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Essays arrive in groups rather than one at a time. The most recent group is below in full, and every earlier one after it, newest first.

Essays arrive in groups rather than one at a time, and a group usually opens up a subject not covered before. Between one group and the next nothing changes, so a reader who has seen the most recent group has seen everything.

21 September 2026

18 essays on splitting the units, when the observations repeat each other, the reference distribution the design supplies, three series, and a count, coverage without a distribution, the distribution itself, what conditioning on a variable does, intervals, counted, what makes it checkable, a standard error for a model that is wrong and hierarchy past one number

What the guess is worth, when it is worth anything. The variance cost of an even split relative to the variance-minimising one for a risk difference, against the first arm's proportion, with the second at 0.3. The cost is a pure number: it does not depend on the trial's size. It is exactly zero at 0.3 and at 0.70, where the two arms have the same p(1 − p); it is 0.19% at a half and 4.36% at a tenth. Across the whole range from a tenth to nine tenths it never exceeds 4.36%, which is what the variance-minimising rule is worth here — and what it is worth is the reason it is safe to use with a guess.

Splitting the units

The arm whose variance is its answer

With a binary outcome the allocation rule is a function of the proportions the trial exists to estimate. It costs at most 4.36% of variance to ignore it anywhere between a tenth and nine tenths, because √(p(1−p)) stays within a factor of two of its peak across 98% of the unit interval.

4 figures

The damage and the warning, against the same dial. Two readings at each persistence. In the darker colour, how often a regression between two independent series of 200 steps is called significant at 5%: 4.9% at φ = 0, 34.2% at 0.8, 52.4% at 0.9, 83.4% at a unit root. In the lighter, how often the standard unit-root test refuses a unit root on one of those series — the chance the analyst is told the series is stationary and may be regressed: 87.2% at φ = 0.9 and 31.9% at 0.95. At φ = 0.9 both are high at once, which is a correct diagnostic licensing a regression that is wrong half the time.

When the observations repeat each other

The cliff that is a slope

A regression between two independent series is called significant 4.9% of the time at no persistence, 52.4% at a lag-one correlation of 0.9, and 83.4% at a unit root. The rule the field offers asks whether the last of those holds, and at 0.9 the unit-root test correctly refuses one 87.2% of the time.

7 figures

An exact test rejecting a true hypothesis a fifth of the time. How often each analysis reports an effect when the average treatment effect is exactly zero and the effect varies between units, at 150 units with 25% treated. The permutation test on the difference in means reads 4.20% where the effect is constant — where the two nulls coincide and its exactness applies — and 22.93% where the effect varies with a standard deviation of 3. The same test on the studentised difference reads 6.27% there, and the ordinary large-sample t, which makes no exactness claim at all, reads 6.60%.

The reference distribution the design supplies

The null the exactness is for

A permutation test is exact under the hypothesis that the treatment changed nothing for anybody. Under the hypothesis it changed nothing on average, with a quarter of the units treated and the effect varying between them, it rejects a true null 22.93% of the time.

5 figures

The bounded error and the unbounded one. How the sequential trace procedure's answer is distributed, against the sample length, for a three-series system with 2 genuine relations. Over-counting — claiming a stationary combination that is a random walk — reads 4.9%, 7.2%, 5.7%, 6.2%, 5.9%, 4.2% across the six lengths, never far from the 5% of a single test. Under-counting reads 69.5%, 40.2%, 14.0%, 0.5%, 0.0%, 0.0%. The procedure is described as a 5% rule and the 5% applies to one of those columns.

Three series, and a count

The rank is a decision

The sequential procedure's 5% bounds one of its two errors. Over-counting reads between 4.2% and 7.2% at every sample length from fifty observations to three hundred; under-counting reads 69.5% at fifty and 0.0% at three hundred, and nothing in the procedure bounds it.

6 figures

One statistic that is right under both hypotheses. Rejection rates for both statistics under both nulls, at 25% of 150 units treated, with the weak-null readings taken at an effect spread of 3. The difference in means is exact under the sharp null and rejects 22.93% of true weak nulls. The studentised difference is exact under the sharp null — 4.07% — and reads 6.27% under the weak one. The repair is a change of statistic inside the same construction: the same re-randomisations, the same fixed outcomes, a different number compared across them.

The reference distribution the design supplies

A statistic that is exact twice

Dividing the difference in means by its own separate-variance standard error before permuting takes the rejection rate under a true weak null from 20.47% to 6.07%, keeps the exactness under the sharp null at 4.07%, and costs 0.8 points of power against a real effect. At an even split it changes nothing at all, in every draw.

6 figures

Six cells, and 5% is the right answer in all of them. How often a regression between two independently generated series is called significant at the 5% level, for two worlds and three treatments, at 200 observations. Every pair is independent by construction, so 5% is correct everywhere and every other reading is a failure. Untreated: 82.9% and 100.0%. With a fitted line removed: 74.2% and 33.5%. Differenced: 5.0% and 5.2%. The treatment that controls the rate in both worlds is the one that discards the level and the trend, which is the quantity a study of trending series was about.

