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eduKateSG · Why Mathematics? · Design-Point Importance Sampling: Shifted Proposals, Likelihood Ratios, Variance Reduction and Failure Modes
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics matters when failures are rare is easy to see in importance sampling. Ordinary Monte Carlo draws mostly safe cases. Design-point importance sampling first uses reliability geometry to locate a likely point on the failure boundary, then shifts a proposal distribution toward that point. Likelihood ratios correct the deliberate change in sampling so the estimate still targets the original failure probability.
This guide develops the density-ratio identity, shifted Gaussian proposals, log weights, effective sample size and multimode limitations. It connects FORM geometry with simulation while keeping their roles separate: the design point guides where to sample, but the final weighted estimator and its diagnostics determine whether the rare-event calculation is credible.
Full chapter index
Chapters 1–5 · Understand the question
Chapters 6–10 · Reproduce the example
Chapters 11–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–25 · Study and verify
Crude Monte Carlo wastes model calls when almost every draw is safe. For probability 10^-5, one million original draws give about ten failures on average. This is the central mathematical mechanism in why move the samples. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Rare counts have high relative uncertainty. Estimate required precision first. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind the importance-sampling identity. For target density f and proposal q with adequate support, p_f=E_q[I(g<=0)f(X)/q(X)]. In the illustrative case, The ratio f/q reverses the deliberate sampling change. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. If q is zero in an important failure region, correction is impossible. Check support before simulation.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat design point as a guide as a model-checking problem rather than a vocabulary exercise. In standard-normal space, the FORM design point is the closest boundary point to the origin. It often identifies a region contributing strongly to probability. Recompute the example from the raw values and label every unit. One local point may not represent several modes. Search from multiple starts.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of shifted gaussian proposal becomes useful only when its mechanism is explicit. A simple proposal keeps unit covariance but shifts its mean from zero toward u*. Consider the worked illustration: If u*=(3,0), draw more samples near that boundary. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. A proposal centred exactly at one point can oversample a narrow local region. Tune covariance and compare alternatives.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://doi.org/10.1016/0167-4730(86)90012-3. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use likelihood ratio for a shift to practise sensitivity analysis. For target N(0,1) and proposal N(m,1), f(x)/q(x)=exp(-mx+m^2/2). With m=3 and x=3.2, the weight is exp(-5.1), about 0.00610. Before accepting the result, identify the quantity that would change the conclusion most. Weights grow rapidly in the opposite tail. Compute in log space.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for unbiased weighted average. With a fixed valid q, the mean of failure indicators times ordinary likelihood ratios is unbiased under standard integrability conditions. The initial example is: Zero contributions from safe draws are still part of the denominator N. Then create a nearby case where the usual shortcut becomes unreliable. Self-normalised weights define a different estimator. Name the exact estimator.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect a one-dimensional tail example to a real decision while protecting the boundary between calculation and judgement. Estimate P(X>3) by drawing X from N(3,1). About half the proposal draws exceed 3, rather than roughly 0.00135 under the target. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The proposal is tuned to this one-sided event. Compare repeated-run variance.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Efficiency depends on failure-weight contributions being reasonably stable. A few enormous weights can dominate the estimate. This is the central mathematical mechanism in weight variability. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Many failures do not guarantee good effective information. Plot log weights and maximum contribution. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind effective sample size. A common weight diagnostic is ESS=(sum w)^2/sum w^2 for nonnegative weights. In the illustrative case, Equal weights give ESS=N; one dominant weight drives ESS toward one. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. ESS does not detect a completely missed region. Combine ESS with mode coverage.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat log-sum-exp arithmetic as a model-checking problem rather than a vocabulary exercise. Log densities prevent underflow and overflow in rare-event tails. Store log f, log q and their difference for every draw. Recompute the example from the raw values and label every unit. Numerical stability cannot repair a poor proposal. Audit extreme values.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of proposal covariance becomes useful only when its mechanism is explicit. Broadening covariance improves exploration but increases weight variability; narrowing it concentrates near the design point. Consider the worked illustration: A covariance factor can be tuned by pilots. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Tuning on one seed can overfit. Repeat across seeds and modes.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use multiple design points to practise sensitivity analysis. Series systems and symmetric failures may have several important boundary points. Use a mixture proposal with a component near each mode. Before accepting the result, identify the quantity that would change the conclusion most. Incorrect mixture weights can inflate variance. Record component labels and weights.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://doi.org/10.1016/0167-4730(89)90003-9. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for mixture density denominator. For a mixture q=sum_j pi_j q_j, every likelihood ratio uses the full mixture density. The initial example is: A sample drawn from component one is still evaluated under all components in q. Then create a nearby case where the usual shortcut becomes unreliable. Using only the generating component gives the wrong ratio. Compute the denominator explicitly.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect finding design points costs money to a real decision while protecting the boundary between calculation and judgement. FORM searches may require gradients and repeated limit-state evaluations before sampling begins. A nonlinear finite-element boundary can make u* expensive to find. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Search cost must be included in efficiency comparisons. Count all model calls.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Importance sampling can remain useful if the centre is approximate, but efficiency may fall. Shift by 2.5 instead of the true threshold 3 and compare. This is the central mathematical mechanism in imperfect design points. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A wrong mode can create severe variance. Run centre sensitivity checks. