Why Mathematics? | Bayes Factors for Bayesian Reliability Models: Marginal Likelihood, Prior Odds, Posterior Odds and Sensitivity

Start with the evidence integral. A Bayes factor asks how much the observed data shift relative support between two declared probabilistic models. Two lifetime models use the same 120 component records. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. It cannot compare an unstated alternative or repair a poor candidate set. Write both models and the direction BF_AB before calculating.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep every model prior, parameter prior, likelihood and candidate owner beside the evidence calculation. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use the model-comparison question 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? A Bayes factor changes posterior odds by multiplication, while posterior probability also depends on every candidate and its prior weight. That is why a careful report includes the construction, not only the final scalar.

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Now keep the odds direction visible. For model M, integrate likelihood p(y|theta,M) against prior p(theta|M) over the full parameter space. Each possible Weibull parameter pair contributes according to prior mass and data fit. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Maximum likelihood alone does not perform this averaging. Show the likelihood, prior and integration measure.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. A posterior model probability is conditional on the declared set and changes when the set changes. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use marginal likelihood as an integral 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.

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The prior is part of the calculation. Exact failures contribute a density and right-censored records contribute survival probabilities inside the evidence integral. A unit surviving to 500 hours contributes S(500|theta). This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Treating censored units as failures changes marginal likelihood. Use one documented observation model for every candidate.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Prior scale sensitivity is part of the scientific result because marginal likelihood averages over prior volume. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use censored reliability likelihood 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.

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Start with the evidence integral. The marginal likelihood is the probability density or mass assigned to the observed data before seeing them, under the model and prior. A concentrated prior near plausible lifetimes can assign more predictive mass than a diffuse prior. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Density values depend on the common measurement scale. Keep data transformations and units identical.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Proper priors and common data units are mathematical requirements, not formatting details. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use prior predictive interpretation 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/10.1080/01621459.1995.10476572. Read the authors’ definition and conditions before carrying the formula into a censored reliability model.

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Now keep the odds direction visible. BF_AB=p(y|M_A)/p(y|M_B); values above one favour A in the defined direction. 0.008 divided by 0.002 equals 4. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Reversing the models produces 1/4, not a contradiction. Print the numerator and denominator beside the result.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Relative evidence still needs predictive and mechanism checks. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use bayes factor direction 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.

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The prior is part of the calculation. Prior model odds equal P(M_A)/P(M_B) and express model-level beliefs before the current data. Equal prior probabilities give odds 1:1. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Equal odds are a modelling choice, not automatic neutrality. State and justify the model prior.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep every model prior, parameter prior, likelihood and candidate owner beside the evidence calculation. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use prior odds 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? A Bayes factor changes posterior odds by multiplication, while posterior probability also depends on every candidate and its prior weight. That is why a careful report includes the construction, not only the final scalar.

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Start with the evidence integral. Posterior odds equal Bayes factor times prior odds. BF 4 times prior odds 1 gives posterior odds 4. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Multiplying probabilities rather than odds gives the wrong result. Convert probabilities to odds before updating.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. A posterior model probability is conditional on the declared set and changes when the set changes. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use posterior odds 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.

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Now keep the odds direction visible. With only A and B and equal prior odds, odds 4:1 convert to probability 4/5. The worked posterior model probability is 0.80. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Adding another candidate changes normalised probabilities. Name the complete candidate set.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Prior scale sensitivity is part of the scientific result because marginal likelihood averages over prior volume. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use posterior model probability 0.80 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.

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The prior is part of the calculation. Logarithms turn evidence ratios into differences and prevent numerical underflow. log BF_AB=log 0.008-log 0.002=log 4≈1.3863. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. A log base must be stated when thresholds are discussed. Compute on the natural-log scale and label it.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Proper priors and common data units are mathematical requirements, not formatting details. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use log 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.

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Start with the evidence integral. Marginal likelihood integrates nuisance parameters rather than fixing them at estimates. Shape, scale and batch frailty are averaged under their joint prior. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Poorly identified nuisance priors can dominate the evidence. Run targeted prior sensitivity checks.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Relative evidence still needs predictive and mechanism checks. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use nuisance parameters 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.

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Now keep the odds direction visible. Models spreading prior mass over large low-likelihood regions can have lower marginal likelihood even if their best fit is high. A flexible mixture fits well at one mode but predicts many implausible datasets beforehand. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Calling this an automatic penalty does not make the prior objective. Explain which prior volume creates the trade-off.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep every model prior, parameter prior, likelihood and candidate owner beside the evidence calculation. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use automatic complexity trade-off 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? A Bayes factor changes posterior odds by multiplication, while posterior probability also depends on every candidate and its prior weight. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

The prior is part of the calculation. Ordinary marginal likelihoods require priors whose normalising constants are defined. A proper lognormal prior on a positive scale parameter has total mass one. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. An improper flat prior can leave a Bayes factor arbitrary. Use proper priors or a specialised justified method.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. A posterior model probability is conditional on the declared set and changes when the set changes. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use proper 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.

Second authoritative source: https://doi.org/10.1214/ss/1009212519. Compare its prediction target, diagnostics and finite-sample cautions with this worked example.

