG Mathematics Learning Hub, eduKateSingapore Mathematics World, eduKate Sengkang Mathematics Hub, Bukit Timah Tutor Mathematics Hub and eduKate Punggol Mathematics Pathway provide wider learning routes. Primary sources: https://doi.org/10.1214/aoms/1177731118, https://www.osti.gov/servlets/purl/976099, https://doi.org/10.2307/2983610.
Frequently asked questions
What does an SPRT compare?
It compares cumulative likelihood under two stated hypotheses as observations arrive.
What are the three possible actions?
Favour the alternative, favour the null, or continue sampling.
Do Wald boundaries guarantee exact error rates?
They are useful approximations; overshoot, truncation, dependence and model mismatch require validation.
Why use log likelihoods?
They turn products into sums and improve numerical stability.
Is repeated ordinary testing an SPRT?
No. A valid SPRT requires a sequential statistic and stopping boundaries designed together.
Extended practice laboratory
Practice case 1: Reproduce increments
Setup: Use p0=.1 and p1=.3.
Checking target: Confirm 1.0986 and −0.2513.
Practice case 2: Trace evidence
Setup: Use 1,0,0,1.
Checking target: Confirm cumulative values.
Practice case 3: Compute boundaries
Setup: Use α=.05,β=.10.
Checking target: Confirm 2.890 and −2.251.
Practice case 4: Reverse sequence
Setup: Use 0,0,0,0.
Checking target: Find the lower-bound timing.
Practice case 5: Stronger alternative
Setup: Set p1=.5.
Checking target: Inspect larger success evidence.
Practice case 6: Truncate
Setup: Stop at four samples.
Checking target: Define and assess a terminal rule.
Practice case 7: Correlate data
Setup: Generate clustered successes.
Checking target: Compare nominal calibration.
Practice case 8: Gaussian shift
Setup: Derive a normal-mean increment.
Checking target: Check units.
Practice case 9: Threshold sweep
Setup: Change α and β.
Checking target: Compare delay and errors.
Practice case 10: Decision memo
Setup: Report no crossing after four.
Checking target: Recommend continued sampling.