When the observations repeat each other

The repair that keeps the question

A regression between two independent trending series is significant 82.9% of the time on random walks and 100.0% on trend-stationary ones. Subtracting a fitted line leaves 74.2% and 33.5%; differencing leaves 5.0% and 5.2% and throws away the trend the study was about.

7 figures

Too small breaks it and too large does not. Coverage of the weighted interval against the factor the true likelihood ratio is multiplied by, at a test population 80.0% drawn from the noisier group and 200 calibration points. The exact weight is the factor of 1 and covers 95.70%. Overstating it costs nothing: 96.13% at sixteen times too large. Understating it costs, and costs steeply below about a half — 94.93% at half, 88.37% at an eighth and 67.90% at a thirtieth. The question this answers was whether a wrong weight degrades smoothly or falls off a cliff, and the answer is that it does neither symmetrically: the curve is smooth and one-sided.

Coverage without a distribution

The weight that has to be estimated

A likelihood ratio sixteen times too large costs 5.5% of interval width and no coverage at all; one a thirtieth of the right size covers 67.90%. The estimate from a batch of five unlabelled covariates covers 95.10% against an exact repair's 95.30%, and the binomial says why.

4 figures

Three contrasts on one dataset, three different splits. The variance-minimising allocation for each of three ways of reporting the same two-arm comparison, against the first arm's proportion, with the second at 0.1. A risk difference wants the arm with the larger p(1 − p) to get more units; a log odds ratio wants it to get fewer, and the two curves are exact reflections of each other in the half line. A log risk ratio wants something else again. At a first-arm proportion of 0.6 they ask for 62.0%, 21.4% and 38.0% of the units. A trial reporting more than one of them cannot be optimal for either.

Splitting the units

Two contrasts, one split

A risk difference wants 62.0% of the units in the first arm, a log risk ratio wants 21.4% and a log odds ratio wants 38.0% — on one dataset, with one pair of proportions. The difference's rule and the odds ratio's are exact reflections of each other, so no split can be near-optimal for both.

4 figures

What each wrong count costs, 4 steps ahead. Squared forecast error 4 steps ahead at each imposed rank, relative to the correctly specified fit, at 200 observations. With 1 genuine relations, imposing 0 costs 13.3% and imposing 2 costs 4.8%. With 2 genuine relations, imposing 1 costs 15.6% and imposing 3 costs 2.5%. Under-counting is the more expensive mistake in both systems, and it is the one the procedure's level does not bound.

Three series, and a count

Which mistake about the rank costs

On a system with two relations, imposing none costs 29.2% of squared forecast error and imposing three costs 2.5%. The expensive mistake is under-counting, which is the error the procedure's 5% does not bound — so the guarantee protects the cheap side.

6 figures

Three detectors for one departure, all at 5%. How often each of three checks on the calibration scores fires, against the size of the drift, with every critical value simulated under no drift so that all three sit at 5.0% exactly. The incumbent — a rank comparison of the first half of the scores against the second — reaches four-in-five power at a growth factor of 4.31. Reading each score's rank against its position reaches it at 2.65, and the largest running departure of the scores from their mean at 2.12. The ordering of the three is the ordering by how much of the sample's arrangement each one uses.

Coverage without a distribution

A detector built for the ordering

The best of three checks for a drifting scale fires at half the growth factor the standard one needs — 2.12 against 4.31 — and still leaves 6.50 points of coverage gone before it does, against 0.51 for serial correlation. The reversal was not a property of the test.

5 figures

The law is the eigenvalues, and nothing else. The mean and the skewness of n(ĝ − g) at the stationary point, measured over 40,000 draws, against the closed forms ½ Σλ and 2√2 Σλ³ ⁄ (Σλ²)^(3⁄2). The worst disagreement anywhere is 0.028. In one variable the second-order law is a single χ² and its sign is the sign of g″; here it is a weighted sum with the Hessian's eigenvalues as weights, so a bowl and a valley differ in both moments and a saddle has both equal to zero.

The distribution itself

A flat point with more than one direction

At a stationary point of a function of several means the second-order law is ½ Z′HZ, so the bias is half the Hessian's trace — 2.008 for a bowl, 5.028 for a valley, and −0.006 for a saddle, where the eigenvalues cancel. The saddle's coverage is the worst of the three.

4 figures

One of these converges and the other does not. Two measurements on the same fits, against the sample length, for a system with 2 genuine relations. The distance from the fitted plane to the true plane falls from 0.1438 at 100 observations to 0.0075 at 1600 — halving with each doubling, which is the 1/n rate this field's estimates converge at. The angle between the leading fitted relation and the leading generating one reads 29.6° and 29.0° at those same lengths, and is flat in between. The plane is an estimate; the relation inside it is not.