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Start with the evidence integral. Widening a parameter prior can lower average predictive density by spreading mass. A hazard-ratio prior with standard deviation 10 assigns much mass to extreme effects. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. A dramatic Bayes factor may reflect a scale choice. Publish a defensible grid of prior scales.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Prior scale sensitivity is part of the scientific result because marginal likelihood averages over prior volume. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use prior scale sensitivity 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.

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Now keep the odds direction visible. With large samples, point-null and continuous-alternative comparisons can disagree with fixed-threshold p-values. A small p-value can coexist with support for a concentrated null under stated priors. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. This is not an arithmetic error; the questions and priors differ. Report both targets without equating them.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Proper priors and common data units are mathematical requirements, not formatting details. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use jeffreys-lindley behaviour 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.

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The prior is part of the calculation. Each model can contain a range of parameter values, integrated under its prior. Model A allows positive ageing shapes while model B fixes constant hazard. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Changing the hypothesis changes the integral. Define parameter restrictions precisely.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Relative evidence still needs predictive and mechanism checks. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use composite reliability hypotheses 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.

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Start with the evidence integral. Fleet and batch effects require integrating population and latent parameters at the intended prediction level. A new-batch comparison marginalises unknown batch frailty. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Conditioning on fitted frailties targets existing batches instead. Match evidence to the scientific focus.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep every model prior, parameter prior, likelihood and candidate owner beside the evidence calculation. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use hierarchical 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.

Did You Know? A Bayes factor changes posterior odds by multiplication, while posterior probability also depends on every candidate and its prior weight. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Now keep the odds direction visible. Analytic evidence is rare, so bridge sampling, nested sampling or other validated methods estimate it. Repeated bridge runs should agree within Monte Carlo error. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. The harmonic-mean estimator can be catastrophically unstable. Use diagnostics and independent repeats.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. A posterior model probability is conditional on the declared set and changes when the set changes. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use numerical estimation 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.

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The prior is part of the calculation. An estimated Bayes factor has simulation uncertainty in addition to modelling uncertainty. Log evidence estimates differ by 0.12 across runs. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Reporting many decimals hides computational error. Repeat seeds and report stability.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Prior scale sensitivity is part of the scientific result because marginal likelihood averages over prior volume. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use monte carlo error 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.

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Start with the evidence integral. Posterior model probabilities depend on which candidates and prior model weights are included. Adding a credible lognormal alternative redistributes probability. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. A two-model 0.80 is not permanent evidence. Treat the candidate list as part of the analysis.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Proper priors and common data units are mathematical requirements, not formatting details. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use model-space sensitivity 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.

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Now keep the odds direction visible. BIC can approximate a log Bayes factor under regular large-sample conditions and particular prior behaviour. A quick BIC screen may agree with a full evidence calculation. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Singularity, boundaries or weak identification can break the approximation. Do not label BIC as an exact Bayes factor.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Relative evidence still needs predictive and mechanism checks. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use bayes factors and bic 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.

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The prior is part of the calculation. Bayes factors assess integrated model evidence; PSIS-LOO estimates predictive performance for held-out units. A strong evidence model can lose on a deployment-focused log score. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. The objectives are not interchangeable. Choose the criterion from the decision question.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep every model prior, parameter prior, likelihood and candidate owner beside the evidence calculation. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use bayes factors versus psis-loo 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? A Bayes factor changes posterior odds by multiplication, while posterior probability also depends on every candidate and its prior weight. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Start with the evidence integral. A model can win relative evidence and still miss important failure-time features. Both candidates underpredict early failures. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Relative preference is not absolute fit. Simulate replicated reliability summaries.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. A posterior model probability is conditional on the declared set and changes when the set changes. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use posterior predictive adequacy 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.

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Now keep the odds direction visible. Later fleets test calibration and transport beyond the development data. A model with BF 4 loses calibration at another factory. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Marginal likelihood does not guarantee transportability. Freeze and validate predictions externally.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Prior scale sensitivity is part of the scientific result because marginal likelihood averages over prior volume. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. Use external 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.

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The prior is part of the calculation. Bayes factors connect ratios, integrals, probability, odds and logarithms. Students can verify 0.008/0.002=4 and 4/(1+4)=0.80. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Arithmetic alone cannot justify priors. Separate calculation from modelling judgement.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Proper priors and common data units are mathematical requirements, not formatting details. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. 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: Mathematics Learning Hub, eduKate Mathematics Learning System, approach to learning mathematics, Bukit Timah Mathematics curriculum overview, cross-subject learning approach, nested likelihood-ratio guide. Choose the destination that matches the next question rather than treating links as a checklist.

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Start with the evidence integral. A Bayes factor is not a p-value, not a universal truth probability and not prior-free. BF_AB=4 means four times the marginal evidence within the setup. This step matters because integrated evidence, prior sensitivity and reliability model comparison require the mathematical object, prediction target and reliability question to stay aligned. Verbal strength labels should not replace context. Read Kass and Raftery, then compare prediction with PSIS-LOO.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Relative evidence still needs predictive and mechanism checks. 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: compute a marginal-likelihood ratio, update prior odds, convert posterior odds to probability and repeat under a different model prior. 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.

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