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind updating form or sorm estimates. Importance sampling can check and refine local reliability approximations using exact limit-state evaluations. In the illustrative case, Use the design point for proposal construction, then estimate probability directly. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Agreement does not validate input distributions. Separate approximation from model-form checks.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat confidence intervals as a model-checking problem rather than a vocabulary exercise. Estimate the variance of weighted contributions, while inspecting heavy tails and repeated runs. Report standard error and coefficient of variation. Recompute the example from the raw values and label every unit. A normal interval may be poor when contributions are skewed. Use replication and robust diagnostics.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of matched-budget comparison becomes useful only when its mechanism is explicit. Compare crude Monte Carlo and importance sampling using equal total true-model calls, including pilots and design-point search. Consider the worked illustration: Show probability estimate, uncertainty and failures observed. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Ignoring setup costs exaggerates savings. Use one declared budget.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use failure-mode discovery to practise sensitivity analysis. A broad pilot or another method may be needed before concentrating on known modes. Random exploratory samples can reveal an unexpected region. Before accepting the result, identify the quantity that would change the conclusion most. The design-point method is confirmatory if discovery is incomplete. Reserve an exploration fraction.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for practice activity. Estimate a normal tail using target draws and shifted draws, then calculate every weight. The initial example is: Repeat with a two-tail event |X|>3 and observe mode loss from a one-sided proposal. Then create a nearby case where the usual shortcut becomes unreliable. Small samples can look erratic. Compare many reproducible runs.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect parent and educator guidance to a real decision while protecting the boundary between calculation and judgement. Importance sampling connects exponentials, normal distributions, averages and fair correction. Ask why biased sampling can still estimate an unbiased target. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. A clever proposal is not permission to ignore missing regions. Use a visible weight table.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
The proposal is not a new physical model and more observed failures do not automatically mean lower error. Likelihood ratios preserve the original target. This is the central mathematical mechanism in common misconceptions. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A design point is not the only possible failure. Demand support and variance diagnostics. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind when not to use it. If the failure domain is unknown, fragmented or poorly represented by design points, adaptive or splitting methods may be safer. In the illustrative case, A discontinuous boundary can defeat local searches. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A forced shift may create extreme weights. Pilot against a known reference.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat research and next reading as a model-checking problem rather than a vocabulary exercise. Harbitz developed an efficient design-point-based sampling method, while later structural-systems work examined importance sampling across multiple modes. Read the primary papers beside FORM and the later cross-entropy owner. Recompute the example from the raw values and label every unit. Historical efficiency claims are problem-specific. Reproduce analytic tail examples.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
Continue with the Mathematics Learning Hub and the previous reliability article. These links own broader probability ideas and the immediately preceding rare-event geometry.
The mathematics of frequently asked questions becomes useful only when its mechanism is explicit. FAQs should cover support, shifts, weights, ESS, mixtures, setup cost and validation. Consider the worked illustration: Each answer should identify f, q and the failure event. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Avoid reporting proposal failure frequency as target probability. Publish the weight diagnostics.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Quick answers
What should I calculate first? Start with the estimand or failure event, then a simple baseline.
Does the advanced method guarantee accuracy? No. It needs input, numerical and model checks.
Which school mathematics matters? Algebra, graphs, probability, functions and careful units all contribute.
How can a student practise safely? Use fictional or open benchmark data and preserve every intermediate value.
What makes a career claim responsible? Describe how the mathematics is used without promising admission, salary or employment.
Extended practice laboratory
Practice case 1: Why move the samples
Setup: Use this chapter’s illustrative case: For probability 10^-5, one million original draws give about ten failures on average.
Task: Estimate required precision first.
Checking target: Rare counts have high relative uncertainty.
Practice case 2: The importance-sampling identity
Setup: Use this chapter’s illustrative case: The ratio f/q reverses the deliberate sampling change.
Task: Check support before simulation.
Checking target: If q is zero in an important failure region, correction is impossible.
Practice case 3: Design point as a guide
Setup: Use this chapter’s illustrative case: It often identifies a region contributing strongly to probability.
Task: Search from multiple starts.
Checking target: One local point may not represent several modes.
Practice case 4: Shifted Gaussian proposal
Setup: Use this chapter’s illustrative case: If u*=(3,0), draw more samples near that boundary.
Task: Tune covariance and compare alternatives.
Checking target: A proposal centred exactly at one point can oversample a narrow local region.
Practice case 5: Likelihood ratio for a shift
Setup: Use this chapter’s illustrative case: With m=3 and x=3.2, the weight is exp(-5.1), about 0.00610.
Task: Compute in log space.
Checking target: Weights grow rapidly in the opposite tail.
Practice case 6: Unbiased weighted average
Setup: Use this chapter’s illustrative case: Zero contributions from safe draws are still part of the denominator N.
Task: Name the exact estimator.
Checking target: Self-normalised weights define a different estimator.
Practice case 7: A one-dimensional tail example
Setup: Use this chapter’s illustrative case: About half the proposal draws exceed 3, rather than roughly 0.00135 under the target.
Task: Compare repeated-run variance.
Checking target: The proposal is tuned to this one-sided event.
Practice case 8: Weight variability
Setup: Use this chapter’s illustrative case: A few enormous weights can dominate the estimate.
Task: Plot log weights and maximum contribution.
Checking target: Many failures do not guarantee good effective information.
Practice case 9: Effective sample size
Setup: Use this chapter’s illustrative case: Equal weights give ESS=N; one dominant weight drives ESS toward one.
Task: Combine ESS with mode coverage.
Checking target: ESS does not detect a completely missed region.
Practice case 10: Log-sum-exp arithmetic
Setup: Use this chapter’s illustrative case: Store log f, log q and their difference for every draw.
Task: Audit extreme values.
Checking target: Numerical stability cannot repair a poor proposal.