Talks

Prediction-Powered Inference: Anastasios Angelopoulos

38 min read

Anastasios Angelopoulos · 2025 · 44:10 · ▶ Watch

Summary

The question. Machine-learning models can now produce cheap “measurements” of things that are expensive to measure: a protein’s 3D shape, the species of a plankton, the shape of a galaxy, the vote marked on a ballot. Can a scientist use those predictions to learn about the world, and still report a confidence interval that is honest?

The thesis. Yes, if you keep a small set of real (“gold-standard”) measurements next to the predictions and use them to measure the model’s own error. Prediction-powered inference (PPI) does this in three steps:

  1. Average the model’s predictions over a large unlabeled pool. This is cheap and low-variance, but it may be biased.
  2. Estimate the bias on the small labeled set, as the average of (true value minus prediction). He later calls this term the rectifier.
  3. Subtract the bias from step 1. The result is unbiased, and its confidence interval has the correct coverage whatever the model is.

If the model is good, the interval is much narrower than one built from the labeled data alone. In his examples that means about half the labels (galaxies), a third (plankton) or a tenth (ballots). If the model is bad, a tuning step (“power tuning”) stops it from being worse than the classical interval.

How he closes. Pointers to follow-on work (active data collection, bootstrap versions, imputed covariates, auto-evaluation of language models at Google DeepMind), then a Q&A on finite samples, non-overlapping intervals, and what the method cannot handle.

His stance. Modest and technical. He calls it “partial progress”. His strongest claim is that the method works “for any predictor”, though it “might not give you gains for any predictor”. The one assumption he calls the main one: the gold-standard labels must come from data that is “IID with the data that’s unlabeled”.

Map

Point
Motivation: generative models as a source of data for scientific inference
Why measurements are expensive: crystallography, plankton, astronomy
AlphaFold as the inspiration; gold standard is rare, predictions are abundant
A real study that uses AlphaFold predictions as data (odds ratio); the joke “we’re replacing all the data with predictions”
What goes wrong: bias, and a confidence interval that misses the truth by a wide margin
Goals of the method (model agnostic, valid, no worse than classical, finite-sample and asymptotic)
Setup: n labeled pairs, N unlabeled points, a fixed black-box model f
Mean estimation: the PPI estimator and its variance
Code: “eight lines”, the ppi_python package
Galaxy Zoo: spiral galaxies, about 2x savings
This is an old idea: survey sampling, causal inference, semiparametrics, semi-supervised learning
Convex estimation (regressions, quantiles) and power tuning
Rectified risk for general targets (generalized linear models)
Plankton counting at Woods Hole: about 3x fewer labels
Electronic ballot counters and election audits: about 10x
Extensions: active statistical inference, prediction-powered bootstrap, imputed covariates, DeepMind auto-raters
Q&A: can two valid intervals fail to overlap?
Q&A: finite-sample version with Hoeffding
Q&A: limits (distribution shift, sequential data)
Q&A: performance gain and KL divergence

Putting it together

The mental model. A model’s prediction is a cheap, biased ruler. You cannot trust the ruler, but you can check it. Measure a few things with the expensive, accurate method; see how far off the ruler is on those; then use the ruler on the many remaining things and subtract the typical error. The error you subtract is estimated from few points, so it has noise. That noise is the price of validity. The gain is that the noise is smaller than the noise of measuring the quantity itself, as long as the error varies much less than the quantity does. Everything in the talk is a version of this: the mean estimator, power tuning (how much of the ruler to trust) and the rectified loss (the same idea for regression).

Tensions to keep open.

  1. Validity versus efficiency. The guarantee holds for every model; the gain depends on the model. A poor model buys you nothing, and the method will not tell you in advance.
  2. Fixed model versus tuned model. The theory treats the predictor as fixed and not trained on the labeled points. In practice people tune, select and fine-tune models on the same data. The talk raises this for the tuning parameter only.
  3. Same population versus convenient population. The labeled and unlabeled points must come from the same distribution. Most real uses (a new site, a new year, a new survey wave) do not quite meet that.
  4. Predictions as help versus predictions as replacement. The joke marks the temptation. PPI’s answer is that predictions can reduce the number of gold-standard labels, but they cannot replace them.

Reading path.

  • Start: the Science paper (Angelopoulos, Bates, Fannjiang, Jordan, Zrnic, 2023) for the idea; then ppi_python and its tutorials.
  • Then: PPI++ (Angelopoulos, Duchi, Zrnic) for power tuning and convex targets.
  • Deeper: Robins, Rotnitzky and Zhao (1994) on augmented estimators, and a survey-sampling text on the generalized regression estimator. These show the idea is 30 to 70 years old.
  • If you care about survey simulation with language models: idea 6 below, and the three 2025 papers linked there.

