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eduKateSG · Why Mathematics? · Intrinsic Bayes Factors for Reliability Models: Minimal Training Samples, Default Priors, Averaging Rules and Prediction
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 reliability models use standard noninformative priors that work for estimation but not for ordinary Bayes factors. Intrinsic Bayes factors construct valid comparisons through minimal training samples and an averaging rule. The training observations turn baseline priors into proper posteriors, while averaging across possible training samples reduces dependence on any single arbitrary subset.
In a teaching illustration, three training samples yield partial Bayes factors A:B of 2, 3 and 6. Their arithmetic mean is 11/3=3.667, while their geometric mean is (2×3×6)^(1/3)=36^(1/3)=3.302. These are different intrinsic-style summaries and must be labelled rather than mixed. This guide develops minimal training samples, propriety, averaging, censoring, computation, model priors and prediction.
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
Start from an eligible minimal training sample. Standard improper priors can support parameter estimation but leave marginal likelihoods arbitrary. A location parameter uses a flat baseline prior. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A proper posterior does not imply a defined Bayes factor. Separate estimation from comparison.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Save eligible training-sample rules, subset indices, averaging variant and factor direction. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use the default-prior problem as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Did You Know? Arithmetic and geometric intrinsic Bayes-factor variants can differ even when built from the same partial factors. That is why a careful report includes the construction, not only the final scalar.
Now calculate its partial factor. Use a small subset that makes each model-specific posterior proper. Several observed failures identify shape and scale. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. An arbitrary subset can favour one model. Consider all eligible subsets.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check posterior propriety and design rank under every candidate for every sampled subset. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use training-sample construction as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Averaging reduces dependence on one subset. Choose the smallest design that produces proper posterior distributions under all candidates. Two or more informative failures may be required. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. The minimum depends on model rank and censoring pattern. Verify propriety model by model.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Inspect influential training sets rather than reporting only the average. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use minimal training sample as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Start from an eligible minimal training sample. After training, compare how the remaining observations update the trained priors. One subset yields partial BF A:B=2. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A partial factor depends on the selected subset. Preserve its index.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat Monte Carlo sampling of large subset collections. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use partial bayes factors as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Primary research: https://doi.org/10.1080/01621459.1996.10476668. Read the authors’ definition and conditions before carrying the formula into a censored reliability model.
Now calculate its partial factor. Aggregate partial factors across eligible minimal samples. Three examples give 2, 3 and 6. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Averaging without naming the rule is ambiguous. State the variant.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep arithmetic, geometric and other intrinsic variants distinctly labelled. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use averaging over training samples as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Averaging reduces dependence on one subset. Add 2, 3 and 6 and divide by three. The arithmetic mean is 11/3=3.667. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A large factor can influence the arithmetic mean strongly. Inspect the distribution.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Save eligible training-sample rules, subset indices, averaging variant and factor direction. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use arithmetic intrinsic example as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Did You Know? Arithmetic and geometric intrinsic Bayes-factor variants can differ even when built from the same partial factors. That is why a careful report includes the construction, not only the final scalar.
| Quantity | Value |
|---|---|
| Partial factors A:B | 2, 3, 6 |
| Arithmetic mean | 3.667 |
| Product | 36 |
| Geometric mean | 3.302 |
Start from an eligible minimal training sample. Average log factors and exponentiate. The geometric mean is 36^(1/3)=3.302. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. The geometric and arithmetic variants are not interchangeable. Report which one is used.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check posterior propriety and design rank under every candidate for every sampled subset. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use geometric intrinsic example as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Now calculate its partial factor. Every partial factor must use the same A:B orientation before averaging. Reciprocal factors correspond to B:A. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Averaging a mixture of directions is meaningless. Label each ratio.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Inspect influential training sets rather than reporting only the average. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use direction of the factor as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Averaging reduces dependence on one subset. Training samples must contain enough failure information; censored rows alone may not identify lifetime parameters. A subset includes two failures and one censored unit. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Random minimal subsets can be invalid under heavy censoring. Define eligibility rules.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat Monte Carlo sampling of large subset collections. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use censored observations as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Start from an eligible minimal training sample. Regression reliability models need training subsets with full-rank covariate information. Two failures at identical stress levels cannot identify a slope. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Sample size alone does not guarantee propriety. Check matrix rank.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep arithmetic, geometric and other intrinsic variants distinctly labelled. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use design matrices and rank as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Now calculate its partial factor. Averaging aims to avoid dependence on observation ordering or one chosen subset. All eligible minimal samples receive a declared treatment. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Convenience sampling can reintroduce subjectivity. Document inclusion probabilities.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Save eligible training-sample rules, subset indices, averaging variant and factor direction. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use exchangeability and ordering as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Did You Know? Arithmetic and geometric intrinsic Bayes-factor variants can differ even when built from the same partial factors. That is why a careful report includes the construction, not only the final scalar.
