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eduKateSG · Why Mathematics? · Tilted Bootstrap: Exponential Weights and Targeted Tail Simulation
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 a bootstrap proposal can be tilted toward a chosen statistic value? The tilted bootstrap changes observation probabilities so resamples concentrate near a desired tail, then uses importance methods to recover the intended inference. Exponential weights turn a mathematical target into an efficient simulation plan.
Did You Know? Exponential tilting multiplies each observation’s baseline weight by exp(lambda times influence). A single parameter lambda can move the proposal expectation smoothly toward a target.
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
Ordinary bootstrap may generate too few resamples near a confidence endpoint. Tilting searches for non-uniform observation probabilities that make the statistic land near chosen values more often.
R tilt.boot runs an initial uniform bootstrap, finds resampling probabilities that place the statistic at required values, then runs importance resampling under those probabilities. Read the official documentation.
Starting from baseline probabilities u_i, a common construction sets p_i(lambda)=u_i exp(lambda L_i)/sum_j u_j exp(lambda L_j), where L_i are empirical influence values.
Use values [1,2,3], equal baselines and lambda=ln 2 applied to x. Unnormalised weights are [2,4,8], so tilted probabilities are [1/7,2/7,4/7].
| Values | Unnormalised | Probabilities |
|---|---|---|
| 1,2,3 | 2,4,8 | 1/7,2/7,4/7 |
The tilted expected value is 1(1/7)+2(2/7)+3(4/7)=17/7≈2.4286, above the uniform mean 2. Positive lambda favours larger values in this illustration.
| Uniform mean | Tilted mean |
|---|---|
| 2 | 17/7 = 2.4286 |
Software chooses lambda or probabilities so a linear approximation or bootstrap quantile matches a requested theta. Root-finding and influence accuracy therefore enter the method.
The initial uniform run estimates target tail values when theta is not supplied. The R documentation warns that extreme alpha values need enough initial replicates.
Empirical influence connects observation probability shifts to statistic movement. Poor influence approximations can tilt in the wrong direction, especially for nonsmooth statistics.
Tilted samples represent the proposal, not the ordinary bootstrap target. Importance weights are used when estimating target distributions or moments.
Lower and upper endpoints may use different proposal distributions, so R accepts a vector of replicate counts: one initial uniform stage plus counts for each tilted target.
The documented tool supports ordinary or balanced simulation within stages. Balancing and tilting solve different Monte Carlo problems and must both be reported.
R tilt.boot can use local frequency smoothing instead of exponential tilting. This requires a positive initial run and a bandwidth choice, creating another sensitivity parameter.
A target far outside the empirical range can demand highly concentrated probabilities, leading to low importance ESS and unstable endpoints. A warning is information, not an obstacle to hide.
Compute log probabilities and normalise with log-sum-exp. Check that probabilities are positive, sum to one and do not collapse onto one observation.
A tilted proposal can focus resamples around a lower or upper confidence limit for a robust location statistic, reducing wasted central simulations.
For an upper failure-rate limit, positive tilting can emphasise influential failures. Scientific dependence and censoring still need a correct resampling model.
Ordinary bootstrap samples uniformly and estimates all regions with one proposal. Tilted bootstrap spends computation near selected statistic values and requires correction.
CHAPTER 18 OF 34 · Check assumptions
18. Comparison with generic importance resampling
Importance sampling is the general proposal-and-weight framework. Tilting is a structured way to choose proposal probabilities using influence and a target statistic.
BCa adjusts which empirical quantiles define endpoints. Tilted bootstrap changes how tail resamples are generated. They address calibration and simulation efficiency at different levels.
Report initial B, target theta or alpha, influence method, lambda/probabilities, stage counts, weight ESS, root convergence and ordinary-bootstrap cross-checks.
Tilted bootstrap can fail with extreme targets, weak influence approximations, nonsmooth statistics, missing support, concentrated weights, dependence or an invalid resampling unit.
Calculate baseline probabilities, influence values, lambda, log weights, normalised tilted probabilities and target expectation. Track each stage and its correction weights.
Use a tested tilt implementation, unit-test probabilities [1/7,2/7,4/7] and expectation 17/7, save solver diagnostics, target values, proposal arrays and seed.
Tilted resampling supports rare-event inference, robust statistics, risk and reliability. Careers require optimisation, numerical stability and substantive model checking.
Simulate a skewed mean and estimate a 97.5% endpoint with uniform and tilted proposals. Compare tail ESS, endpoint error and computation across seeds.
Tilting is not changing the scientific data, a proposal expectation is not the final target estimate, and a concentrated proposal can be worse despite reaching the tail often.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The importance-resampling guide explains correction weights. The BCa guide explains adjusted tail levels.
Students should normalise three exponential weights by hand. Parents can ask what target the proposal was designed to reach and how inference was corrected afterward.
- Derive [1/7,2/7,4/7].
- Calculate 17/7.
- Explain lambda sign.
- Diagnose concentrated probabilities.
- Compare tilting with BCa.
### Does tilting change selection probabilities? Yes.
### Is correction needed? Usually yes.
### What guides the tilt? Influence and target values.
### Can it target both tails? Yes, with separate stages.
Report initial run, target values, influence estimator, tilt rule, lambda or probabilities, stage B, seed, solver status, importance ESS and endpoint stability.
Exponential tilting often finds a probability distribution close to the baseline while satisfying an approximate target constraint. This balances tail focus against excessive distortion.
Confirm achievable targets, stable influence, positive normalised probabilities, converged solver, adequate importance ESS, ordinary cross-check and transparent multi-stage accounting. Preserve each stage separately so proposal-specific weights remain auditable. Re-run the root solver from more than one starting value, compare achieved and requested targets, and report when a tail proposal concentrates on only a few observations. Numerical convergence is necessary but does not establish statistical validity.
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: Set baseline probabilities
Setup: Start from the article’s tilted-bootstrap 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: Estimate influence
Setup: Start from the article’s tilted-bootstrap 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: Choose lambda
Setup: Start from the article’s tilted-bootstrap 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: Compute log weights
Setup: Start from the article’s tilted-bootstrap 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: Exponentiate weights
Setup: Start from the article’s tilted-bootstrap 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: Normalise probabilities
Setup: Start from the article’s tilted-bootstrap 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: Calculate target expectation
Setup: Start from the article’s tilted-bootstrap 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: Solve lambda
Setup: Start from the article’s tilted-bootstrap 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: Run initial bootstrap
Setup: Start from the article’s tilted-bootstrap 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: Choose theta
Setup: Start from the article’s tilted-bootstrap 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: Run lower-tail stage
Setup: Start from the article’s tilted-bootstrap 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: Run upper-tail stage
Setup: Start from the article’s tilted-bootstrap 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: Compute correction weights
Setup: Start from the article’s tilted-bootstrap 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: Calculate ESS
Setup: Start from the article’s tilted-bootstrap 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: Add extreme target
Setup: Start from the article’s tilted-bootstrap 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: Add boundary
Setup: Start from the article’s tilted-bootstrap 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 balanced stage
Setup: Start from the article’s tilted-bootstrap 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: Use smoothing
Setup: Start from the article’s tilted-bootstrap 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 tilted-bootstrap 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 tilted-bootstrap 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: Compare BCa
Setup: Start from the article’s tilted-bootstrap 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 solver
Setup: Start from the article’s tilted-bootstrap 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: Audit support
Setup: Start from the article’s tilted-bootstrap 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 tilted-bootstrap 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 tilted-bootstrap 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.