Part 1: Before you read

Key terms

  • Gold-standard label. The slow, accurate measurement: a crystallographic structure, an expert annotation, a hand-checked ballot, a real person’s answer. Written Y.
  • Prediction. The model’s guess at the label for the same item, written f(X). X is whatever the model sees (a protein sequence, an image).
  • Labeled set (n points) and unlabeled set (N points). Both are random draws from the same population; N is much bigger than n. The unlabeled set has X and the prediction, but no Y.
  • Black-box model. The method does not look inside the model, and does not ask how accurate it is. It must be fixed in advance: “we’re not averaging over the randomness of the learning there”.
  • θ* (theta star). The thing you want to learn about the population: its mean, a quantile, a regression coefficient, an odds ratio. “You should think about theta star as a function of the distribution”.
  • Confidence interval, coverage. A 95% interval is a recipe that, over repeated samples, contains θ* at least 95% of the time. The interval is the random thing; θ* is fixed. “The probability, theta star, lies in the interval, which is random, is 95%“.
  • Valid. The coverage promise holds. A narrow interval that misses the truth is worse than a wide one that covers it.
  • Power. Here: how narrow the valid interval is for a given number of labels. More power means fewer labels for the same width.
  • Rectifier. The correction term: the average of (Y minus f(X)) over the labeled points. It “fixes” the bias of the prediction average.
  • Odds ratio. In his words, “the ratio of the probability that an event would happen in one group over another”. (Precisely, a ratio of odds, where odds are p/(1-p).)
  • Control variate. A cheap quantity with a known average that you subtract from an estimator to reduce its variance. See idea 2.

People named

PersonWhy they come upStart with
Anastasios AngelopoulosSpeaker; lead author of PPIThe Science paper (2023)
Stephen Bates, Clara Fannjiang, Michael Jordan, Tijana ZrnicCo-authors of the original paperSame paper
Tijana Zrnic, John DuchiCo-authors of the convex-estimation / power-tuning work (PPI++)PPI++ (arXiv 2311.01453)
Heidi Sosik, Rob OlsonBuilt the Imaging FlowCytobot at Woods HoleTheir plankton science paper
DeepMind researchersUse PPI to calibrate auto-raters for language modelsSee idea 5
Galaxy Zoo teamCrowdsourced galaxy labelsGalaxy Zoo 2 data release

The speaker’s own frame

Angelopoulos was a PhD student at UC Berkeley, then a postdoc; the moderator says he is about to take a faculty post. The moderator also credits him with co-developing Chatbot Arena, the language-model comparison platform, and with a monograph on conformal prediction. His working theme is turning black-box predictions into statements with guarantees. PPI sits next to conformal prediction in that programme: both give valid uncertainty statements without trusting the model.

The current debate

  • PPI is established and growing. The original paper appeared in Science in November 2023 (Berkeley news, arXiv 2301.09633). It claims validity for means, quantiles, and linear and logistic regression coefficients, with demonstrations in proteomics, astronomy, genomics, remote sensing, census analysis and ecology. PPI++ added computational tools and power tuning, and the authors report it is “essentially never worse” than classical or plain PPI (arXiv 2311.01453).
  • Language-model “survey simulation” is the hot application. A February 2025 paper turns simulated survey responses into confidence sets for human population parameters, and chooses how much simulated data to use based on how far the model disagrees with real people (Huang, Wu and Wang, arXiv 2502.17773). A NeurIPS 2025 paper studies synthetic data from foundation models, and finds the gain is large only when the synthetic data’s errors are predictive of the real data’s errors (Byun et al., NeurIPS 2025). Another late-2025 paper notes that PPI and similar post-hoc corrections have not been rigorously tested for large-scale survey simulation, and compares them with fine-tuning and prompting (arXiv 2510.11408). I found no strong published critique of PPI itself in this search, so read the open questions as open, not as settled.
  • Extensions the speaker names. Active statistical inference, prediction-powered bootstrap, imputed covariates. Papers are cited by the speaker only by description; look them up by those names.

The ideas

Six strands meet in this talk. Only some are named by the speaker.

 SURVEY SAMPLING ───────┐                    ┌──── SEMIPARAMETRIC STATISTICS
 Horvitz-Thompson       │                    │     Robins-Rotnitzky-Zhao
 model-assisted (GREG)  ▼                    ▼     influence functions
                 ┌─────────────────────────────────┐
 CONTROL VARIATES│  PPI: use a biased cheap        │ SEMI-SUPERVISED LEARNING
 Monte Carlo     │  predictor, correct it with a   │ labeled + unlabeled data
 variance        │  few real labels, keep validity │
 reduction       └─────────────────────────────────┘
                   ▲               ▲             ▲
        CONFIDENCE │   M-ESTIMATION│             │ DISTRIBUTION SHIFT
        INTERVALS  │   (regression,│             │ covariate shift, reweighting;
        CLT,       │   quantiles)  │             │ surrogate outcomes
        Hoeffding

Where each is covered:

  • Why the plug-in answer fails, and the estimator in one line: idea 1 and idea 2
  • Control variates, survey sampling, semiparametrics: idea 2
  • Confidence intervals, power tuning, finite samples: idea 3
  • Beyond the mean: M-estimation and semi-supervised learning: idea 4
  • Applications and the IID assumption: idea 5
  • Model predictions standing in for human answers (📚 context, not the talk): idea 6

1. Why plugging predictions in fails

In short. If you replace measurements with model predictions and run your usual analysis, two things go wrong: the estimate is biased, and the confidence interval does not account for that bias. The interval is narrow because there are many predictions, but it is centred on the wrong number. The point of PPI is to keep the cheap data without giving up the guarantee.