Averaging reduces dependence on one subset. The number of minimal training samples can be enormous. Choose m records from n across many combinations. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Exhaustive averaging may be impossible. Use justified Monte Carlo sampling.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check posterior propriety and design rank under every candidate for every sampled subset. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use large combinatorial sets as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Second authoritative source: https://doi.org/10.1080/01621459.1996.10476668. Compare its prediction target, diagnostics and finite-sample cautions with this worked example.
Start from an eligible minimal training sample. Randomly sample eligible training sets and estimate the averaging functional. Ten thousand subsets approximate the geometric mean. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Rare influential subsets may be missed. Repeat and stratify.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Inspect influential training sets rather than reporting only the average. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use monte carlo over subsets as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Now calculate its partial factor. Each trained model can still require marginal-likelihood computation. Laplace or bridge sampling evaluates a partial factor. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Training-sample averaging does not remove numerical error. Propagate both uncertainties.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat Monte Carlo sampling of large subset collections. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use numerical integration as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Averaging reduces dependence on one subset. The intrinsic construction begins from specified standard noninformative priors. Location and scale baselines use declared measures. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Changing parameterisation can alter default constructions. State the measure.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep arithmetic, geometric and other intrinsic variants distinctly labelled. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use baseline prior measure as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Start from an eligible minimal training sample. Intrinsic factors can support nested or nonnested candidate models. Weibull and lognormal families are compared. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. The method does not prove one family is true. Keep the candidate set open.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Save eligible training-sample rules, subset indices, averaging variant and factor direction. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use nested and nonnested comparisons as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Did You Know? Arithmetic and geometric intrinsic Bayes-factor variants can differ even when built from the same partial factors. That is why a careful report includes the construction, not only the final scalar.
Now calculate its partial factor. Coherent posterior model probabilities require model priors and compatible pairwise calculations. Four lifetime families enter a model average. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Separate pairwise summaries may not normalise cleanly. Use a common reference model.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check posterior propriety and design rank under every candidate for every sampled subset. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use multiple models as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Averaging reduces dependence on one subset. Different minimality or eligibility definitions can change finite-sample evidence. Requiring at least one failure per stress level changes the pool. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Automatic does not mean assumption-free. Publish rule sensitivity.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Inspect influential training sets rather than reporting only the average. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use sensitivity to training rules as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Start from an eligible minimal training sample. A surprising failure inside a tiny training set can create an extreme partial factor. One early failure yields BF 20. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Blind averaging can hide influential subsets. Plot subset factors.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat Monte Carlo sampling of large subset collections. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use outlying training samples as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Now calculate its partial factor. Fractional factors use a likelihood power; intrinsic factors average explicit training-sample comparisons. Both address weak-prior model comparison. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. They distribute training information differently. Use them as complementary checks.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep arithmetic, geometric and other intrinsic variants distinctly labelled. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use relation to fractional bayes factors as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Averaging reduces dependence on one subset. The intrinsic approach can be interpreted through priors intended for model comparison and prediction. Posterior predictions average over candidate lifetime families. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Good default evidence does not guarantee field calibration. Validate predictions.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Save eligible training-sample rules, subset indices, averaging variant and factor direction. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use prediction and intrinsic priors as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Did You Know? Arithmetic and geometric intrinsic Bayes-factor variants can differ even when built from the same partial factors. That is why a careful report includes the construction, not only the final scalar.
