Empirical bootstrap laws and support limits

Build exact resampling laws, preserve repeated-record weights and expose what unseen outcomes resampling cannot recover.

40 minutes · intermediate · Original Poker4Quant lesson

What you will learn

Derive an empirical bootstrap law conditional on observed records, reconcile its bias and variance, and construct a support-based witness separating resampling precision from population accuracy.

Prerequisites: Exact laws of estimators · Shrinkage, bias and squared error · Exact confidence sets by test inversion · From outcomes to random variables

Separate the experiment from its empirical replacement

You have one recorded dataset and want to understand how its statistic might vary if data collection were repeated. The true sampling distribution depends on the unknown population law F. An empirical bootstrap substitutes the observed record distribution for F and repeats the statistic calculation there. Write E* and P* for expectations and probabilities conditional on the fixed records. An exact P* calculation can still approximate the wrong population mechanism. The star is a reminder of which experiment supplies the randomness, not a badge of accuracy. This lesson computes a tiny conditional experiment completely so that simulation noise cannot hide that distinction.

Assume the original observations were independent, identically distributed numerical measurements from one stable population. Ordinary record resampling is aimed at that setting; dependent hands within sessions, changing trading regimes and selected success logs require further design. For the workshop the observed sample size is three, each resample also has size three, and the statistic is either the arithmetic mean or maximum. These small records are an arithmetic microscope, not a recommendation that three observations suffice for dependable inference. We make no interval-coverage claim and supply no prior over unobserved populations.

Keep repeated records as separate tickets

Label the observed records A, B and C. In the worked dataset their values are −2, −2 and 4. The empirical distribution assigns one third to each record label, so the numerical value −2 has probability two thirds and 4 has probability one third. Collapsing identical values into one equally weighted category would replace this distribution with another one. Duplicate measurements are legitimate observations; accidentally duplicating a database row is a different data-quality problem. The resampling calculation cannot determine which history produced the records.

A replacement resample is three independent uniform index draws from {A,B,C}. There are 3³=27 equally weighted ordered paths. Path AAC is legal and has mean zero and maximum four. Path AAA is also legal and has mean and maximum −2. A permutation mechanism instead has six ordered paths using every label once. For either symmetric statistic every permutation gives the original statistic exactly. Its zero variance reflects the wrong resampling mechanism for this purpose. Reordering the same records cannot imitate the change in composition that new observations would produce.

Derive the full conditional mean law

Let K count selections of record C. Under replacement, K has probabilities (8,12,6,1)/27 for counts zero through three. One way to see this without invoking a named distribution is to choose the positions of C and then fill each remaining position with either A or B. There are choose(3,k)2^(3−k) such paths. The resample sum is −2(3−K)+4K=−6+6K, giving mean T*=−2+2K. Its support is therefore {−2,0,2,4}. These probabilities belong to resampling the observed records, rather than repeatedly sampling a hypothetical known population.

The conditional mean is [−2×8+0×12+2×6+4×1]/27=0, which equals the observed mean T=0. Thus E*[T*]−T=0 is the conditional bootstrap bias estimate. Its variance is [4×8+0×12+4×6+16×1]/27=72/27=8/3. Independently, one empirical draw has second moment (4+4+16)/3=8 and mean zero. Independence of three replacement draws makes their mean variance 8/3. This exact conditional variance uses all 27 paths; a B−1 denominator belongs to estimating a variance from B simulated statistic values, a separate numerical task.

Change the statistic and expose support limits

For the maximum M*, all three draws must miss C to produce −2. That has probability 8/27. Every other path produces 4, so P*(M*=4)=19/27. The original maximum is M=4, and E*[M*]=(−16+76)/27=20/9. The conditional bootstrap bias is 20/9−4=−16/9. Its variance follows from the two-point gap: 6²(8/27)(19/27)=608/81. The same empirical records can give zero bootstrap bias for their mean and nonzero bootstrap bias for their maximum. Neither result by itself establishes the actual bias under F.