🎙 In the talk.

  • The pull: “Gold standard measurements are rare. Predictions are abundant.”
  • The temptation, with a joke: “We thought it was data science. Actually, we’re replacing all the data with predictions.”
  • The failure: the model is wrong exactly where it matters. “this is a protein that is intrinsically disordered. And exactly in the region where it’s intrinsically disordered, the model makes an error”. Errors are correlated with the thing being studied.
  • The caveat that justifies the whole method: “gold-standard measurements are the gold standard for a reason”.
  • The test case: he took a perturbed subset of the real study where the labels are known, and compared three intervals. The imputation-only interval was not just off-centre; it was too small by “a factor of 6 or something”. The classical interval (labels only) was valid but wide. PPI aims at “a happy medium”: “an actually valid confidence interval that’s smaller”.
  • The goals for the method: use both data sets; model agnostic (“It’s just a black-box model”); small intervals when the model is good; “no worse than the classical algorithm” when it is bad; finite-sample valid if possible.

📚 Deeper theory (marked as context, not his claims)

  • Two separate failures. Imputing all labels gives you bias (the model’s average error is not zero) and overconfidence (the interval treats predictions as if they were independent real observations, so it ignores model error). Fixing one does not fix the other. The odds-ratio example is a ratio of nonlinear functions, so the bias can be large even when per-item accuracy looks high.
  • Why correlated errors matter. If errors were random noise with mean zero, averaging many predictions would cancel them. They are not: a model systematically misreads a type of case (disordered regions), and the target is exactly about that type. This is the same logic as selection bias.
  • Accuracy is not the right measure. A model that is 99% accurate can still bias an estimate when you study the 1% where it fails. Ballots are the sharpest case in the talk: a 0.5% error can “flip an election”.
  • This is also a version of Goodhart/proxy problems†: a proxy is trusted most where it was never tested.

Where others push back.

  • A scientist may say the gold-standard measurement is also imperfect (expert annotators disagree; X-ray structures have uncertainty). PPI treats the labels as truth. Noisy gold labels are outside the talk.
  • Some argue that for exploratory work a biased but cheap estimate is fine. PPI is for the case where you must report a claim with an error bar.

What they leave out. What to do when there are no gold-standard labels at all. PPI needs some.

Check yourself

  • Why can a 99% accurate model still give a biased estimate?
  • In the odds-ratio example, what did the imputation-only interval get wrong: its centre, its width, or both?
  • Why is “no worse than the classical interval” a meaningful goal?

2. The estimator: imputed mean plus rectifier

In short. To estimate an average, take the average of the model’s predictions on the big unlabeled set, then add a correction: the average gap between truth and prediction on the small labeled set. The result is unbiased for any model. Its variance depends on how variable the gap is, not how variable the quantity is. That is where the gain comes from.

🎙 In the talk.

  • The setup: n labeled pairs; N unlabeled points; the model f is fixed. “the coverage of our confidence interval should be 1 minus alpha at least.”
  • The classical estimator is the plain average of the labeled Y’s. The prediction-powered one “is going to do one over big N” times the sum of the predictions, “possibly biased, because the predictions are not necessarily any good. So what we’re going to do is we’re going to add a debiasing term”.
  • Why it is unbiased: take expectations; the prediction terms on the labeled set and on the unlabeled set “cancel because they’re the same”.
  • Variance: “1 over n variance of y hat prime i plus 1 over little n variance of Yi minus y hat i”. Because the two pieces use independent data, the variances add.
  • Why it is good: think of N as huge, so only the second term matters. “If y minus y hat is substantially smaller than the variation… than the variation that we would see in y alone”, the interval is much tighter.
  • A clarification from the room: the labels and predictions are “coupled through x. And that’s part of the juice.”
  • Practical: “this is like eight lines of code” and there is a package.

