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eduKateSG · Why Mathematics? · Integrated Brier Score for Reliability Survival Curves: Time Integration, IPCW Error, Horizons and Model Comparison
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 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes practical when a reliability model predicts an entire survival curve and performance must be summarised across a useful time window. The Integrated Brier Score, or IBS, aggregates time-specific squared probability errors rather than judging one chosen horizon. With right censoring, each Brier score needs a valid censoring adjustment before numerical integration.
This guide develops Brier-score curves, integration limits, trapezoidal quadrature, time weights, censoring support, benchmarks and validation. If Brier scores are 0.10, 0.14 and 0.18 at 0, 1 and 2 time units, trapezoidal integration over the two-unit interval gives IBS 0.14. Lower is better under the same definition. IBS is an average predictive-loss summary, not a calibration curve, ranking probability, causal claim or maintenance decision rule.
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 21–25 · Study and verify
A model may be accurate early and weak late, so one horizon can misrepresent its curve-wide behaviour. Brier loss rises from .10 to .18 over two time units. This is the central mathematical mechanism in why integrate over time. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. An average still hides local peaks. Plot the full loss curve. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind the time-specific brier score. At each t, squared error compares I(T greater than t) with predicted survival S-hat(t). In the illustrative case, A failure by t has outcome zero. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Early censoring makes status unknown. Use a censoring-aware estimator.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat the integration interval as a model-checking problem rather than a vocabulary exercise. IBS must declare lower and upper limits that reflect operations and data support. Integrate from 0 to 1,000 hours. Recompute the example from the raw values and label every unit. Different intervals answer different questions. Pre-specify tau.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of normalising by interval length becomes useful only when its mechanism is explicit. A common IBS convention divides the integrated loss by the interval length, producing a time average. Consider the worked illustration: Area .28 over length 2 gives .14. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Some software uses other weights. Report the formula.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://doi.org/10.1002/(SICI)1097-0258(19990915/30)18:17/18%3C2529::AID-SIM274%3E3.0.CO;2-5. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use a three-point example to practise sensitivity analysis. Use BS(0)=.10, BS(1)=.14 and BS(2)=.18 on equally spaced times. Two trapezoids have areas .12 and .16. Before accepting the result, identify the quantity that would change the conclusion most. Three points are instructional, not adequate numerical integration. Show each trapezoid.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for trapezoidal quadrature. For adjacent times, approximate area by width times the average of endpoint losses. The initial example is: (.10+.14)/2 times 1=.12. Then create a nearby case where the usual shortcut becomes unreliable. Coarse grids miss sharp changes. Refine the grid.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect completing the calculation to a real decision while protecting the boundary between calculation and judgement. Add .12 and .16 to get .28, then divide by 2. IBS=.14. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Rounding before integration can change close comparisons. Keep full precision.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Each trapezoid must use its actual width. Times 0,1,3 give widths 1 and 2. This is the central mathematical mechanism in unequal time spacing. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A simple arithmetic mean of scores weights grid density. Use time-aware quadrature. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind time-weighted variants. A chosen weighting function can emphasise operationally important periods. In the illustrative case, Late-life failures receive more weight. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Changing weights changes the estimand. Name and justify weights.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat ipcw at each time as a model-checking problem rather than a vocabulary exercise. Right-censored data commonly use inverse probability of censoring weights separately at each t. A unit observed through t contributes with 1/G(t). Recompute the example from the raw values and label every unit. Unstable G creates unstable tail loss. Inspect weight curves.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of conditional censoring becomes useful only when its mechanism is explicit. When censoring depends on measured covariates, G may need conditional estimation. Consider the worked illustration: High-load machines leave observation earlier. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Unmeasured dependence remains a threat. Audit the observation process.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use administrative censoring to practise sensitivity analysis. Known study-end censoring can create special structure not well represented by pooled Kaplan–Meier G. Late entrants have shorter potential follow-up. Before accepting the result, identify the quantity that would change the conclusion most. Generic IPCW may be biased when covariates predict censoring time. Consider the administrative score.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://pubmed.ncbi.nlm.nih.gov/10474158/. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for choosing tau by support. The upper limit should stay inside a region with adequate event and censoring information. The initial example is: Few units remain observable after 1,200 hours. Then create a nearby case where the usual shortcut becomes unreliable. Integrating into an unsupported tail adds noise. Restrict tau.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect compare models fairly to a real decision while protecting the boundary between calculation and judgement. Models must share test data, interval, grid, censoring method and outcome definition. Model A .14 and B .16 use the same setup. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Different grids can reverse results. Use paired evaluation.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