Three series, and a count

A space is not a relation

The fitted plane approaches the true one at rate 1/n — 0.1438 at a hundred observations and 0.0075 at sixteen hundred. The angle between the leading fitted relation and the leading generating one reads 29.6° and 29.0° at those same lengths, and never moves.

5 figures

What conditioning on a variable does

Adjusting for a shadow

A covariate that is 80% signal removes 68.85% of the confounding, not 80% — the share is λ(1 − ρ²)/(1 − λρ²) and it is below the reliability everywhere. The residual bias is 0.1084 against an effect of 0.5, and at 25,600 rows it is 17.6 standard errors wide.

4 figures

Three promises, and no procedure keeps all three. Average coverage and worst-case coverage for four 95% intervals for a proportion at n = 40, computed exactly. Their expected widths are 0.2418, 0.2417, 0.2472, 0.2641 in the same order. The textbook interval and the score interval have the same expected width to four digits — 0.2418 and 0.2417 — and worst-case coverages of 55.31% and 92.21%. The exact interval never breaks its promise and is 9.3% wider than the score interval to do it. Each of the three columns orders the four procedures differently.

Intervals, counted

An interval that covers and says nothing

A procedure returning the whole line 95% of the time and the empty set otherwise has coverage exactly 95% at every parameter value. Two real intervals at forty observations have expected widths of 0.2418 and 0.2417 and worst-case coverages of 55.31% and 92.21%.

4 figures

When the observations repeat each other

How slow a return a sample can see

At two hundred observations the test finds a gap that halves in five steps four times in five, one that halves in eight 37.3% of the time, and one that halves in fifty 4.95% of the time — which is the rate at which it finds pairs with no mechanism at all. The boundary moves with the sample, not with its square root.

6 figures

Two companions on one simulation, two hundredfold apart. How many times as many draws each companion is worth, on the same 4,000 simulated samples of 40 observations. The coverage of the interval is estimated with the observed count as its companion, whose expectation is 12 exactly; they correlate at 0.2665 and the companion is worth 1.08 times the draws. The expected width is estimated with p̂(1 − p̂) as its companion, whose expectation is 0.20475 exactly; they correlate at 0.9977 because the width is a monotone function of it, and the companion is worth 214 times the draws — 856 thousand simulated samples' worth of precision from four thousand.

What makes it checkable

The check worth more than the check

The same exactly known companion that verifies a simulation can sharpen it. On one set of four thousand draws, one companion is worth 1.08 times the draws and another is worth 214 times them, and the factor is 1 − ρ² with nothing else in it.

3 figures

One estimator, three answers, and only the reference changes. Coverage of the cluster-robust 95% interval for the slope against the number of clusters, at 30 rows in each. The estimator is identical in all three curves; what differs is the number it is compared against. At 5 clusters it covers 75.05% against a normal, 85.30% against a t on 4 degrees of freedom and 87.95% against a t on 3. At 80 clusters the three agree to within a point. The correction costs nothing: the same standard error, a different table.

A standard error for a model that is wrong

The reference the sandwich is read against

The cluster-robust interval covers 75.05% at five clusters and 93.58% at eighty. The same estimate read against a t on G − 2 covers 87.95% at five, and the estimator is unchanged — three hundred rows grouped into five clusters cover 74.28% where the same three hundred grouped into seventy-five cover 94.63%.

5 figures

Each interval covers one question and not the other. Coverage of each interval for the overall mean, scored against both estimands, over 20,000 two-site studies of 10 observations apiece. The fixed-effect interval covers the mean of the two sites in hand 96.37% of the time and the population mean 54.77%. The random-effects interval covers the population mean 94.96% — exactly its level, from one degree of freedom — and over-covers the two sites in hand at 98.25%. Both are correct; they are answers to different questions printed in the same place.

Hierarchy past one number

What a two-unit study should report

The fixed-effect interval covers the mean of the two sites in hand 96.37% of the time and the population mean 54.77%. The random-effects interval covers the population mean 94.96% — exactly its level, from one degree of freedom — and is 11.6 times wider.

4 figures

Before that

Everything published earlier, newest first. Titles only — the cards are on the full listing.