The estimator, written out (his notation, with my spelling-out of the symbols; the formulas follow the talk and the paper):

Labeled:   (X_i, Y_i),  i = 1..n         predictions f(X_i)
Unlabeled: X~_j,        j = 1..N         predictions f(X~_j)

Classical:        mean_hat = (1/n) * sum_i Y_i

Prediction-powered:
  theta_PP = (1/N) * sum_j f(X~_j)          <- imputed mean (cheap, maybe biased)
           + (1/n) * sum_i ( Y_i - f(X_i))  <- rectifier (debiasing)

Variance:   s2_f / N  +  s2_Delta / n
  s2_f     = variance of the predictions f(X~)
  s2_Delta = variance of the gap  Y - f(X)

CI (95%):   theta_PP  +/-  1.96 * sqrt( s2_f / N + s2_Delta / n)

The classical interval has half-width 1.96 times the standard deviation of Y over the square root of n. PPI replaces the standard deviation of Y by the standard deviation of the gap Y minus f(X), plus a small term for N.

A small worked example (for illustration; not from the talk). Say Y has variance 1, n = 100 labels, N = 10,000 unlabeled points, and the gap Y minus f(X) has variance 0.25, so the model explains about three quarters of the variance of Y and f(X) has variance about 0.75. Classical half-width: 1.96 x sqrt(1/100) = 0.196. PPI half-width: 1.96 x sqrt(0.25/100 + 0.75/10,000) ≈ 0.099. To get that width, the classical method would need about 390 labels instead of 100, roughly 4 times as many.

📚 Deeper theory

  • Why bias cancels exactly. E[f(X~)] equals E[f(X)] because both sets are draws from the same population. So the expected value of the estimator is E[f(X)] + E[Y] - E[f(X)] = E[Y]. The model could be arbitrary; it only needs to be fixed. This is the whole validity argument for the mean.
  • The gain is about the gap, not accuracy. A model with a constant offset (always 5 units too high) is useless as a point predictor but, for the mean, costs nothing: the gap has zero variance. The model that helps is one whose errors are similar from item to item, or whose predictions track Y. A scale error costs little; random scatter costs a lot.
  • A hidden trade-off. The estimator adds the variance of the imputed mean (the s2_f over N term). With N finite and a bad model, this term plus the larger gap variance can make PPI wider than classical. That is why idea 3 exists.
  • It is an average, so the CLT applies. The interval is built from the central limit theorem, so it is asymptotic. Finite-sample versions come from concentration bounds (idea 3).
  • The package.† The Python package is ppi_python, installed with pip install ppi-python; the speaker’s spoken “PPI PI” is a caption/speech slip for that name.

📚 Roots: Control variates and model-assisted survey sampling

The problem. You want an average over a population. Measuring everyone is too costly. A cheap proxy is available for everyone. How can the proxy help without being trusted?

The point. This is exactly the problem the survey statisticians solved. The speaker says so: “this idea of mean estimation, using this sort of control variate thing, is not that new”, and gives the tree-height example, where surveyors eyeball 95% of trees and measure 5% with a ruler. PPI is the same design with a neural network as the eyeballer.

The chain of thinkers

  1. Neyman, 1934†. Showed that probability sampling lets you make valid claims about a population without modelling it. Design-based inference is where “the randomness comes from the sampling, not from a model” starts. PPI’s guarantee has the same flavour: it is valid whatever the model.
  2. Horvitz and Thompson, 1952†. Weighting each sampled unit by the inverse of its selection probability gives an unbiased estimate of a total. This is the first bias-removing correction built from sampling probabilities.
  3. Cochran, Sampling Techniques, and the ratio and regression estimators (1950s)†. Use an auxiliary variable known for the whole population to sharpen the estimate from a sample. The regression estimator is PPI’s mean estimator with a linear model as f and tuning (idea 3).
  4. Särndal, Swensson and Wretman, Model Assisted Survey Sampling (1992)†. The “generalized regression estimator”: a model supplies predictions for everyone; sample residuals correct them; the claim is validity even if the model is wrong. This is the closest ancestor. The speaker names “model-assisted survey sampling”.
  5. Control variates in Monte Carlo (Hammersley and Handscomb, 1964)†. Subtract a correlated quantity whose mean is known; variance falls by the squared correlation. The best coefficient is the covariance over the variance, the same shape as power tuning’s best weight.
  6. PPI (2023). Replaces the linear or parametric model by any machine-learning model, and extends to quantiles and regression.

Where PPI differs. In survey sampling the auxiliary data (say, the size of every farm) is known exactly and the model is usually simple and fit on the sample. In PPI the model is a large pre-trained system, fixed and not fit on the labels; the labeled sample can be small; and the target can be a regression coefficient or an odds ratio, not only a total.

📚 Roots: Semiparametric statistics and missing data

The problem. If some data are missing, how do you estimate a quantity efficiently without assuming the whole distribution?

The point. PPI is a missing-data problem: Y is observed for n points and missing for N. The speaker lists “a big field of causal inference that has to do with missing data” and “semiparametric statistics” among the relatives.