A marginal survival curve provides a useful reference for interpreting absolute loss. Null IBS .18 versus model .14. This is the central mathematical mechanism in compare with a null model. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. The reference is population-specific. Report both scores. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind link to ipa. An integrated prediction-accuracy index can rescale model IBS against null IBS. In the illustrative case, 1-.14/.18=.2222. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. This inherits both loss estimates and interval choices. Keep absolute IBS visible.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat calibration and discrimination as a model-checking problem rather than a vocabulary exercise. IBS combines probability calibration and separation across time. A monotone score transformation may preserve ranking but alter IBS. Recompute the example from the raw values and label every unit. The score does not locate the source of error. Add calibration and AUC curves.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of internal validation becomes useful only when its mechanism is explicit. Training-set IBS is optimistic after model selection and tuning. Consider the worked illustration: A flexible curve follows training noise. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. IPCW does not correct overfitting. Resample the full pipeline.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use external validation to practise sensitivity analysis. Evaluate the frozen model over a prespecified interval in a new fleet. A supplier model is tested under field use. Before accepting the result, identify the quantity that would change the conclusion most. Case mix and censoring can change IBS. Describe transport conditions.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for uncertainty. Inference should repeat model fitting, censoring estimation and integration. The initial example is: A unit-level bootstrap recomputes the entire curve. Then create a nearby case where the usual shortcut becomes unreliable. Treating time points as independent is wrong. Use paired resampling.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect decision utility to a real decision while protecting the boundary between calculation and judgement. Low average loss does not encode intervention cost, capacity or harm. IBS .14 does not choose a threshold. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Good prediction may not improve outcomes. Add decision analysis.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Reproduce .12+.16=.28 and divide by two, then change the final loss to .30. The revised second trapezoid is .22 and IBS .17. This is the central mathematical mechanism in practice calculation. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. The toy grid is deliberately simple. Predict direction before computing. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind parent and educator guidance. IBS connects squared error, graphs, area, averages and numerical integration. In the illustrative case, Ask why unequal time gaps cannot use a plain mean. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Students may confuse area with final score. Draw shaded trapezoids.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat common misconceptions as a model-checking problem rather than a vocabulary exercise. IBS is not simply the average of arbitrarily sampled Brier scores, not calibration alone and not comparable across different intervals. Lower is better only under a shared definition. Recompute the example from the raw values and label every unit. Tail censoring can dominate. Report grid, tau and weights.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
Continue with the Mathematics Learning Hub, the eduKate Mathematics Learning System, the approach to learning mathematics, the Singapore Mathematics curriculum overview, the cross-subject learning approach and the previous reliability article. Each destination has its own role: subject navigation, learning method, school-to-advanced progression, or continuity with the latest reliability mechanism.
The mathematics of research and next reading becomes useful only when its mechanism is explicit. Graf and colleagues developed censoring-adjusted prediction-error measures that underpin time-integrated Brier evaluation. Consider the worked illustration: Read the primary paper beside the trapezoid worksheet. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Production use needs valid censoring support. Reproduce .14 first.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Quick answers
What should I calculate first? Start with the estimand or failure event, then a simple baseline.
Does the advanced method guarantee accuracy? No. It needs input, numerical and model checks.
Which school mathematics matters? Algebra, graphs, probability, functions and careful units all contribute.
How can a student practise safely? Use fictional or open benchmark data and preserve every intermediate value.
What makes a career claim responsible? Describe how the mathematics is used without promising admission, salary or employment.
Extended practice laboratory
Practice case 1: Why integrate over time
Setup: Use this chapter’s illustrative case: Brier loss rises from .10 to .18 over two time units.
Task: Plot the full loss curve.
Checking target: An average still hides local peaks.
Practice case 2: The time-specific Brier score
Setup: Use this chapter’s illustrative case: A failure by t has outcome zero.
Task: Use a censoring-aware estimator.
Checking target: Early censoring makes status unknown.
Practice case 3: The integration interval
Setup: Use this chapter’s illustrative case: Integrate from 0 to 1,000 hours.
Task: Pre-specify tau.
Checking target: Different intervals answer different questions.
Practice case 4: Normalising by interval length
Setup: Use this chapter’s illustrative case: Area .28 over length 2 gives .14.
Task: Report the formula.
Checking target: Some software uses other weights.
Practice case 5: A three-point example
Setup: Use this chapter’s illustrative case: Two trapezoids have areas .12 and .16.
Task: Show each trapezoid.
Checking target: Three points are instructional, not adequate numerical integration.
Practice case 6: Trapezoidal quadrature
Setup: Use this chapter’s illustrative case: (.10+.14)/2 times 1=.12.
Task: Refine the grid.
Checking target: Coarse grids miss sharp changes.
Practice case 7: Completing the calculation
Setup: Use this chapter’s illustrative case: IBS=.14.
Task: Keep full precision.
Checking target: Rounding before integration can change close comparisons.
Practice case 8: Unequal time spacing
Setup: Use this chapter’s illustrative case: Times 0,1,3 give widths 1 and 2.
Task: Use time-aware quadrature.
Checking target: A simple arithmetic mean of scores weights grid density.
Practice case 9: Time-weighted variants
Setup: Use this chapter’s illustrative case: Late-life failures receive more weight.
Task: Name and justify weights.
Checking target: Changing weights changes the estimand.
Practice case 10: IPCW at each time
Setup: Use this chapter’s illustrative case: A unit observed through t contributes with 1/G(t).
Task: Inspect weight curves.
Checking target: Unstable G creates unstable tail loss.