The support restriction is stronger than any numerical precision claim: no replacement resample can contain a measurement above four. Consider an alternative population placing masses 1/2, 1/4 and 1/4 at −2, 4 and 10. It can produce the observed ordered dataset with probability (1/2)²(1/4)=1/16. Yet a fresh three-observation maximum exceeds four with probability 1−(3/4)³=37/64, while the empirical resampling probability is zero. This finite witness does not prove bootstrap failure for every statistic or every large-sample setting. It proves that these records and perfect resampling arithmetic cannot identify an unseen tail.

Audit what more computation actually buys

The workshop lets you change each record value, switch statistics and select replacement or permutation. Cycle the three draw labels to construct a concrete path. Inspect the grouped law, then highlight the bucket equal to the original statistic. Audit the constructed statistic, the mass at that original value, conditional bias and conditional variance. Keep the subtraction centers straight: bias uses the original statistic, while variance uses the resampling mean. A maximum law with negative bootstrap bias is a useful test of that distinction. Fractions and finite decimals are accepted within one millionth.

If every record has the same value, both displayed statistics have conditional variance zero under either protocol. That describes this empirical distribution; it does not prove that the unknown population is constant. Increasing the number B of simulated resamples would reduce numerical error around the same conditional law. Collecting more independent observations changes the empirical distribution itself and might reveal missing support. Enlarging each resample beyond n changes the statistic’s sampling scale and is not equivalent to increasing B. Before applying ordinary record bootstrap reasoning to poker or market logs, identify the independent collection unit, sampling selection, target statistic and the inferential claim that still needs validation.

Worked example

Observed records A=−2, B=−2, C=4 are resampled three times with replacement. Derive the mean and maximum laws and audit an unseen-tail alternative.

Group the 27 index triples by C count. Counts are 8,12,6,1, obtained from choose(3,k)2^(3−k). Their sum is 27.
Map K to mean −2+2K. Weighted first moment is (−16+0+12+4)/27=0. Weighted squared moment is (32+0+24+16)/27=8/3.
Check independently: empirical one-draw variance is (4+4+16)/3=8, divided by resample size three gives 8/3. The original mean and conditional bias are zero.
Map the same paths to maximum: 8 paths give −2 and 19 give 4. Its mean is 20/9, bias −16/9, and variance 36×8×19/27²=608/81.
For a population with masses (1/2,1/4,1/4) at (−2,4,10), the observed ordered records have probability 1/16. A fresh maximum exceeds four with probability 1−27/64=37/64, whereas no empirical resample can exceed four.
Without replacement, all six permutations have mean zero and maximum four. The resulting zero conditional variance diagnoses a different mechanism.

All resampling totals are exact for the empirical law. Unknown-population accuracy and confidence coverage require additional justification.

Hands-on: Build an empirical resampling law

Edit record values, construct index paths, contrast replacement with permutation and audit conditional moments without claiming unknown-population accuracy.

Build an empirical resampling law

Three fixed observed records, three draws. Labels remain distinct when values match. Exact conditional arithmetic does not establish the unknown population law.

Edit observed records

Record A: -2

Record B: -2

Record C: 4

Choose the experiment

Construct an index path

Press a draw to cycle A → B → C → A. Without replacement every label must appear once. Original statistic: 0.000000.

Path AAC: legal under selected protocol.

Group the complete conditional law

All 27 legal ordered paths have equal mass. Select every bucket whose statistic equals the original statistic. Highlight and pressed state mean selected.

Reconcile moments and scope

Enter fractions or finite decimals within 0.000001. Bias subtracts the original statistic; variance centers on the resampling mean. These are exact law moments, not simulation estimates.

Construct a resample and audit its conditional law. Draft lasts for this visit.

Common mistakes

Giving equally likely weight to distinct values rather than observed records.
Permuting records without replacement and interpreting constant statistics as uncertainty estimates.
Centering variance at the original statistic when the bootstrap statistic has nonzero conditional bias.
Calling an exact empirical resampling law the true sampling law.
Increasing B and claiming that missing population support has been recovered.

Keep these ideas

Condition explicitly on the observed records.
Preserve multiplicities and resample n indices with replacement for the ordinary bootstrap.
Group legal index paths before calculating statistic moments.
Distinguish conditional bootstrap bias from true population bias.
More computation improves resampling precision; it cannot discover unseen outcomes.