Why Mathematics? | Sequential Monte Carlo Samplers for Bayesian Reliability Models: Particle Weights, Effective Sample Size, Resampling and Evidence

Begin with a weighted particle cloud. SMC samplers approximate a path of distributions defined on a common parameter space. Power posteriors move from a proper prior to a reliability posterior. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. The sequence index is algorithmic, not calendar time. Define every target.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record targets, incremental potentials, particle counts, ESS thresholds, resampling schemes and kernels. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use a sequence of target distributions 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? Resampling can equalise particle weights while simultaneously reducing the number of distinct ancestors. That is why a careful report includes the construction, not only the final scalar.

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Now adapt when weights become uneven. Represent each target by many parameter particles and weights. Two thousand particles carry Weibull shape and scale values. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Particles are not automatically independent posterior draws. Preserve ancestry and weights.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep ancestry so high current ESS cannot hide genealogical collapse. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use particle population 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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Resampling and rejuvenation serve different jobs. When the target changes, update each weight by the appropriate density ratio. A beta increase multiplies by likelihood to the beta difference. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Omitting proposal or backward-kernel terms biases a general SMC sampler. Derive the exact ratio.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Validate every evidence contribution and weighted summary before discarding weights. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use incremental importance weights 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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Begin with a weighted particle cloud. Divide positive unnormalised weights by their sum. Four weights become 0.10, 0.20, 0.30 and 0.40. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Rounding can make displayed weights fail to sum exactly to one. Calculate with full precision.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat full runs with independent seeds and compare adaptive temperature ladders. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use normalising the weights 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/10.1111/j.1467-9868.2006.00553.x. Read the authors’ definition and conditions before carrying the formula into a censored reliability model.

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Now adapt when weights become uneven. Compute one divided by the sum of squared normalised weights. 1/(.01+.04+.09+.16)=3.333. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. ESS is a diagnostic equivalent count, not the number of distinct parameter values. Report the formula.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check mode survival, parameter movement and prior sensitivity together. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use worked effective sample size 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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Resampling and rejuvenation serve different jobs. As targets separate, a few particles may receive nearly all weight. One particle owns 70 percent after a large beta step. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Increasing nominal particle count does not guarantee a healthy ESS. Adapt the next step.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record targets, incremental potentials, particle counts, ESS thresholds, resampling schemes and kernels. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use weight degeneracy 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? Resampling can equalise particle weights while simultaneously reducing the number of distinct ancestors. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Begin with a weighted particle cloud. Resample when ESS falls below a chosen threshold. Trigger at half the particle count. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Resampling at every tiny step adds variance and computation. State the threshold.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep ancestry so high current ESS cannot hide genealogical collapse. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use resampling decision 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 adapt when weights become uneven. Systematic, stratified and residual resampling can reduce variance relative to multinomial choices. Systematic resampling uses one random offset. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. No scheme restores modes already lost. Record the method and seed.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Validate every evidence contribution and weighted summary before discarding weights. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use multinomial and lower-variance schemes 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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Resampling and rejuvenation serve different jobs. Resampling copies high-weight particles and discards low-weight ones. The 0.40 particle may receive two descendants. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Equal weights after resampling do not mean diverse states. Track unique ancestors.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat full runs with independent seeds and compare adaptive temperature ladders. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use ancestral duplication 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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Begin with a weighted particle cloud. Apply a kernel invariant to the new target after resampling. Several Metropolis steps move duplicated Weibull particles. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. A low-acceptance kernel leaves clones unchanged. Diagnose acceptance and movement.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check mode survival, parameter movement and prior sensitivity together. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use mcmc rejuvenation 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 adapt when weights become uneven. Choose the next beta so predicted ESS reaches a target. A bisection search finds a beta increment. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Data-dependent schedules complicate exact replication without saved states. Archive every chosen beta.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record targets, incremental potentials, particle counts, ESS thresholds, resampling schemes and kernels. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use adaptive tempering 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? Resampling can equalise particle weights while simultaneously reducing the number of distinct ancestors. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Resampling and rejuvenation serve different jobs. Each particle evaluates failures with densities and censored units with survival probabilities. Log likelihood is cached across beta updates. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. A coding inconsistency affects every weight. Test one failure and one censoring case.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep ancestry so high current ESS cannot hide genealogical collapse. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. 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.

Second authoritative source: https://doi.org/10.1111/j.1467-9868.2006.00553.x. Compare its prediction target, diagnostics and finite-sample cautions with this worked example.

