Why Mathematics? | Bayesian Neural Networks for Predictive Maintenance: Weight Uncertainty, Posterior Prediction, Aleatoric and Epistemic Variance

Begin with a probabilistic neural network. A maintenance model must distinguish confidence from a sharp guess. Sparse data in a new regime should widen uncertainty. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. A narrow interval can still be wrong. Validate coverage.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Save architecture, priors, likelihood, inference method and seeds. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use why uncertainty 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.

Did You Know? Predictive variance adds expected conditional variance and variance across posterior predictive means, not their standard deviations. That is why a careful report includes the construction, not only the final scalar.

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Now average predictions across plausible weights. Layers combine weighted inputs and nonlinear activations. Sensor features map to failure risk. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Architecture alone does not specify uncertainty. Define a probabilistic model.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Recalculate aleatoric and epistemic variance before adding them. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use neural-network mapping 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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Variance decomposition separates two uncertainty sources. Bayesian treatment assigns priors and a posterior to network weights. Many weight settings remain plausible. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Weight uncertainty is parameterisation-dependent. Focus on predictive consequences.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Compare approximate posterior methods on simulated benchmarks. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use weight 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.

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Begin with a probabilistic neural network. Priors control function complexity and weight scale. Large weights are discouraged unless supported. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. A generic normal prior may induce odd functions. Run prior predictive checks.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Monitor calibration and covariate shift after deployment. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use prior regularisation 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/10.1162/neco.1992.4.3.448. Read the authors’ definition and conditions before carrying the method into a censored reliability model.

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Now average predictions across plausible weights. The likelihood connects network output to observed labels or measurements. A Bernoulli likelihood models horizon failure. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Wrong noise assumptions distort uncertainty. Match the data type.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Keep uncertainty estimates distinct from safety guarantees. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use 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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Variance decomposition separates two uncertainty sources. Average likelihood predictions across posterior weight draws. Each network gives a slightly different risk. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Using one best network discards posterior uncertainty. Integrate over draws.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Save architecture, priors, likelihood, inference method and seeds. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use posterior prediction 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? Predictive variance adds expected conditional variance and variance across posterior predictive means, not their standard deviations. That is why a careful report includes the construction, not only the final scalar.

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Begin with a probabilistic neural network. Conditional variance represents irreducible or modelled observation noise. Repeated measurements vary even at the same state. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. It can be heteroskedastic and learnable. State the noise model.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Recalculate aleatoric and epistemic variance before adding them. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use aleatoric uncertainty 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 average predictions across plausible weights. Variation among posterior predictive means reflects uncertainty about the function. New operating conditions produce disagreement. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Approximate inference can understate it. Check against simulation and shift tests.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Compare approximate posterior methods on simulated benchmarks. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use epistemic uncertainty 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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Variance decomposition separates two uncertainty sources. Total variance equals expected conditional variance plus variance of conditional means. Separate noise from model uncertainty. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. The decomposition is conditional on the model. Label both terms.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Monitor calibration and covariate shift after deployment. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use law of total variance 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 probabilistic neural network. Add 0.20, 0.40 and 0.50 and divide by three. The mean is 0.3667. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Three draws are too few for inference. Use them only for arithmetic.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Keep uncertainty estimates distinct from safety guarantees. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use worked average mean 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 average predictions across plausible weights. Average squared deviations from 0.3667. The population variance is about 0.0156. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Sample-variance denominator would differ. State the convention.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Save architecture, priors, likelihood, inference method and seeds. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use worked epistemic variance 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? Predictive variance adds expected conditional variance and variance across posterior predictive means, not their standard deviations. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Variance decomposition separates two uncertainty sources. Add average conditional variance 0.09 and 0.0156. Total variance is about 0.1056. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Variances, not standard deviations, add. Convert only after summing.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Recalculate aleatoric and epistemic variance before adding them. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use worked total variance 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://authors.library.caltech.edu/records/8n76z-g9k64. Compare its reference distribution, diagnostics and finite-simulation cautions with this worked example.