17 September 2026

18 essays on past the first term of the normal approximation, what partial pooling does to one group, to the set, and to a ranking, a proportion's interval near the boundary, and the coin, an interval read beside something else, what a sample-size calculation was given, shape, and what it does to a two-sample test and the interval that holds observations, not a mean

A bound written for a coin — past the first term of the normal approximation
A group from the population's own tail — what partial pooling does to one group, to the set, and to a ranking
A hole no sample size fills — a proportion's interval near the boundary, and the coin
Five times in six — an interval read beside something else
The spread a pilot supplies — what a sample-size calculation was given
A correction that goes below zero — past the first term of the normal approximation
Estimates that are too alike — what partial pooling does to one group, to the set, and to a ranking
The chance a trial succeeds — what a sample-size calculation was given
Two intervals that overlap — an interval read beside something else
What a guaranteed minimum costs — a proportion's interval near the boundary, and the coin
A league table of a hundred — what partial pooling does to one group, to the set, and to a ranking
An approximation built at the threshold — past the first term of the normal approximation
An outcome cut in two — what a sample-size calculation was given
Significant in one, not in the other — an interval read beside something else
The coin that makes it exact — a proportion's interval near the boundary, and the coin
The skewness of a difference — shape, and what it does to a two-sample test
All of the next ten — the interval that holds observations, not a mean
The side a bound is read from — shape, and what it does to a two-sample test

16 September 2026

20 essays on the analyses that were available and not run, what a summary of a scatter is a property of, what a diagnostic plot is showing, when the stratified answer and the pooled one disagree, two tests, a threshold, and the rate they are read against, the interval that holds observations, not a mean and shape, and what it does to a two-sample test

How many analyses there really were — the analyses that were available and not run
R² is a property of the design — what a summary of a scatter is a property of
The band the eye was standing in for — what a diagnostic plot is showing
The reversal a coin cannot prevent — when the stratified answer and the pooled one disagree
The second test that is not a second opinion — two tests, a threshold, and the rate they are read against
Two standard deviations of what — the interval that holds observations, not a mean
Where the two tails disagree — shape, and what it does to a two-sample test
A degrees of freedom that is not a count — shape, and what it does to a two-sample test
A tenth as wide, and both of them right — the interval that holds observations, not a mean
The change that is not confounding — when the stratified answer and the pooled one disagree
The plot is about the wrong quantity — what a diagnostic plot is showing
The t statistic wearing different clothes — what a summary of a scatter is a property of
The test is a point somebody chose — two tests, a threshold, and the rate they are read against
What naming it in advance costs — the analyses that were available and not run
Conditioning on what the treatment caused — when the stratified answer and the pooled one disagree
Ninety-three observations, and nothing assumed — the interval that holds observations, not a mean
Residuals are not the errors — what a diagnostic plot is showing
The correction that makes the estimate worse — the analyses that were available and not run
The prevalence the test has to estimate — two tests, a threshold, and the rate they are read against
The summary that was meant to work — what a summary of a scatter is a property of

15 September 2026

20 essays on comparing two forecasters, a design chosen rather than looked up, groups that borrow, the prior, doing visible work, the surface between the corners, decided before the data and hierarchy past one number

Correcting the forecast instead — comparing two forecasters
How many places a design goes — a design chosen rather than looked up
The fewest groups that can borrow — groups that borrow
The shortest interval, and the one that does not move — the prior, doing visible work
The sign the curvature has — the surface between the corners
The word a fraction costs — decided before the data
Two groupings that cross — hierarchy past one number
A level with two units — hierarchy past one number
An interval for something else — the prior, doing visible work
Augmenting a design that has already run — a design chosen rather than looked up
The correction that leaves the region — comparing two forecasters
The design that refuses the corners — decided before the data
When the best setting is outside the region — the surface between the corners
Where the borrowing goes — groups that borrow
One population, or two — groups that borrow
The criterion with no derivative — a design chosen rather than looked up
The run that confirms it — the surface between the corners
The run that did not happen — decided before the data
What the interval is short by — comparing two forecasters
When the prior is confident and wrong — the prior, doing visible work

13 September 2026

20 essays on stopping rules, when a fixed width is reached, corrections, and what each controls, the interval, studentised, a forecast that is a probability, when the data stops early, a variable that moves one thing only, the value that is not there, weighting one sample into another, the tail past the last observation, reversals that are not errors, tests, and the second number and regression, and what the summary hides

The effect a stopped trial reports — stopping rules
The trials that stopped early — when a fixed width is reached
A width rule on skewed outcomes — when a fixed width is reached
False discoveries that arrive together — corrections, and what each controls
The outcomes a trial could have stopped with — stopping rules
A boundary for giving up — stopping rules
Estimating how many nulls are true — corrections, and what each controls
The interval with no resampling in it — the interval, studentised
A look the trend asked for — stopping rules
An order that spends the error rate — corrections, and what each controls
A forecaster that rounds — a forecast that is a probability
A horizon chosen after looking — when the data stops early
A standard error that knows about the instruments — a variable that moves one thing only
Intervals for the findings — corrections, and what each controls
The assumption that identifies the mechanism — the value that is not there
The moments a balance is told — weighting one sample into another
The run length a declustering chooses — the tail past the last observation
The slope of a density nobody can see — reversals that are not errors
The smallest of three combinations — tests, and the second number
The start an efficient robust line inherits — regression, and what the summary hides

