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eduKateSG · Why Mathematics? · Importance Resampling Bootstrap: Proposal Distributions and Correction Weights
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 8–14
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 15–21
Check assumptions
See where design and calibration matter.Route 4 · Chapters 22–28
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 29–34
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when rare but influential bootstrap regions are too inefficient to reach by uniform resampling? Importance resampling deliberately samples from a proposal distribution that visits those regions more often, then reweights replicates back to the target bootstrap distribution.
Did You Know? The importance weight of a bootstrap replicate depends on its whole frequency vector: every time an observation appears, its target-to-proposal probability ratio contributes another factor.
Full chapter index
Chapters 1–7 · Understand the question
Chapters 8–14 · Reproduce the example
Chapters 15–21 · Check assumptions
Chapters 22–28 · Explore applications
Chapters 29–34 · Study and verify
Confidence limits live in distribution tails. Ordinary resampling may spend most computation near the centre. Importance resampling changes selection probabilities to visit a relevant tail more often.
R imp.weights states that for frequency vector f, proposal probabilities p and target probabilities q, the replicate weight is product((q/p)^f). Read the official documentation.
The proposal p generates resamples. The target q defines the bootstrap distribution for inference, often uniform q_i=1/n. Results are not interpreted under p until likelihood-ratio weights correct the change.
Let q=(.25,.25,.25,.25), p=(.10,.20,.30,.40), and a size-four replicate have frequencies f=(1,0,1,2). It contains observation 1 once, 3 once and 4 twice.
| q | p | f |
|---|---|---|
| .25,.25,.25,.25 | .10,.20,.30,.40 | 1,0,1,2 |
The weight is (0.25/0.10)^1(0.25/0.20)^0(0.25/0.30)^1(0.25/0.40)^2≈0.8138. The zero frequency removes observation 2’s ratio from this product.
| Weight |
|---|
| 0.8138 |
Independent selections multiply probabilities. A frequency vector compresses the ordered sample; raising each ratio to f_i reproduces the likelihood ratio between target and proposal resampling laws.
Weighted expectations are often computed as sum(w_b h_b)/sum(w_b). This removes an unknown common scale but introduces finite-sample bias. The exact estimator must be named.
If a few weights dominate, effective computation is far smaller than B. A common diagnostic is ESS=(sum w)^2/sum(w^2), interpreted as a weight-concentration measure rather than real sample size.
Every resample possible under target q must have positive proposal probability p. If p_i=0 while q_i>0, some target samples are unreachable and weights cannot repair the missing support.
A proposal should increase probability of resamples that move the statistic toward a desired endpoint. Poor targeting can increase variance rather than reduce it.
The R documentation supports defensive mixture weights when several proposal distributions are used. Including some ordinary target sampling can protect against a badly chosen tail proposal.
Products of many ratios can underflow or overflow. Calculate log w=sum f_i log(q_i/p_i), subtract the maximum log weight, exponentiate and normalise.
Weighted empirical distribution functions and weighted quantiles can estimate target tail locations. Quantile conventions and effective tail weight must be recorded.
Importance resampling can make tail endpoint estimation more efficient, but interval coverage still depends on the bootstrap approximation, statistic regularity and correct importance correction.
Rare failure-pattern resamples may drive an upper uncertainty limit. A proposal can oversample high-influence failures, then weights recover the ordinary-bootstrap target.
Extreme-loss configurations are scarce under uniform resampling. Importance sampling can focus computation, but a poor proposal or unbounded weights produces unstable results.
Importance resampling probabilities are a computational device. Survey design weights represent population selection. They can interact but are not interchangeable.
Balanced bootstrap coordinates inclusion counts while retaining the target marginal law. Importance resampling changes proposal probabilities and requires correction weights.
Tilted bootstrap constructs proposal probabilities to place the statistic near chosen values. Importance sampling is the broader correction framework used after sampling from those proposals.
Verify p and q sum to one, check support, reproduce one weight, inspect ESS and maximum normalised weight, compare weighted estimates across proposals and include ordinary sampling.