The chain of thinkers

  1. Rubin, 1976†. Classified missing data (completely at random, at random, not at random). PPI’s assumption that labeled data are “IID with the data that’s unlabeled” is the “completely at random” case.
  2. Bickel, Klaassen, Ritov and Wellner, Efficient and Adaptive Estimation for Semiparametric Models (1993)†. The theory of the best achievable variance and the influence function.
  3. Robins, Rotnitzky and Zhao, 1994†. The augmented inverse-probability-weighted (AIPW) estimator: use a model for the missing outcome, then add an inverse-weighted residual correction. It is consistent if either the model or the missingness mechanism is right, and is efficient when both are. PPI is the case where the missingness probabilities are known (a simple random subset is labeled), so validity does not depend on the model.
  4. Chernozhukov et al., “double/debiased machine learning” (2018)†. Brings AIPW-style correction to flexible machine-learning models, with sample splitting. The data-splitting point matters for the tuning question.
  5. Zhang, Brown and Cai (2019) and later work on semi-supervised inference†. The unlabeled-data version of the same arguments.

Where others push back. Experts in this literature note that PPI’s estimator is a known special case, and that the value of the paper is the clean packaging (black-box model, finite-sample variants, many targets, software) and the demonstrations. The speaker agrees the idea has “been floating around”.

Check yourself

  • Why does the correction term have expected value equal to the model’s bias?
  • If a model’s predictions were always exactly 5 units too high, what would happen to the PPI interval for the mean?
  • What is the one thing the survey-sampling tree example and the plankton example have in common?

3. Guarantees: power tuning and finite samples

In short. The basic estimator is valid for any model but can be worse than the classical one if the model is bad. Power tuning blends the two: it adds a weight lambda between 0 and 1 on the predictions, picked to make variance as small as possible. A finite-sample version of the interval also exists, from Hoeffding’s inequality, at the cost of a wider interval.

🎙 In the talk.

  • The risk: “if the predictions are actually worse, if they’re random, we’re actually increasing our variance”.
  • The fix: “You’re better than both the classical solution and the imputed solution and the PPI solution with no tuning”. At lambda = 0 it is the classical average; at lambda = 1 it is the basic PPI estimator. There is “an explicit expression” for the best lambda, involving covariances that “can be estimated”.
  • On bias: “for any fixed lambda, this is unbiased”. If lambda is fitted on the same data, “Yes, but it is consistent”, so the argument is asymptotic; for finite samples, data-splitting is the answer.
  • Can two valid intervals fail to overlap? Yes, by chance. “There’s no high probability statement here, because the only randomness that we’re averaging over is the interval”. One interval fails with probability 5%, the other with a smaller probability, and both failing in opposite directions is rare.
  • Finite samples: assume Y is bounded in [0, 1]. The gap Y minus f is bounded in [-1, 1], so its range is 2; the prediction average has range 1. You take a Hoeffding interval for each piece and “smash these together” for a total of “three Hoeffding” widths (2 + 1). He adds that there are “more advanced ways of doing it where you’re coupling these into one average”.

Power tuning, written out (the form follows the paper; check it against the paper before relying on it):

theta_lambda = lambda * (1/N) sum_j f(X~_j)  +  (1/n) sum_i ( Y_i - lambda * f(X_i))

lambda = 0  ->  classical average of Y
lambda = 1  ->  basic prediction-powered estimator
lambda*     ->  the value that minimises variance  (≈ Cov(Y, f) / ((1 + n/N) * Var(f)))<span class="unchecked" title="From background knowledge, not checked against a source">†</span>

If the predictions carry no information about Y, lambda* is near 0 and you recover the classical interval. If they track Y closely, lambda* approaches 1.

📚 Deeper theory

  • Same shape as a control-variate coefficient. The best weight is a regression slope of Y on f. This is why the improvement depends on correlation: the variance reduction factor is about one minus the squared correlation between Y and f, adjusted for n and N.
  • Asymptotic versus finite-sample validity. The CLT-based interval is valid only as n grows. Hoeffding’s inequality (1963) gives a bound valid at every n for bounded variables, at the price of ignoring the actual variance and so being wider. Variance-adaptive bounds (empirical Bernstein, as an audience member asks) narrow the gap, and there are sharper “betting” confidence sequences†.
  • Why a Hoeffding union is conservative. Splitting the error budget alpha between the two pieces (alpha/2 each, by the union bound) costs a little coverage efficiency†. The speaker’s “three Hoeffding” counts widths: the two pieces have ranges 2 and 1.
  • Tuning on the same data. Estimating lambda from the data makes the estimator nonlinear in the data, so exact unbiasedness is lost, although it stays consistent. This is the familiar “winner’s curse” of tuning, and the reason to split data or to cross-fit.
  • Intervals can disagree without a contradiction. Two 95% intervals from different methods on the same data are each random sets; they fail together rarely. A gap between them is evidence that something is off (the model, the assumptions, or luck), and the speaker treats it as a rare random event. A reader may want to treat repeated disagreement as a diagnostic signal.