11 September 2026

20 essays on what makes it checkable, when the data stops early, reversals that are not errors, tests, and the second number, regression, and what the summary hides, the distribution itself, weighting one sample into another, a variable that moves one thing only and a forecast that is a probability

A coverage table with its own error — what makes it checkable
The interval at the end of the curve — when the data stops early
A dropout the data cannot see — when the data stops early
The same draws for both methods — what makes it checkable
Two analyses of one baseline — reversals that are not errors
A simulation that stops when it looks settled — what makes it checkable
One minus Kaplan–Meier is not a risk — when the data stops early
The measurement that got them enrolled — reversals that are not errors
The p-value a replication gets — tests, and the second number
Two points that hide each other — regression, and what the summary hides
Where the derivative is zero — the distribution itself
A lead that a heavy tail keeps — reversals that are not errors
A ratio whose interval has to be the whole line — the distribution itself
A robust loss and a far x — regression, and what the summary hides
A weight fitted to balance — weighting one sample into another
Leaving each row out of its own first stage — a variable that moves one thing only
The draws aimed at the tail — what makes it checkable
The hazard ratio the follow-up chose — when the data stops early
The liar with two answers — a forecast that is a probability
Two ways to combine p-values — tests, and the second number

7 September 2026

50 essays on weighting one sample into another, a forecast that is a probability, coverage without a distribution, what conditioning on a variable does, a variable that moves one thing only, the value that is not there, the tail past the last observation, a standard error for a model that is wrong, what decides whether a tuning list decides, the interval, studentised, the same table at seven correlations, a probe from what the rule blocks and a charge that is not a straight line

A score that balances — weighting one sample into another
An identity in three terms — a forecast that is a probability
Coverage from exchangeability alone — coverage without a distribution
One arithmetic, three decisions — what conditioning on a variable does
The assumption nothing tests — a variable that moves one thing only
Three mechanisms and one dataset — the value that is not there
Three shapes, one limit — the tail past the last observation
What a wrong model estimates — a standard error for a model that is wrong
A curve that is a binning — a forecast that is a probability
Dropping the incomplete rows — the value that is not there
How many observations a weight leaves — weighting one sample into another
The bread and the filling — a standard error for a model that is wrong
The maximum converges slowly — the tail past the last observation
The two worlds that look the same — what conditioning on a variable does
Weak, and back where it started — a variable that moves one thing only
What the split costs — coverage without a distribution
Adjusting for everything — what conditioning on a variable does
Calibrated and useless — a forecast that is a probability
Marginal is not conditional — coverage without a distribution
One imputation is not an observation — the value that is not there
Robust is not free — a standard error for a model that is wrong
The region with no comparison — weighting one sample into another
The threshold is a dial — the tail past the last observation
What the first stage does not know — a variable that moves one thing only
A collider before the treatment — what conditioning on a variable does
A level with no data in it — the tail past the last observation
A score that rewards lying — a forecast that is a probability
A step that is not a ratio — what decides whether a tuning list decides
Either model, but not neither — weighting one sample into another
The count or the length — the interval, studentised
The other dial — the same table at seven correlations
The score is the modelling — coverage without a distribution
The variance between imputations — the value that is not there
Three corrections and a leverage — a standard error for a model that is wrong
Whose effect it is — a variable that moves one thing only
A set of pairs, not a vector — a probe from what the rule blocks
An imputation model the analysis does not contain — the value that is not there
Right for the wrong reason — a standard error for a model that is wrong
The estimated weight is the better one — weighting one sample into another
The miscalibration a perfect forecaster shows — a forecast that is a probability
The sample is a condition — what conditioning on a variable does
Two instruments that disagree — a variable that moves one thing only
Two intervals for one return level — the tail past the last observation
What the correction assumes — a charge that is not a straight line
When the order matters — coverage without a distribution
A charge that reads the draw — a charge that is not a straight line
The clustering the tail has — the tail past the last observation
The count that is not the rows — a standard error for a model that is wrong
The mechanism the data cannot see — the value that is not there
The variable the treatment caused — what conditioning on a variable does

2 September 2026

20 essays on the same table at seven correlations, the interval, studentised, what decides whether a tuning list decides, a charge that is not a straight line, overlap and complementarity, separated and a probe from what the rule blocks

A margin that turns over — the same table at seven correlations
An interval that carries its scale — the interval, studentised
The eighth that was not a constant — what decides whether a tuning list decides
The width a band is measured in — a charge that is not a straight line
Two effects in one number — overlap and complementarity, separated
What the rule blocks — a probe from what the rule blocks
A lag the sample has less of — a charge that is not a straight line
A model and a count — a probe from what the rule blocks
The zero that was a crossing — the same table at seven correlations
Two factors pointing opposite ways — what decides whether a tuning list decides
What a zero is made of — overlap and complementarity, separated
What studentising costs — the interval, studentised
A line that beats two curves — a charge that is not a straight line
A quantity that loses to a heuristic — a probe from what the rule blocks
A split that depends on the order — overlap and complementarity, separated
A table and a list — what decides whether a tuning list decides
An answer that changes — the same table at seven correlations
The ordering reverses again — the interval, studentised
Counting it exactly does not help — a probe from what the rule blocks
What a better charge buys — a charge that is not a straight line

