Why Mathematics? | Laplace Approximation for Bayesian Reliability Models: Posterior Modes, Hessians, Gaussian Curvature and Marginal Likelihood

Start with the integral near its mode. Bayesian evidence integrates likelihood times prior over every parameter value. A censored Weibull model integrates over log scale and log shape. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A plug-in likelihood at the mode omits posterior volume. Write the target integral first.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the optimisation target, mode, gradient norm, parameter scale and full Hessian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use the integral behind the method 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? Laplace approximation combines a mode’s height with a curvature-based volume, so a taller peak need not have larger integrated mass. That is why a careful report includes the construction, not only the final scalar.

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Now read curvature as local width. Optimise the log joint density rather than likelihood alone. The prior shifts a weakly identified ageing parameter toward plausible values. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A local optimiser can stop at the wrong mode. Use gradients and dispersed starts.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a stable log determinant and confirm the negative Hessian is positive definite. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use posterior mode 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 Gaussian volume corrects peak height. A second-order Taylor expansion approximates the log joint near the mode. The linear term vanishes when the gradient is zero. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. The approximation is local and can miss skewness or boundaries. Inspect the expansion region.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep likelihood, prior and Jacobian terms in the same parameterisation. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use quadratic expansion 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 integral near its mode. The negative second derivative or Hessian measures local curvature at the mode. H=4 gives a local variance near 1/4 in one dimension. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A non-positive Hessian signals no valid local Gaussian peak. Check eigenvalues numerically.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Compare against quadrature or simulation on a tractable case. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use negative hessian 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/10.1080/01621459.1986.10478240. Read the authors’ definition and conditions before carrying the formula into a censored reliability model.

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Now read curvature as local width. The evidence combines peak height with the width of the peak. A lower but wider posterior can integrate to more mass than a sharper peak. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Peak fit alone is not model evidence. Separate height and volume.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check skewness, boundaries and every plausible mode before trusting a local Gaussian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use gaussian volume 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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The Gaussian volume corrects peak height. Use h=-10 and H=4 in the one-dimensional formula. Half log 2pi is 0.91894 and half log 4 is 0.69315. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Dropping the determinant term overstates evidence. Reproduce every term on the log scale.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the optimisation target, mode, gradient norm, parameter scale and full Hessian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use worked log evidence -9.7742 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? Laplace approximation combines a mode’s height with a curvature-based volume, so a taller peak need not have larger integrated mass. That is why a careful report includes the construction, not only the final scalar.

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Start with the integral near its mode. Subtract approximate log evidences before exponentiating. -9.7742-(-11)=1.2258 and exp(1.2258) is about 3.407. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Approximation errors from both models enter the difference. Report direction and numerical uncertainty.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a stable log determinant and confirm the negative Hessian is positive definite. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use approximate bayes 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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Now read curvature as local width. For d parameters, use d/2 log(2pi) minus one-half log determinant of the negative Hessian. A three-parameter reliability model uses a 3 by 3 curvature matrix. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Multiplying raw eigenvalues can overflow or underflow. Use a stable log determinant.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep likelihood, prior and Jacobian terms in the same parameterisation. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use multivariate determinant 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 Gaussian volume corrects peak height. Off-diagonal Hessian terms describe local coupling between parameters. Weibull shape and scale form a tilted ellipse. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Ignoring correlation changes the Gaussian volume. Plot or inspect the covariance matrix.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Compare against quadrature or simulation on a tractable case. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use parameter correlation 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 integral near its mode. Failures contribute density and censored records contribute survival probability before differentiation. A right-censored unit changes gradients through log survival. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Treating censoring as failure changes the mode and Hessian. Unit-test both record types.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check skewness, boundaries and every plausible mode before trusting a local Gaussian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use censored likelihood contributions 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 read curvature as local width. Modern software can calculate exact derivatives of the coded log density. Autodiff supplies a Hessian for a hierarchical failure model. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Correct derivatives of incorrect code remain wrong. Compare with finite differences on a small case.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the optimisation target, mode, gradient norm, parameter scale and full Hessian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use automatic differentiation 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? Laplace approximation combines a mode’s height with a curvature-based volume, so a taller peak need not have larger integrated mass. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

The Gaussian volume corrects peak height. Numerical derivatives depend on step size and parameter scale. A step that works for log scale fails for a tiny rate parameter. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Subtraction error and truncation error pull in opposite directions. Repeat with several steps.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a stable log determinant and confirm the negative Hessian is positive definite. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use finite-difference 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/10.1080/01621459.1986.10478240. Compare its prediction target, diagnostics and finite-sample cautions with this worked example.