Importance resampling can fail through missing support, extreme weights, a weak proposal, numerical overflow, small tail ESS, dependence or a wrong statistical bootstrap model.
List p, q, frequencies and log-ratio contributions by observation. Sum log weights, normalise across replicates, calculate ESS and compare weighted with ordinary estimates.
Implement log weights from the frequency matrix, unit-test 0.8138 up to common scaling, use stable log-sum-exp normalisation and reject proposals with missing target support.
Importance sampling is used in risk, reliability, rare-event simulation, Bayesian computation and uncertainty quantification. Practitioners need probability, numerical analysis and domain knowledge.
Estimate a rare bootstrap quantile using uniform and targeted proposals at equal B. Compare endpoint error, ESS and maximum weight across repeated seeds.
Larger raw B does not imply large weighted ESS, unequal proposal probabilities are not survey weights, and reweighting cannot recover target regions with zero proposal support.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The balanced-bootstrap guide coordinates counts without likelihood ratios. The percentile guide explains tail endpoints.
Students should calculate one likelihood ratio and one ESS. Parents can ask why rare cases were sampled more often and how the correction returns to the intended target.
- Recompute 0.8138.
- Explain frequency exponents.
- Calculate ESS.
- Diagnose p_i=0.
- Compare proposal and target.
### Which distribution generates samples? p.
### Which distribution defines inference? q.
### Why reweight? To correct the proposal change.
### Can weights degenerate? Yes.
Report target q, proposal p, support check, B, seed, frequency convention, log-weight method, normalisation, ESS, maximum weight and tail stability.
If all weights are equal, ESS=B. If one weight dominates, ESS approaches 1. This is a computational diagnostic for weighted replicates, not a replacement for the original data sample size.
Confirm target/proposal distinction, common support, stable log weights, adequate tail ESS, defensive mixture when needed, reproducibility and no confusion with substantive sampling weights. Preserve the proposal probabilities and frequency matrix so another analyst can reconstruct every weight. Compare weighted estimates with an ordinary-bootstrap benchmark and explain any material difference. If tail ESS is small, widen the simulation plan or redesign the proposal before interpreting extra decimal places.
Continue through the Mathematics Learning Hub. Rebuild one resampling or diagnostic calculation by hand before trusting a large automated run.
Extended practice laboratory
Practice case 1: Define target q
Setup: Start from the article’s importance-resampling mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Define proposal p
Setup: Start from the article’s importance-resampling mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Check support
Setup: Start from the article’s importance-resampling mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Build frequency vector
Setup: Start from the article’s importance-resampling mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Calculate ratios
Setup: Start from the article’s importance-resampling mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Apply exponents
Setup: Start from the article’s importance-resampling mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Multiply weight
Setup: Start from the article’s importance-resampling mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Use log weight
Setup: Start from the article’s importance-resampling mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Normalise weights
Setup: Start from the article’s importance-resampling mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Calculate weighted mean
Setup: Start from the article’s importance-resampling mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Calculate ESS
Setup: Start from the article’s importance-resampling mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Inspect maximum weight
Setup: Start from the article’s importance-resampling mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Change proposal
Setup: Start from the article’s importance-resampling mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Add defensive mixture
Setup: Start from the article’s importance-resampling mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Target lower tail
Setup: Start from the article’s importance-resampling mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Target upper tail
Setup: Start from the article’s importance-resampling mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Use weighted quantile
Setup: Start from the article’s importance-resampling mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Increase B
Setup: Start from the article’s importance-resampling mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Change seed
Setup: Start from the article’s importance-resampling mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Compare ordinary run
Setup: Start from the article’s importance-resampling mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Add zero support
Setup: Start from the article’s importance-resampling mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Audit numerical stability
Setup: Start from the article’s importance-resampling mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Separate survey weights
Setup: Start from the article’s importance-resampling mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Name limits
Setup: Start from the article’s importance-resampling mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write conclusion
Setup: Start from the article’s importance-resampling mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the probability or weighting identity, and state one diagnostic before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.