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Begin with a weighted particle cloud. The ratio of adjacent normalising constants can be estimated by the weighted mean of incremental potentials. Stage means 0.75 and 0.80 multiply to 0.60. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. A product of normalised weights is not the same estimator. Store each stage contribution.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Validate every evidence contribution and weighted summary before discarding weights. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use evidence from incremental means 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 adapt when weights become uneven. Take the log of 0.60 or sum log 0.75 and log 0.80. The result is about -0.51083. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Products underflow across long paths. Accumulate log ratios.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat full runs with independent seeds and compare adaptive temperature ladders. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use worked log ratio -0.51083 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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Resampling and rejuvenation serve different jobs. Repeated SMC runs quantify variability from particles, resampling and moves. Ten runs have log-evidence standard deviation 0.08. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. One large population does not reveal all instability. Repeat with independent seeds.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check mode survival, parameter movement and prior sensitivity together. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. 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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Begin with a weighted particle cloud. Repeated resampling can leave the final cloud descended from very few early particles. Two thousand particles trace to twelve prior ancestors. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Current ESS can look high while path diversity is poor. Inspect ancestry.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record targets, incremental potentials, particle counts, ESS thresholds, resampling schemes and kernels. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use genealogical degeneracy 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? Resampling can equalise particle weights while simultaneously reducing the number of distinct ancestors. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Now adapt when weights become uneven. A particle population can represent several modes if tempering and movement preserve them. Particles split between two failure mechanisms. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Resampling can accidentally eliminate a smaller mode. Use enough particles and mode-aware moves.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep ancestry so high current ESS cannot hide genealogical collapse. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. 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.

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Resampling and rejuvenation serve different jobs. Importance weights become more variable as dimension grows. Many frailty effects make a beta step difficult. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. More particles alone may scale poorly. Use blocked kernels and careful paths.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Validate every evidence contribution and weighted summary before discarding weights. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use high-dimensional 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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Begin with a weighted particle cloud. Weight calculations and particle moves often parallelise. Each worker updates a subset of particles. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Resampling and global ESS still require coordination. Design deterministic aggregation.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat full runs with independent seeds and compare adaptive temperature ladders. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use parallel computation 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 adapt when weights become uneven. The starting population and evidence depend on the proper prior. A diffuse latent-variance prior broadens early particles. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. SMC does not create an objective evidence scale. Repeat defensible priors.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check mode survival, parameter movement and prior sensitivity together. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. 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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Resampling and rejuvenation serve different jobs. Particle filters usually advance latent states through observation time; SMC samplers can advance through artificial distributions on a fixed parameter space. Both use weights and resampling. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Calling every SMC sampler a time-series filter is misleading. Name the sequence being traversed.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Record targets, incremental potentials, particle counts, ESS thresholds, resampling schemes and kernels. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use relation to particle filters 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? Resampling can equalise particle weights while simultaneously reducing the number of distinct ancestors. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Begin with a weighted particle cloud. AIS uses independent weighted annealing paths, while SMC adds population resampling and often adaptive schedules. Both may use identical power posteriors and MCMC kernels. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Resampling changes dependence and diagnostics. Match claims to mechanics.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Keep ancestry so high current ESS cannot hide genealogical collapse. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use relation to ais 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 adapt when weights become uneven. Weighted particles estimate posterior means, survival curves and model evidence. Mission survival is a weighted average before resampling. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Discarding weights too early biases summaries. Use the correct stage weights.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Validate every evidence contribution and weighted summary before discarding weights. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. Use predictive summaries 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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Resampling and rejuvenation serve different jobs. SMC connects fractions, squares, averages, probability, algorithms and simulation. Students can reproduce ESS 3.333 and log 0.60. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. The central idea is keeping a population useful as the target changes. Move counters across a sequence of weighted boxes.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Repeat full runs with independent seeds and compare adaptive temperature ladders. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. 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: Bukit Timah Mathematics curriculum overview, cross-subject learning approach, nested likelihood-ratio guide, Mathematics Learning Hub, eduKate Mathematics Learning System, approach to learning mathematics. Choose the destination that matches the next question rather than treating links as a checklist.

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Begin with a weighted particle cloud. Sequential Monte Carlo samplers are not restricted to time-series filtering and ESS is not a guarantee of mode coverage. Resampling controls weights but can reduce ancestry. This step matters because weighted particle populations, adaptive resampling and evidence require the mathematical object, prediction target and reliability question to stay aligned. Valid kernels and proper density ratios remain essential. Read Del Moral, Doucet and Jasra and compare AIS.

A reproducible audit should record variables, units, observation boundaries, censoring codes, likelihood factorisation and software settings. Check mode survival, parameter movement and prior sensitivity together. 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: normalise four particle weights, calculate ESS, then multiply two incremental normalising-constant estimates. 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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