31 August 2026

20 essays on the rate and the size of a disagreement, a charge for a covariance's own dimension, the block length read on a quantile, a probe chosen rather than picked, both halves of the dependence at once and two searches over different features

A rate times a size — the rate and the size of a disagreement
The charge nobody derived — a charge for a covariance's own dimension
The instrument and the reading — the block length read on a quantile
The part the rule already took — a probe chosen rather than picked
Two failures that cancel — both halves of the dependence at once
Two searches that share nothing — two searches over different features
A length for each instrument — the block length read on a quantile
A probe chosen from the design — a probe chosen rather than picked
A search that is already the other — two searches over different features
A symmetry that was not enough — both halves of the dependence at once
The quarrel that changes the winner — the rate and the size of a disagreement
What a window leaves free — a charge for a covariance's own dimension
A copula that halves a marginal — both halves of the dependence at once
A width that moves and an error that does not — a charge for a covariance's own dimension
How often it matters — the rate and the size of a disagreement
The reversal that was the instrument's — the block length read on a quantile
Three quarters of the way to one search — two searches over different features
What a chosen probe finds — a probe chosen rather than picked
The zero that survives both — both halves of the dependence at once
What the interval covers — the block length read on a quantile

30 August 2026

20 essays on a covariance with no parameter, the other half of the dependence, how long the list is, a block length chosen from the data, the diagnostic after the trial and two searches over one sample

A family before a fit — a covariance with no parameter
A zero that is arithmetic — the other half of the dependence
The comparison that was not made — how long the list is
The length nobody has — a block length chosen from the data
The statistic the p-value is about — the diagnostic after the trial
Two searches, one sample — two searches over one sample
A probe nobody chose — the diagnostic after the trial
An ordering that depends on the rule — a block length chosen from the data
The charge that is not a sum — two searches over one sample
The plug-in and the maximum — a covariance with no parameter
The symmetry the marginals could not show — the other half of the dependence
The volume a whitening moves — how long the list is
A charge that depends on the rule — two searches over one sample
A list is not a rule — how long the list is
Half a reference distribution — the diagnostic after the trial
Nothing in the fit picks the width — a covariance with no parameter
What choosing the length costs — a block length chosen from the data
Which tail the cut sits in — the other half of the dependence
Before the trial and after — the diagnostic after the trial
What fitting them together buys — a covariance with no parameter

29 August 2026

20 essays on paying for a search, fitted together, or fitted after, what a chain cannot report, a guarantee that needed a symmetry and where a taper's case begins

A break that was looked for — paying for a search
A dependence fitted with the line — fitted together, or fitted after
A test rather than a survey — what a chain cannot report
A zero that rests on a symmetry — a guarantee that needed a symmetry
The gap a sample shows — where a taper's case begins
A split survives what a mean does not — a guarantee that needed a symmetry
Bias is not the whole of it — where a taper's case begins
The fit that takes the memory out — fitted together, or fitted after
The statistic that changes sign — what a chain cannot report
What a search costs in parameters — paying for a search
A defect that is about size — what a chain cannot report
Choosing whether to break — paying for a search
Iterating is not maximising — fitted together, or fitted after
Measuring a variance rather than a quantile — where a taper's case begins
The cut that is not a quantile — a guarantee that needed a symmetry
A second break on a flat profile — paying for a search
A window for every candidate — fitted together, or fitted after
Balancing a skewed covariate — a guarantee that needed a symmetry
The diagnostic at two hundred — what a chain cannot report
The error no window repairs — where a taper's case begins

28 August 2026

12 essays on a block, weighted inside itself, the shape a dependence has, what a dictionary buys and what it costs and when a fixed width is reached

A block weighted inside itself — a block, weighted inside itself
A dependence with a shape — the shape a dependence has
A dictionary that is neither — what a dictionary buys and what it costs
A width the trial has to stop for — when a fixed width is reached
A taper and a critical value — a block, weighted inside itself
Stopping on the arms — when a fixed width is reached
What the extra function buys — what a dictionary buys and what it costs
Where the generality runs out — the shape a dependence has
The set a dictionary leaves — what a dictionary buys and what it costs
The window a whitening wants — the shape a dependence has
The order the tail is drawn at — the shape a dependence has
The walk that cannot cross — what a dictionary buys and what it costs

26 August 2026

12 essays on estimating the dependence, not naming it, a cut point, at a correlation and what a block may vary