Start from an eligible minimal training sample. Save eligible-set rules, sampled subsets, seeds, integration settings and every partial factor. Another analyst reproduces the arithmetic mean. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A final average alone cannot be audited. Archive subset-level output.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check posterior propriety and design rank under every candidate for every sampled subset. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use reproducibility as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Now calculate its partial factor. Model comparison remains conditional on the observed fleet and candidate set. A model with larger intrinsic BF performs poorly on a later fleet. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Evidence is not a warranty. Report external scores.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Inspect influential training sets rather than reporting only the average. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use later-data validation as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Averaging reduces dependence on one subset. The method connects combinations, averages, logarithms, ratios and sample design. Students can reproduce 3.667 and 3.302. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. The central idea is averaging away an arbitrary training choice. Compare arithmetic and geometric means.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat Monte Carlo sampling of large subset collections. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use parent and student learning route as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Useful next reading: approach to learning mathematics, Bukit Timah Mathematics curriculum overview, cross-subject learning approach, nested likelihood-ratio guide, Mathematics Learning Hub, eduKate Mathematics Learning System. Choose the destination that matches the next question rather than treating links as a checklist.
Start from an eligible minimal training sample. Intrinsic Bayes factors are not created by one convenient holdout split. Eligibility, averaging variant and direction must be explicit. This step matters because training-sample averaging, default priors and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Default priors still require a defined parameterisation. Read Berger and Pericchi.
A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep arithmetic, geometric and other intrinsic variants distinctly labelled. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to support a model comparison.
Try a small learning experiment: calculate arithmetic and geometric means of partial factors two, three and six, then explain why the answers differ. Use common misconceptions and next reading as the focus. Predict the direction of change before using software, alter one defensible input, and finish with two sentences: what the calculation supports, and what remains uncertain. This habit turns mathematics education into transferable problem-solving rather than formula recall.
Quick answers
What must be fixed before comparing models? Use the same observations, censoring mechanism, time scale, likelihood convention and predictive unit.
Does a preferred value prove a model is true? No. The comparison is conditional on its candidates, data and assumptions.
Which mathematics matters? Algebra organises identities, logarithms stabilise products, probability represents uncertainty, variance measures instability and graphs expose influential cases.
How should a student practise? Preserve intermediate quantities, predict directions before calculating and change one input at a time.
What follows the comparison? Diagnostics, uncertainty analysis, posterior or predictive checks, later-data validation and a separate decision analysis when costs or safety matter.
Extended practice laboratory
Practice case 1: The default-prior problem
Setup: Use this chapter’s illustrative case: A location parameter uses a flat baseline prior.
Task: Separate estimation from comparison.
Checking target: A proper posterior does not imply a defined Bayes factor.
Practice case 2: Training-sample construction
Setup: Use this chapter’s illustrative case: Several observed failures identify shape and scale.
Task: Consider all eligible subsets.
Checking target: An arbitrary subset can favour one model.
Practice case 3: Minimal training sample
Setup: Use this chapter’s illustrative case: Two or more informative failures may be required.
Task: Verify propriety model by model.
Checking target: The minimum depends on model rank and censoring pattern.
Practice case 4: Partial Bayes factors
Setup: Use this chapter’s illustrative case: One subset yields partial BF A:B=2.
Task: Preserve its index.
Checking target: A partial factor depends on the selected subset.
Practice case 5: Averaging over training samples
Setup: Use this chapter’s illustrative case: Three examples give 2, 3 and 6.
Task: State the variant.
Checking target: Averaging without naming the rule is ambiguous.
Practice case 6: Arithmetic intrinsic example
Setup: Use this chapter’s illustrative case: The arithmetic mean is 11/3=3.667.
Task: Inspect the distribution.
Checking target: A large factor can influence the arithmetic mean strongly.
Practice case 7: Geometric intrinsic example
Setup: Use this chapter’s illustrative case: The geometric mean is 36^(1/3)=3.302.
Task: Report which one is used.
Checking target: The geometric and arithmetic variants are not interchangeable.
Practice case 8: Direction of the factor
Setup: Use this chapter’s illustrative case: Reciprocal factors correspond to B:A.
Task: Label each ratio.
Checking target: Averaging a mixture of directions is meaningless.
Practice case 9: Censored observations
Setup: Use this chapter’s illustrative case: A subset includes two failures and one censored unit.
Task: Define eligibility rules.
Checking target: Random minimal subsets can be invalid under heavy censoring.
Practice case 10: Design matrices and rank
Setup: Use this chapter’s illustrative case: Two failures at identical stress levels cannot identify a slope.
Task: Check matrix rank.
Checking target: Sample size alone does not guarantee propriety.