📚 Roots: Confidence intervals and concentration

The problem. What does “95% confident” mean, and how do you guarantee it?

The chain of thinkers

  1. Neyman, 1937†. Defined confidence intervals as random sets with guaranteed long-run coverage. The “probability” is over repeated samples, not over the parameter. This is the sense the speaker uses.
  2. Laplace, de Moivre and the central limit theorem (18th-19th c.)†. Averages of many independent terms are approximately normal. PPI’s asymptotic interval is plus or minus 1.96 standard errors.
  3. Hoeffding, 1963†. For bounded independent variables, the average deviates from its mean by more than t with probability at most 2·exp(-2nt²/range²). Valid for every n.
  4. Bernstein, Bennett and the empirical Bernstein bound (Maurer and Pontil, 2009)†. Use the observed variance to sharpen the bound.
  5. Stein, shrinkage and the idea of “borrowing strength”†. Blending an estimate with a prior or auxiliary estimate with a data-driven weight is the statistical cousin of power tuning.

Where others push back. Some statisticians prefer Bayesian treatments. A Bayesian PPI paper exists (Hofer et al.); it combines labeled data with machine-labeled data in a model-based way, trading the model-free guarantee for sharper use of prior information.

What they leave out. Intervals for many quantities at once (multiplicity) and valid inference after selecting a model. PPI++ reports it can do inference on single regression coefficients without multiplicity correction (arXiv 2311.01453).

Check yourself

  • What does lambda = 0 correspond to, and what does lambda = 1 correspond to?
  • Why can two valid 95% intervals still not overlap, and how often?
  • In the Hoeffding version, why are the widths added up as 2 + 1?

4. Beyond the mean: rectified risk

In short. The mean is the easy case. Many scientific questions are about regression coefficients, quantiles or odds ratios, which are defined by minimising a loss. PPI handles these by rectifying the loss function itself: the same prediction-average-plus-correction idea, applied to the loss, so that for every candidate parameter value the rectified loss is unbiased for the true loss.

🎙 In the talk.

  • The scope: “In mean estimation that would be like quadratic loss. In quantile estimation that would be like the pinball loss. Ordinary least squares and logistic regression, these are sort of defined by optimization problems anyway”.
  • The method: if θ* is the minimiser of an expected loss, “we can solve what we call a rectified optimization problem” where the debiasing term is the rectifier.
  • The guarantee: “for any value of theta it’s unbiased for the ground truth loss”; the estimator is consistent and follows the central limit theorem.
  • The limit: “at least for generalized linear models—you can’t do this for all losses. But there’s a special trick for generalized linear models”.
  • This is joint work with Zrnic and Duchi. (Captions garble the names.)

📚 Deeper theory

  • What “defined by minimising a loss” means. The mean minimises squared error; the median minimises absolute error; the q-th quantile minimises the “pinball” loss; the least-squares line minimises squared residuals; logistic regression minimises log loss. A parameter estimated this way is an M-estimator.
  • How the rectified loss works (a sketch following the paper, not shown in the talk): for each candidate θ, estimate the loss as (1/N) sum of loss(θ, f(X~)) over unlabeled points, plus (1/n) times the sum of [loss(θ, Y) minus loss(θ, f(X))] over labeled points. For each fixed θ this is unbiased for the true expected loss; minimise over θ to get the estimate. For the mean with squared loss it reduces to the estimator in idea 2.
  • Why the odds ratio example matters. The AlphaFold study’s target is an odds ratio from a logistic regression. That is a rectified-risk target, which is why the talk spends time on the mean first, but cares about this.
  • Why GLMs get special treatment. Their gradient is linear in the data in a way that lets the variance of the estimator be written in closed form, so intervals are straightforward. For other losses, the intervals are harder.

📚 Roots: M-estimation and semi-supervised learning

The problem. A good estimator for a complicated target, using labeled and unlabeled data.

The chain of thinkers

  1. Huber, 1964, 1967†. The asymptotic theory of M-estimators: estimators defined as minimisers of sums, with a sandwich-form variance. PPI’s intervals for regression coefficients use this form.
  2. Koenker and Bassett, 1978†. Quantile regression and the “pinball” loss named in the talk.
  3. Nelder and Wedderburn, 1972†. Generalized linear models.
  4. Semi-supervised learning (Chapelle, Schölkopf and Zien, 2006; and theory papers on semi-supervised inference by Zhang, Brown, Cai and by Chakrabortty and Cai)†. Use unlabeled data to improve estimation; the speaker cites semi-supervision as a relative. The new thing in PPI is a black-box predictor, fixed in advance.
  5. Prediction-powered extensions. PPI++ (Angelopoulos, Duchi, Zrnic), and the other work the speaker lists.

Where others push back. Rectified losses require the model to be fixed. If the model is trained on data that includes the labeled points (for example a large language model trained on the web, which may include the benchmark), the labeled residuals can understate the true error on new points. This is a data-leakage concern that the talk does not discuss.