A covariance with no parameter in it — estimating the dependence, not naming it
A cut is not a polynomial, and it does not have to be — a cut point, at a correlation
A proposal that moves more than two units — what a block may vary
Stationary is not convergent — what a block may vary
The arcsine that closes it, and the error that was overstated — a cut point, at a correlation
The window that has to be chosen, and the term that was dropped — estimating the dependence, not naming it
The triangle that was not the multiplier's — estimating the dependence, not naming it
The zero that survives a cut — a cut point, at a correlation
Where the gain is, and where the decision is — what a block may vary
A ratio that changes between blocks — what a block may vary
Errors generated from a fitted model — estimating the dependence, not naming it
The bias that lands in the slope — what a block may vary

24 August 2026

12 essays on counting what is independent, when the two are not independent and the weights the corner needs

A penalty is a trace — counting what is independent
The fourth moment that was missing — when the two are not independent
Weights that need only a ratio — the weights the corner needs
A zero that was an assumption — when the two are not independent
Blinded, and still exact — the weights the corner needs
One number for a table of candidates — counting what is independent
The condition that cannot be dropped — the weights the corner needs
The repair that was exact and made it worse — counting what is independent
Walking the admissible set — when the two are not independent
Draws that repeat each other — when the two are not independent
The residuals are not the errors — counting what is independent
What a multiplier cannot keep — counting what is independent

23 August 2026

12 essays on scoring a search without spending data, when the set is too large to walk and a promise about two arms

A criterion is a prediction of the hold-out — scoring a search without spending data
A dictionary that is a product — when the set is too large to walk
A width promised for a difference — a promise about two arms
Where the enumeration stops — when the set is too large to walk
Where the two searches cross — scoring a search without spending data
Which weights are the inverse variances — a promise about two arms
A count that has to be estimated — when the set is too large to walk
The degrees of freedom in the sums — a promise about two arms
Two defects and one resampling — scoring a search without spending data
How long a block a multiplier shares — scoring a search without spending data
What a reference distribution costs to sample — when the set is too large to walk
What a two-arm rule may not pool — a promise about two arms

22 August 2026

12 essays on choosing what the rule reads, the block size as a schedule and a search with no fixed point

A basis is a subspace — choosing what the rule reads
A block size that changes — the block size as a schedule
When the benchmark is a candidate — a search with no fixed point
The displacement is a parameter count — a search with no fixed point
Two degrees of freedom, one total — the block size as a schedule
Which shapes are worth protecting — choosing what the rule reads
Residuals that keep their own variance — a search with no fixed point
What a schedule actually buys — the block size as a schedule
Where the guarantee is exactly zero — choosing what the rule reads
A schedule that reads the mean — the block size as a schedule
The corner the test is calibrated at — a search with no fixed point
When the constraints run out — choosing what the rule reads

21 August 2026

12 essays on searching among fitted models, the shape the covariate enters by and what the procedure may not read

A table of nested models — searching among fitted models
Balanced on the wrong function — the shape the covariate enters by
The guess with two numbers in it — what the procedure may not read
A null with a model in it — searching among fitted models
A threshold in the tail — the shape the covariate enters by
The worst case in two directions — what the procedure may not read
The rule that cannot see the mean — what the procedure may not read
The weight that is a vector — searching among fitted models
Three functions of one number — the shape the covariate enters by
The analysis and the shape — the shape the covariate enters by
What the blindfold costs — what the procedure may not read
When every null is true — searching among fitted models

19 August 2026

12 essays on balancing what has no levels, what the design is asked to guarantee and the best of a set, and what the search costs

A covariate with no levels — balancing what has no levels
An efficiency that is a ratio — what the design is asked to guarantee
What the other forecast adds — the best of a set, and what the search costs
Eight forecasters and one benchmark — the best of a set, and what the search costs
Protecting one parameter over a range — what the design is asked to guarantee
The rule that reads the number — balancing what has no levels
Stopping when it is precise enough — what the design is asked to guarantee
The models that were never in the running — the best of a set, and what the search costs
What the balanced trial is worth — balancing what has no levels
A distribution drawn from the null — the best of a set, and what the search costs
Balancing more than one number — balancing what has no levels
The interval after a stop it chose — what the design is asked to guarantee

18 August 2026

12 essays on a design that assumes less, more arms than two and comparing two forecasters

The design for the worst case — a design that assumes less
Three arms and three scores — more arms than two
Which forecast is better — comparing two forecasters
Balancing towards unequal targets — more arms than two
When one model contains the other — comparing two forecasters
Where the minimum is attained — a design that assumes less
Correcting the persistence — comparing two forecasters
The analysis after three arms — more arms than two
The design that stops guessing — a design that assumes less
Guessing one arm in three — more arms than two
The repair that moves the wrong number — comparing two forecasters
What a design chosen from the data costs — a design that assumes less

17 August 2026

12 essays on balancing on what was recorded first, the criterion, and what it assumes and the observation that has not happened

