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eduKateSG · Why Mathematics? · Right-Censored Log-Likelihood for Reliability Models: Density, Survival Contributions, Proper Scoring and Model Validation
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
The benefits of learning mathematics become concrete when a reliability model must be rewarded for assigning probability to what was actually observed. Under independent right censoring, an observed failure contributes its predicted event density f(t), while a censored record contributes its predicted survival probability S(c). The right-censored log-likelihood sums log f for events and log S for censorings.
This guide develops likelihood contributions, hazards, survival functions, log scores, propriety, censoring assumptions and validation. In a two-unit example, an event with density 0.08 and a censored unit with survival 0.70 give log-likelihood log(0.08)+log(0.70) ≈ −2.8824 and negative log-likelihood 2.8824. Rindt and colleagues proved that the right-censored log-likelihood is a proper scoring rule in their setting, while several popular censored alternatives are not. A good likelihood score still needs model checks, calibration diagnostics and decision analysis.
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
Likelihood measures how much probability density or survival mass a model assigns to the observations. A failure near a high-density time receives more support. This is the central mathematical mechanism in what a likelihood represents. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Density is not a probability at a point. Keep units explicit. 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 event contribution. An exactly observed failure at t contributes f(t) under the event-time model. In the illustrative case, Density .08 at the failure time enters the product. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Continuous densities carry inverse-time units. Use consistent time scale.
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 censoring contribution as a model-checking problem rather than a vocabulary exercise. A right-censored record at c contributes S(c), the probability of surviving beyond c. S(c)=.70 means the observation is compatible with seventy-percent survival beyond c. Recompute the example from the raw values and label every unit. Censoring is not a failure. Use survival rather than density.
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 indicator bookkeeping becomes useful only when its mechanism is explicit. With delta=1 for an event, the contribution is f(y)^delta S(y)^(1-delta). Consider the worked illustration: The indicator selects f or S. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Reversing delta corrupts the likelihood. Test one event and one censoring.
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://proceedings.mlr.press/v151/rindt22a.html. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use log-likelihood to practise sensitivity analysis. Taking logs converts products into sums: delta log f plus (1-delta) log S. Two records give log(.08)+log(.70). Before accepting the result, identify the quantity that would change the conclusion most. Zero predicted values yield minus infinity. Use numerically stable bounds.
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 a two-record example. Use f=.08 for the event and S=.70 for the censored unit. The initial example is: The likelihood is .056. Then create a nearby case where the usual shortcut becomes unreliable. The value has density units because one event contributes f. Compare only compatible models and units.
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 log calculation to a real decision while protecting the boundary between calculation and judgement. Natural logs are about -2.5257 and -.3567. Their sum is -2.8824; NLL is 2.8824. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Rounding can affect close models. 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.
Because f(t)=h(t)S(t), an event contribution can be written as hazard times survival. High hazard alone is insufficient if survival to t is tiny. This is the central mathematical mechanism in hazard-survival form. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Hazard and density are different. Check both components. 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 why logs help. Summing logs avoids numerical underflow from multiplying many small contributions. In the illustrative case, A thousand probabilities would underflow directly. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Log scale changes interpretation. Report average NLL when useful.
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 proper scoring rule meaning as a model-checking problem rather than a vocabulary exercise. A proper score has optimal expected value at the true predictive distribution under its assumptions. Systematic false distributions should not improve expected log score. Recompute the example from the raw values and label every unit. Finite samples can still rank incorrectly. Separate expectation from one dataset.
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 right-censoring assumptions becomes useful only when its mechanism is explicit. Event and censoring processes need the independence conditions used to factor out the censoring mechanism. Consider the worked illustration: Administrative closure independent of latent failure can qualify. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Informative withdrawal violates the simple form. State the censoring model.
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 conditional independence to practise sensitivity analysis. Censoring may be independent only after conditioning on measured covariates. Site and workload explain follow-up. Before accepting the result, identify the quantity that would change the conclusion most. Unmeasured causes remain a threat. Justify the conditioning set.
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://proceedings.mlr.press/v151/rindt22a/rindt22a.pdf. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for event-time density validity. Parametric or flexible models must produce nonnegative densities integrating to one with coherent survival. The initial example is: A monotone survival curve yields f=-S prime when differentiable. Then create a nearby case where the usual shortcut becomes unreliable. Arbitrary outputs may not define a distribution. Validate normalisation.
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 monotone survival to a real decision while protecting the boundary between calculation and judgement. S(t) must stay within zero and one and never increase. S at later time cannot exceed earlier S. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Invalid curves can produce deceptive scores. Check constraints.
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.