Check yourself

  • Which of mean, median, regression slope and odds ratio are defined by minimising a loss?
  • Why does the rectified loss need the model to be fixed?
  • What is the role of the unlabeled data in the rectified loss?

5. Applications, and the assumption that carries everything

In short. Three examples show typical savings (2x, 3x, 10x) and one pattern: PPI helps where labels are slow or expensive and a decent model exists. The Q&A then names the assumption that matters most: the labeled and unlabeled data must come from the same distribution.

🎙 In the talk.

  • Galaxy Zoo: citizens count spiral arms; the model predicts the same. Prediction-powered intervals are narrower than classical ones and the model-only interval is off. “So approximately a factor of 2.”
  • Plankton: a robotic imaging system at Woods Hole generates huge numbers of images and “they hand label over 400,000 images of plankton per year” with three people. With PPI: label a few hundred, use the model for the rest. “You need to label half or a third of the data… So 3x fewer expensive measurements needed”.
  • Ballot counters: scanners mislabel a small share (the example here: a misprinted ballot where a circle detector “latched on to” stray lines). “the error rate is like half a percent or something. But yeah, that’s enough to flip an election, weirdly.” With a model at “99-plus%” accuracy, a few hundred or thousand hand-checked ballots could be enough, “a factor of 10”.
  • Auto-raters at DeepMind: language-model evaluations need labels for factuality and creativity; human labels are costly, so “they’re incentivized in order to create auto raters that give synthetic labels… But then they’re biased. So they’re using prediction-powered inference in order to adjust for this bias, actually in the production pipeline.”
  • Other extensions: active statistical inference, where the model’s own uncertainty guides which points a human should label; prediction-powered bootstrap; imputed covariates.
  • Limits: “it will work for any predictor. It might not give you gains for any predictor.” The main assumption: “the ground truth labels are able to be collected on data that is IID with the data that’s unlabeled.” If there is a domain shift, “likelihood ratio, reweighting” tricks apply, “but there’s a gap that needs to be accounted for”. For sequential data, it can still work with martingale structure, slowly varying data, or a known reweighting. On whether the gain has a clean characterisation in terms of KL divergence: “Not that I know of”.

📚 Deeper theory

  • Savings are not magic. The label savings come from the squared correlation between the label and prediction (idea 2). The three examples have strong models on easy-to-judge images. A model that is hardly better than guessing would give almost nothing.
  • What “IID with the unlabeled” rules out. Labeled data drawn from a different place, season or instrument. In the plankton example, a model trained in one year and labeled in another is a shift. In audits, labeling only the ballots that the scanner flagged would break the assumption; audits sample at random on purpose.
  • Random labeling is the design. The label subset must be chosen by chance (or by known probabilities, as in active inference). If experts choose “interesting” cases to label, the rectifier is biased. This is the survey-sampling lesson.
  • A decent description of what PPI changes. It does not make the model better. It changes what you can say about a quantity when you use the model.
  • Auto-evaluation of language models.† The DeepMind work is, I believe, “AutoEval Done Right: Using Synthetic Data for Model Evaluation” (Boyeau et al., 2024), with Angelopoulos among the authors, but the speaker only says that it is “a paper that I of wrote on that topic”.
  • Chatbot Arena is mentioned only by the moderator. It collects human preference votes for model comparisons. Using auto-judges plus a few human votes has the same structure as PPI.

📚 Roots: Covariate shift and surrogate outcomes

The problem. The labeled data and the target population differ. What can you still say?

The chain of thinkers

  1. Heckman, 1979, selection bias†. Data selected on something related to the outcome cannot be taken at face value.
  2. Shimodaira, 2000, covariate shift†. If P(Y|X) is stable and only P(X) changes, importance weights (the “likelihood ratio” the speaker names) restore correct estimates, at the cost of higher variance.
  3. Prentice, 1989, surrogate endpoints†. When can a cheap or early outcome stand in for the true clinical outcome? Prentice’s criteria say the surrogate must capture the whole effect of the treatment on the outcome; in practice many surrogates fail. A model’s prediction is a surrogate in this sense, and PPI’s rectifier is a way to avoid having to prove that criterion.
  4. Athey, Chetty, Imbens and Kang, “surrogate index” (2019)†. Combine many short-term outcomes into an index predicted from a labeled sample, then use it for long-run effects. Same structure as PPI, with an assumption (comparability) that plays the role of IID.
  5. Domain adaptation and conformal under shift (Tibshirani et al., 2019; Barber et al., 2023)†. The speaker’s own earlier work area, conformal prediction, has weighted versions for shift.

Where others push back. A common complaint about auto-evaluation is that the judge model shares blind spots with the model under test, so errors are positively correlated with the thing evaluated. PPI’s correction handles that only if the labeled sample is large and representative enough to estimate it. This is why choosing the labeled subset well matters more than the choice of model.