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Begin with a probabilistic neural network. Variational inference, Laplace methods or MCMC approximate difficult weight posteriors. Each method trades accuracy and cost. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. A fast approximation is not automatically calibrated. Benchmark it.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Compare approximate posterior methods on simulated benchmarks. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use posterior approximation 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 average predictions across plausible weights. Dropout at prediction can approximate an uncertainty procedure under particular assumptions. Repeated masks produce a distribution. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. It is not identical to exact Bayesian inference. Use precise language.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Monitor calibration and covariate shift after deployment. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use monte carlo dropout 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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Variance decomposition separates two uncertainty sources. Independent trained networks can capture useful predictive diversity. Ensemble spread flags some uncertain cases. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. It is a different construction from a weight posterior. Compare calibration empirically.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Keep uncertainty estimates distinct from safety guarantees. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use deep ensembles comparison 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 probabilistic neural network. Time-to-event maintenance requires a survival likelihood or discrete-time hazards. Units still operating contribute censored information. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Binary labels at arbitrary cutoffs waste time information. Choose the estimand.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Save architecture, priors, likelihood, inference method and seeds. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use censoring 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? Predictive variance adds expected conditional variance and variance across posterior predictive means, not their standard deviations. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Now average predictions across plausible weights. Rare failures can make accuracy misleading. A model predicts survival for almost everything. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Reweighting changes the effective objective. Use proper scores.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Recalculate aleatoric and epistemic variance before adding them. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use class imbalance 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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Variance decomposition separates two uncertainty sources. Factory or season changes can move inputs beyond training support. New vibration ranges appear. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Bayesian uncertainty need not rise reliably under every shift. Monitor distance and calibration.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Compare approximate posterior methods on simulated benchmarks. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use covariate shift 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 probabilistic neural network. Simulate outcomes and compare failure rates, tails and subgroup patterns. Predicted alarms match observed counts. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Passing checks does not prove the network true. Use multiple diagnostics.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Monitor calibration and covariate shift after deployment. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use posterior predictive 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.

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Now average predictions across plausible weights. Assess probability and interval calibration on later or held-out data. Nominal 90% intervals cover near 90%. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Adaptive deployment can change the data stream. Recheck over time.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Keep uncertainty estimates distinct from safety guarantees. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use calibration 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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Variance decomposition separates two uncertainty sources. Combine predictive distributions with downtime and safety costs. Inspect when expected loss is lower. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. A probability threshold is context-dependent. Document it.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Save architecture, priors, likelihood, inference method and seeds. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use decision thresholds 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? Predictive variance adds expected conditional variance and variance across posterior predictive means, not their standard deviations. That is why a careful report includes the construction, not only the final scalar.

Contents · Previous chapter · Next chapter

Begin with a probabilistic neural network. Route high-uncertainty cases to engineers rather than treating automation as certainty. A new regime triggers review. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Human judgement also needs records and feedback. Build a learning loop.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Recalculate aleatoric and epistemic variance before adding them. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use human review 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 average predictions across plausible weights. Save architecture, priors, likelihood, inference settings, seeds and calibration results. A reviewer recreates 0.1056. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Saying Bayesian neural network is insufficient. Archive models and data definitions.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Compare approximate posterior methods on simulated benchmarks. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. Use reproducible reporting 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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Variance decomposition separates two uncertainty sources. BNNs connect averages, variance, functions and computing. Verify the total-variance decomposition. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Standard deviations do not add directly. Plot the three means.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Monitor calibration and covariate shift after deployment. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. 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: eduKate Mathematics Learning System, approach to learning mathematics, Bukit Timah Mathematics curriculum overview, cross-subject learning approach, previous Why Mathematics article, Mathematics Learning Hub. Choose the destination that matches the next question rather than treating links as a checklist.

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Begin with a probabilistic neural network. Bayesian neural networks do not guarantee calibrated uncertainty, causal explanations or safe operation outside training data. They average over a specified uncertainty model. This step matters because uncertainty decomposition, nonlinear predictive maintenance and responsible AI interpretation require the mathematical object, prediction target and reliability question to stay aligned. Independent validation is required. Read MacKay and modern benchmarks.

A reproducible audit should record variables, units, observation boundaries, censoring codes, model version and software settings. Keep uncertainty estimates distinct from safety guarantees. Rebuild the displayed number from saved inputs before interpreting it. A result that cannot be reproduced is not ready to guide reliability work.

Try a small learning experiment: average means 0.20, 0.40 and 0.50, calculate their population variance, and add conditional variance 0.09. 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 checking a model or algorithm? Define the dataset unit, censoring mechanism, time scale, parameterisation, simulation design and diagnostic rule.

Does a satisfactory diagnostic prove the model is true? No. It supports a particular check under stated assumptions and simulation conditions.

Which mathematics matters? Probability defines the reference distribution, simulation makes it observable, proportions summarise tail events, variance describes uncertainty and graphs expose structured mismatch.

How should a student practise? Preserve intermediate quantities, predict directions before calculating, change one input at a time and explain the result in ordinary language.

What follows an internal check? Sensitivity analysis, independent computation, held-out or later-data prediction and a separate decision analysis when cost or safety matters.

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