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Start with the integral near its mode. Log transforms move positive parameters to an unconstrained space and add Jacobians. eta=exp(z) contributes z to the transformed log density. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Using a Hessian on z with a density on eta is inconsistent. Keep one parameterisation throughout.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep likelihood, prior and Jacobian terms in the same parameterisation. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use transforming positive 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 read curvature as local width. Gaussian symmetry is poor when a mode lies near zero or another constraint. A variance component is estimated near zero. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Half the Gaussian mass may lie outside the valid space. Use a better transform or another method.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Compare against quadrature or simulation on a tractable case. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use boundary modes 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 Gaussian volume corrects peak height. A quadratic approximation is symmetric even when the posterior has a long tail. Sparse failures produce a right-skewed shape parameter. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A sharp mode can hide substantial tail mass. Compare with posterior simulation.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check skewness, boundaries and every plausible mode before trusting a local Gaussian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use skewness 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 integral near its mode. One local Gaussian covers one mode, not a separated collection. Two failure mechanisms create distinct peaks. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. The largest local mode can omit much of the integral. Find and combine justified modal contributions.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the optimisation target, mode, gradient norm, parameter scale and full Hessian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use multimodality 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? Laplace approximation combines a mode’s height with a curvature-based volume, so a taller peak need not have larger integrated mass. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Now read curvature as local width. Regular posteriors often become more Gaussian as information grows. A large fleet yields a concentrated interior mode. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. More records do not repair non-identifiability or mixture symmetry. Check regularity rather than assuming it.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a stable log determinant and confirm the negative Hessian is positive definite. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use sample-size 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.

Contents · Previous chapter · Next chapter

The Gaussian volume corrects peak height. Integrating many latent effects can make the Hessian large and sparse. Machine-specific frailties add one dimension per unit. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. A dense determinant can be expensive and unstable. Exploit sparse structure or integrate in stages.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep likelihood, prior and Jacobian terms in the same parameterisation. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use hierarchical random effects 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 integral near its mode. Evidence requires proper priors with all constants included. A normal prior contributes its scale constant at the mode. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Improper priors make Bayes factors arbitrary. Audit prior densities before optimisation.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Compare against quadrature or simulation on a tractable case. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use prior normalisation 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 read curvature as local width. Changing prior scale changes both the peak and the volume penalty. A wider degradation-rate prior lowers approximate evidence. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Fast computation does not make the prior objective. Repeat plausible prior choices.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check skewness, boundaries and every plausible mode before trusting a local Gaussian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use prior 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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The Gaussian volume corrects peak height. Compare the approximation with quadrature, bridge sampling or nested sampling on a tractable model. Two-dimensional numerical integration supplies a benchmark. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Agreement in one dataset does not guarantee every regime. Build a simulation suite.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record the optimisation target, mode, gradient norm, parameter scale and full Hessian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use accuracy checks 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? Laplace approximation combines a mode’s height with a curvature-based volume, so a taller peak need not have larger integrated mass. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Start with the integral near its mode. BIC is a large-sample simplification related to Laplace expansion under regular assumptions. The full approximation retains prior and observed curvature terms. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. BIC and a finite-sample Laplace calculation are not identical. Label the formula actually used.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Use a stable log determinant and confirm the negative Hessian is positive definite. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use relation to 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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Now read curvature as local width. Approximate evidence does not directly measure future calibration or maintenance value. A model with higher evidence misses a new operating regime. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Model comparison and decision analysis answer different questions. Add later-data validation.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep likelihood, prior and Jacobian terms in the same parameterisation. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. Use prediction still matters 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 Gaussian volume corrects peak height. The method connects quadratics, logarithms, matrices, optimisation and area. Students can reproduce -9.7742 and explain why width matters. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. The deepest idea is that area depends on height and shape. Sketch two peaks with equal height and different width.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Compare against quadrature or simulation on a tractable case. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. 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 integral near its mode. Laplace approximation is not merely evaluating the likelihood at its maximum. The Hessian supplies the volume correction. This step matters because local Gaussian integration, curvature and reliability evidence require the mathematical object, prediction target and reliability question to stay aligned. Boundaries, skewness and multiple modes require caution. Read Tierney and Kadane and compare simulation-based evidence methods.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check skewness, boundaries and every plausible mode before trusting a local Gaussian. 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: combine h=-10, half log 2pi and half log 4, then compare two peaks with equal height but different curvature. 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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