What they leave out. Noisy labels, labels that depend on who annotates, and non-random labeling in practice.

Check yourself

  • Why must the labeled ballots be chosen at random?
  • What does the speaker call “the main assumption” and what does he say can be done when it fails?
  • Why is the rectifier a substitute for a surrogate-validity argument?

6. 📚 Context (not in the talk): PPI and model predictions in place of human answers

Everything in this section is outside the talk. The speaker does not discuss surveys, personas or “synthetic respondents”. The link is mine.

In short. A growing practice replaces survey respondents or study participants with language-model “respondents” or personas. PPI describes the one honest way to use such outputs inside an inference: treat the model’s answer as the prediction f(X), keep real human answers as the labels Y, and report an interval whose width reflects how well the model tracks people.

📚 How the pieces map

In the talkIn a survey with model respondents
Gold-standard label YA real person’s answer
Prediction f(X)The model’s answer for a respondent with the same profile or question
Labeled set (n)People who were actually surveyed
Unlabeled set (N)People for whom you have the profile (covariates) but no answer, each with the model’s prediction
Fixed modelA fixed model, prompt and persona recipe, not tuned on the labeled answers
RectifierThe average gap between real answers and the model’s answers on the surveyed people
θ*The population answer: a share who agree, a mean rating, a regression coefficient

What this implies (my reading, derived from the talk’s statements, not claims made in it)

  1. Model answers alone do not give a valid interval. The first failure mode in idea 1 applies directly: if you average many model answers, you get a narrow interval around a possibly biased number. The odds-ratio demonstration is a warning against this.
  2. Some real humans are needed, and they set the validity. The labeled people are what turn the model’s output into a statement about a population. Without them there is no rectifier to estimate.
  3. The savings are bounded by how well the model tracks people item by item. The variance depends on the gap Y minus f(X). A model that gets the overall average right but is poor at predicting individual answers (it ignores who the person is) gives a large gap variance and a small saving. Matching the average is not the same as being useful. The gap also depends on the question: a model may track people on one topic and not another.
  4. The labeled sample must be a random sample of the same population. The talk’s main assumption. A convenience panel of 200 online respondents used to calibrate a model that is then applied to a national population is the distribution shift case; the reweighting tricks and the “gap” apply.
  5. The unlabeled units must be real draws from the target population. PPI needs actual covariates X for actual members of the population. Personas that are invented (rather than sampled from a real frame) are not draws from P, so the unlabeled average is not an estimate for any real population.
  6. The model must be fixed independently of the labeled answers. If the prompt, persona recipe or fine-tuning is adjusted until it matches the labeled answers, the residuals are optimistic. Use separate data for tuning and for the rectifier. Likewise, a model trained on public survey data may have seen the same answers.
  7. It is about aggregate quantities. PPI gives intervals for population parameters, not for what a particular person would say. It does not license reading an individual model respondent as a person.
  8. Different question, different calibration. The rectifier is estimated for one question and population. A new question needs new human labels, or an argument that the model’s gap carries over, which the method does not supply.
  9. What counts as honest reporting. State the number of real respondents, the number of model respondents, the width of the interval, and what it would have been without the model.

📚 The current research (recent preprints and a conference paper, not endorsements)

  • Huang, Wu and Wang (Feb 2025): converts simulated responses into confidence sets for human parameters, and picks the simulated sample size from the observed disagreement between model and humans; claims to apply to any model whatever its fidelity.
  • Byun et al., NeurIPS 2025: studies valid inference with synthetic samples from foundation models with a generalized-method-of-moments estimator. They find gains only when the synthetic data’s errors are predictive of the real data’s errors, a restatement of the gap point in idea 2.
  • arXiv 2510.11408: compares fine-tuning, prompting and post-hoc correction for survey simulation, and says PPI-style corrections have not been rigorously evaluated at large scale in this setting.
  • The first search found no paper by the speaker on language-model respondents. The nearest of his applications is the auto-rater work.

Where others push back (position-level, from the research above and common critiques; sources vary)

  • Some argue that language models can never stand in for people because training text over-represents some groups. PPI neither answers nor needs to answer this: it takes the model as a black box and prices its error.
  • A practical objection: if you need real respondents anyway, and the saving is a factor of 2 to 4 in the best case, the economic case for model respondents may be weaker than the headlines suggest. This is the speaker’s own point that the method “might not give you gains for any predictor”, applied to this setting.
  • A deeper objection: for exploratory work (generating hypotheses, testing question wording) no population claim is made, and the validity machinery is not needed. The line is between exploring with model respondents and reporting a population number from them.

Check yourself

  • In a survey using model respondents, which quantity plays the role of the rectifier?
  • Why does an invented persona break the unlabeled-set assumption?
  • Why can a model that matches the national average still give almost no saving?

Q&A

(Add questions and answers here only if worth keeping.)