Why Mathematics? | Savage–Dickey Density Ratios for Nested Reliability Models: Point Nulls, Prior Compatibility, Posterior Density and Bayes Factors

Start with a genuinely nested point null. A complex reliability model reduces to a simpler one when one parameter equals a fixed value. A time-trend parameter delta equals zero under constant hazard. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Models that cannot be linked by a point restriction need another comparison. Write the nesting map.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the exact nested hypotheses, ratio orientation and common parameter measure. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use the nested-model 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 Savage–Dickey ratio uses density height at a point, not the posterior probability of that exact continuous parameter value. That is why a careful report includes the construction, not only the final scalar.

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Now evaluate both densities on one scale. The null assigns probability to a model with delta fixed, while the alternative gives delta a continuous prior. H0 sets delta=0 and H1 estimates delta. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A continuous alternative gives zero probability to exactly one point. Distinguish model probability from parameter density.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Write nuisance priors under both models and verify the compatibility condition. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use point null and alternative 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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Prior compatibility makes the shortcut valid. Under compatible priors, BF_01 equals posterior density of delta at zero divided by prior density there. Both densities are evaluated on the same scale. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. The formula direction reverses for BF_10. Label numerator and denominator.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat posterior-ordinate estimation across density methods and bandwidths. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use the density-ratio identity 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 a genuinely nested point null. Divide posterior density 0.80 by prior density 0.20. BF_01 equals 4. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Subtracting densities gives 0.60 and is not a Bayes factor. Check units cancel.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep prior-scale sensitivity separate from Monte Carlo error. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use worked ratio four 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/10.1080/01621459.1995.10476554. Read the authors’ definition and conditions before carrying the formula into a censored reliability model.

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Now evaluate both densities on one scale. Multiply prior odds by BF_01 and convert odds to probability. Equal prior odds give posterior odds 4 and probability 4/5=0.80. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A Bayes factor is not itself a posterior probability. Report prior odds.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a generalised correction when the simple ratio conditions do not hold. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use posterior 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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Prior compatibility makes the shortcut valid. A continuous density can exceed one because probability is area over an interval. A narrow posterior around zero has density 0.80 at the point. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Reading density height as an 80 percent chance is wrong. Use interval sketches.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the exact nested hypotheses, ratio orientation and common parameter measure. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use density is not mass 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 Savage–Dickey ratio uses density height at a point, not the posterior probability of that exact continuous parameter value. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Start with a genuinely nested point null. The nuisance-parameter prior under the null must match the conditional prior implied under the alternative in the standard identity. Scale-prior choices agree when delta is fixed at zero. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Convenient but incompatible priors can invalidate the shortcut. Write both model priors.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Write nuisance priors under both models and verify the compatibility condition. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use prior compatibility 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 evaluate both densities on one scale. Verdinelli and Wasserman show a general Bayes factor can include a correction to the simple density ratio. Posterior simulation estimates the ratio and correction. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Assuming the correction is one without checking conditions biases evidence. Derive the implemented identity.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat posterior-ordinate estimation across density methods and bandwidths. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use generalised correction 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.

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Prior compatibility makes the shortcut valid. Both models must evaluate identical failure and censoring records. Failures use density and censored units use survival probability. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Changing censoring treatment alters the posterior density at the null. Freeze observation coding.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep prior-scale sensitivity separate from Monte Carlo error. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use censored 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 a genuinely nested point null. Estimate the ordinate at zero from posterior draws or an analytic conditional. A kernel density estimate uses draws of delta. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Bandwidth choice can change the density at one point. Repeat several estimators.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a generalised correction when the simple ratio conditions do not hold. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use posterior density 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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Now evaluate both densities on one scale. Evaluate the fully normalised prior at the null. A normal prior with scale sigma has density 1/(sigma sqrt(2pi)) at zero. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Dropping the normalising constant changes the Bayes factor. Calculate symbolically and numerically.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the exact nested hypotheses, ratio orientation and common parameter measure. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use prior density evaluation 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 Savage–Dickey ratio uses density height at a point, not the posterior probability of that exact continuous parameter value. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Prior compatibility makes the shortcut valid. A wider alternative prior lowers prior density at zero and can change evidence substantially. Doubling sigma halves the normal density at zero. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A default wide prior is not neutral. Publish a prior-scale grid.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Write nuisance priors under both models and verify the compatibility condition. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use 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.