Balancing what is known in advance — balancing on what was recorded first
The family behind the letters — the criterion, and what it assumes
What the model says next — the observation that has not happened
The interval that forgets it estimated — the observation that has not happened
The rule that can be guessed — balancing on what was recorded first
The two terms anybody wanted — the criterion, and what it assumes
Choosing the order — the observation that has not happened
The analysis has to know the rule — balancing on what was recorded first
The design that needs the answer — the criterion, and what it assumes
The design that hedges — the criterion, and what it assumes
The interval after the choice — the observation that has not happened
The reference the covariates supply — balancing on what was recorded first

15 August 2026

12 essays on a design chosen rather than looked up, the reference distribution the design supplies and three series, and a count

A design is a number — a design chosen rather than looked up
The experiments that could have happened — the reference distribution the design supplies
Three series and a count — three series, and a count
The test that needs the rule — the reference distribution the design supplies
The theorem that says when to stop — a design chosen rather than looked up
Which series goes on the left — three series, and a count
Counting what is still wandering — three series, and a count
Four letters and two camps — a design chosen rather than looked up
The plus one and the round number — the reference distribution the design supplies
The design that has to be integers — a design chosen rather than looked up
What the exactness buys — the reference distribution the design supplies
Which series does the moving — three series, and a count

14 August 2026

12 essays on designs that change while they run, splitting the units and the surface between the corners

Choosing n after looking — designs that change while they run
Not half and half — splitting the units
The design that cannot see a curve — the surface between the corners
Randomising towards the winner — designs that change while they run
The cost of a unit — splitting the units
Walking up the gradient — the surface between the corners
Dropping the losers — designs that change while they run
One control, many arms — splitting the units
Three levels, and the ring where the design says the same thing — the surface between the corners
Allocating on a guess — splitting the units
The estimate after the choice — designs that change while they run
The optimum is a ratio, and its interval is sometimes the whole line — the surface between the corners

11 August 2026

12 essays on series that move together, hierarchy past one number and the spread, and its own uncertainty

The regression that is not spurious — series that move together
The slope that borrows — hierarchy past one number
What the plug-in forgets — the spread, and its own uncertainty
A prior on the spread — the spread, and its own uncertainty
Pooling a proportion — hierarchy past one number
The test with no table — series that move together
Borrowing towards a line — hierarchy past one number
The model that corrects its error — series that move together
When the spread estimates to zero — the spread, and its own uncertainty
The cost of differencing a pair — series that move together
The interval that integrates — the spread, and its own uncertainty
Two levels at once — hierarchy past one number

10 August 2026

12 essays on groups that borrow, when the observations repeat each other and decided before the data

Eight groups, one population — groups that borrow
The observations that repeat each other — when the observations repeat each other
The variance removed before the data — decided before the data
Randomisation is not balance — decided before the data
The weight that decides — groups that borrow
Two walks and a finding — when the observations repeat each other
One factor at a time — decided before the data
The prior the data estimates — groups that borrow
What differencing costs — when the observations repeat each other
How many subjects — decided before the data
The check before the standard error — when the observations repeat each other
When borrowing goes wrong — groups that borrow

8 August 2026

15 essays on when the data stops early, regression, and what the summary hides, the prior, doing visible work, corrections, and what each controls and stopping rules

The data that stops early — when the data stops early
The line that one point drew — regression, and what the summary hides
What a prior is worth — the prior, doing visible work
What the correction corrects — corrections, and what each controls
When the looking happens — stopping rules
Four datasets, one summary — regression, and what the summary hides
Spending the error rate — stopping rules
The curve that survives censoring — when the data stops early
Two different promises — corrections, and what each controls
What a credible interval covers — the prior, doing visible work
R² is not a measure of fit — regression, and what the summary hides
The price of control — corrections, and what each controls
Where the two schools agree — the prior, doing visible work
The base rate was always Bayes — the prior, doing visible work
Twenty residual plots — regression, and what the summary hides

6 August 2026

19 essays on tests, and the second number, reversals that are not errors, the distribution itself, what makes it checkable and intervals, counted

A p-value that is not flat is not a p-value — tests, and the second number
Simpson's reversal is a region, not a table — reversals that are not errors
Sums of almost anything — the distribution itself
The seed is part of the figure — what makes it checkable
What the 95% refers to — intervals, counted
More data is not monotonically better — intervals, counted
The tail converges last — the distribution itself
Two routes to every number — what makes it checkable
What a p-value does not say — tests, and the second number
What a positive test is worth — reversals that are not errors
Regression to the mean — reversals that are not errors
The winner's curse — tests, and the second number
Twenty intervals and one expected miss — intervals, counted
What normal actually looks like — the distribution itself
The shape, and where its mass is — the distribution itself
The shortest interval is the one that misses — intervals, counted
Twenty analyses of nothing — tests, and the second number
Where the bootstrap lies — intervals, counted
The correction for not knowing the spread — intervals, counted

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