Changing from hours to days changes density values and adds a Jacobian term to log likelihood. Density .08 per hour is not .08 per day. This is the central mathematical mechanism in time units and transformations. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Raw log scores across units are incomparable. Fix units and transformation. 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 average versus total score. Total log likelihood grows with sample size, so average negative log-likelihood aids same-design comparison. In the illustrative case, NLL 288 over 100 units averages 2.88. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Censoring mix still matters. Report sample composition.
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 compare models as a model-checking problem rather than a vocabulary exercise. Models should be scored on the same observations with the same time scale and censoring definition. Model A NLL 2.88 and B 3.10 use identical records. Recompute the example from the raw values and label every unit. Different support invalidates direct comparison. Check every finite contribution.
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 popular alternatives and propriety becomes useful only when its mechanism is explicit. Rindt and colleagues showed several commonly used right-censored scores, including studied versions of survival-CRPS and integrated Brier score, are not proper in their setting. Consider the worked illustration: An incorrect distribution may receive better expected alternative score. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. This does not make every practical use worthless. State theoretical scope.
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 internal validation to practise sensitivity analysis. Training log likelihood is optimistically high after fitting. A flexible density memorises event times. Before accepting the result, identify the quantity that would change the conclusion most. Propriety does not prevent overfitting. Use held-out or corrected performance.
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 external validation. Evaluate the frozen distribution on new units under compatible observation assumptions. The initial example is: A model trained on bench tests is scored in field data. Then create a nearby case where the usual shortcut becomes unreliable. Censoring and time support may differ. Describe the validation design.
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 calibration and diagnostics to a real decision while protecting the boundary between calculation and judgement. A strong average log score does not show where survival curves are miscalibrated or which covariates fail. Rare extreme predictions dominate NLL. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. One number hides failure modes. Add calibration curves and residuals.
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.
Multiply .08 by .70, then take the natural log or add the separate logs. Likelihood .056 gives log score -2.8824 and NLL 2.8824. 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. This tiny sample teaches bookkeeping only. Verify both routes agree. 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. The method connects products, logarithms, functions, derivatives and conditional observation. In the illustrative case, Ask why a censored unit contributes survival rather than event density. 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 call density a probability. Use interval probability examples.
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. Log likelihood is not calibration alone, a higher density is not always a better survival curve, and proper does not mean unbiased under violated censoring assumptions. Negative log-likelihood depends on time units. Recompute the example from the raw values and label every unit. One outlier can dominate. Report assumptions and diagnostics.
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. Rindt and colleagues analysed survival scoring rules and proved right-censored log-likelihood proper in their framework. Consider the worked illustration: Read the PMLR paper beside the two-record worksheet. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The event model must define a valid distribution. Reproduce -2.8824 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: What a likelihood represents
Setup: Use this chapter’s illustrative case: A failure near a high-density time receives more support.
Task: Keep units explicit.
Checking target: Density is not a probability at a point.
Practice case 2: Event contribution
Setup: Use this chapter’s illustrative case: Density .08 at the failure time enters the product.
Task: Use consistent time scale.
Checking target: Continuous densities carry inverse-time units.
Practice case 3: Censoring contribution
Setup: Use this chapter’s illustrative case: S(c)=.70 means the observation is compatible with seventy-percent survival beyond c.
Task: Use survival rather than density.
Checking target: Censoring is not a failure.
Practice case 4: Indicator bookkeeping
Setup: Use this chapter’s illustrative case: The indicator selects f or S.
Task: Test one event and one censoring.
Checking target: Reversing delta corrupts the likelihood.
Practice case 5: Log-likelihood
Setup: Use this chapter’s illustrative case: Two records give log(.08)+log(.70).
Task: Use numerically stable bounds.
Checking target: Zero predicted values yield minus infinity.
Practice case 6: A two-record example
Setup: Use this chapter’s illustrative case: The likelihood is .056.
Task: Compare only compatible models and units.
Checking target: The value has density units because one event contributes f.
Practice case 7: Completing the log calculation
Setup: Use this chapter’s illustrative case: Their sum is -2.8824; NLL is 2.8824.
Task: Keep full precision.
Checking target: Rounding can affect close models.
Practice case 8: Hazard-survival form
Setup: Use this chapter’s illustrative case: High hazard alone is insufficient if survival to t is tiny.
Task: Check both components.
Checking target: Hazard and density are different.
Practice case 9: Why logs help
Setup: Use this chapter’s illustrative case: A thousand probabilities would underflow directly.
Task: Report average NLL when useful.
Checking target: Log scale changes interpretation.
Practice case 10: Proper scoring rule meaning
Setup: Use this chapter’s illustrative case: Systematic false distributions should not improve expected log score.
Task: Separate expectation from one dataset.
Checking target: Finite samples can still rank incorrectly.