Second authoritative source: https://doi.org/10.1080/01621459.1995.10476554. Compare its prediction target, diagnostics and finite-sample cautions with this worked example.

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Start with a genuinely nested point null. Density changes with a Jacobian under nonlinear transformation. delta becomes phi=exp(delta). This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Prior density on delta and posterior density on phi cannot be divided directly. Keep a common measure.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat posterior-ordinate estimation across density methods and bandwidths. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use parameter transformations 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 evaluate both densities on one scale. A null on the edge of parameter space can violate the ordinary interior-point setup. A variance component equals zero. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Symmetric density estimators spill mass outside the valid space. Use boundary-aware theory.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep prior-scale sensitivity separate from Monte Carlo error. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use boundary nulls 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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Prior compatibility makes the shortcut valid. Restricting delta to positive values changes prior normalisation and the hypothesis. Only increasing hazard trends are permitted. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A two-sided formula cannot be reused unchanged. State the support.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a generalised correction when the simple ratio conditions do not hold. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use one-sided alternatives 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 a genuinely nested point null. Posterior density at the null can be hard to estimate when simulation misses modes. Latent failure mechanisms create separated delta regions. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A smooth density estimate from one mode is misleading. Use multiple chains.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the exact nested hypotheses, ratio orientation and common parameter measure. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use multimodal posteriors 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 Savage–Dickey ratio uses density height at a point, not the posterior probability of that exact continuous parameter value. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Now evaluate both densities on one scale. Finite posterior draws make the ordinate uncertain. Independent runs give BF_01 values 3.7 and 4.3. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A single rounded value hides computation error. Report repeated-run spread.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Write nuisance priors under both models and verify the compatibility condition. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. 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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Prior compatibility makes the shortcut valid. The ratio is a Bayes-factor identity, not a replacement for model definitions. Both models still have likelihoods and priors. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A shortcut cannot rescue an incoherent prior specification. Audit the full models.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat posterior-ordinate estimation across density methods and bandwidths. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use relation to marginal 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 a genuinely nested point null. Likelihood-ratio tests compare maximised fit; Savage–Dickey uses integrated Bayesian evidence. The same null can yield different summaries. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. P-values and Bayes factors answer different probability questions. Keep interpretations separate.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep prior-scale sensitivity separate from Monte Carlo error. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use relation to likelihood-ratio tests 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 evaluate both densities on one scale. Nuisance frailties make prior compatibility especially important. Machine effects share a variance under both models. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Conditioning on realised frailties can change the comparison target. Define the population level.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a generalised correction when the simple ratio conditions do not hold. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use hierarchical reliability 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.

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Prior compatibility makes the shortcut valid. Testing several fixed values requires explicit model priors and careful multiplicity interpretation. Three degradation thresholds define three submodels. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Separate pairwise ratios do not automatically form coherent model probabilities. Normalise over the candidate set.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the exact nested hypotheses, ratio orientation and common parameter measure. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use multiple point 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.

Did You Know? A Savage–Dickey ratio uses density height at a point, not the posterior probability of that exact continuous parameter value. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Start with a genuinely nested point null. Vary prior scales, density estimators and parameterisations. BF_01 stays between 3.5 and 4.5 under defensible choices. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Robustness is evidence about stability, not truth. Publish the range.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Write nuisance priors under both models and verify the compatibility condition. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use sensitivity and robustness 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 evaluate both densities on one scale. A Bayes factor for delta=0 does not measure later-fleet calibration directly. The simpler model underpredicts a new heat regime. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Model evidence and deployment performance differ. Check future data.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat posterior-ordinate estimation across density methods and bandwidths. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. Use predictive 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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Prior compatibility makes the shortcut valid. The method connects ratios, densities, odds, probability and graphs. Students can verify 0.8/0.2=4 and 4/5=0.8. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. The deepest lesson is that conditions make a shortcut valid. Draw prior and posterior curves at zero.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep prior-scale sensitivity separate from Monte Carlo error. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. 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 a genuinely nested point null. Savage–Dickey is not valid for every nested model and density height is not point probability. Prior compatibility and orientation are essential. This step matters because nested hypotheses, density ratios and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Boundary cases may need different theory. Read Verdinelli and Wasserman on the generalised ratio.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a generalised correction when the simple ratio conditions do not hold. 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: divide posterior density at the null by prior density, update equal prior odds and convert odds four to probability. 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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