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Final StateOut of Distribution: A Map of Where the Data Lived
VOL. I  ·  NODE 114▢  ATLAS

THE SURVEYOR'S VALLEY

Out of Distribution: A Map of Where the Data Lived

Out of distribution is a relative label: a case falls outside a specified reference distribution.

OUTSIDE THE REGION

A map of where the data lived, not of the world

Reference distribution with one case outside it and a separate definition of distribution shift.The figure separates a case-level OOD relation from a change between reference and deployment distributions without treating either as a performance verdict.TRAINING DATAREFERENCEDISTRIBUTIONOODCASEOOD CASERELATIVETO REF.SHIFTDEPLOYMENTVS REF.
  • OOD is relative: a case falls outside a specified reference distribution
  • Distribution shift means the deployment distribution differs from the reference distribution
  • Neither label alone establishes failure, novelty, or lack of training exposure

OOD describes a case relative to a reference distribution; distribution shift compares distributions. Performance under either condition must be measured rather than inferred from the label.

GEOMETRY, NOT A VERDICT

Outside the convex hull means extrapolation. Nothing more.

Points inside and outside a dataset's convex hull labeled as geometric interpolation and extrapolation.The figure presents extrapolation neutrally as location outside the convex hull and explicitly withholds any reliability verdict or model-mechanism claim.INTERPOLATIONINSIDE HULLNEW INPUTEXTRAPOLATEOUTSIDECONVEX HULLNEUTRAL GEOMETRYNOT A VERDICTON OOD / SHIFTNOVELTY / FAILURE
Source: Balestriero, Pesenti & LeCun, “Learning in High Dimension Always Amounts to Extrapolation,” arXiv:2110.09485, 2021.

Convex-hull extrapolation is a geometric relation to a finite dataset. It is not equivalent to probabilistic OOD, distribution shift, novelty, or model failure.

  • Interpolation: the input lies inside the dataset's convex hull
  • Extrapolation: the input lies outside that geometric support
  • In more than 100 dimensions, the paper finds interpolation almost surely never occurs

An epistemic limit, not another definition of OOD

No reliable reference class means the evidence does not support stable odds or a defensible comparison set. That can occur with or without a formally OOD input.

SUREST WHERE IT IS BLINDEST

Confidence can stay high where support is lowest

Confidence-frontier chart showing confidence staying high while support drops at the edge.The schematic separates falling evidence support from expressed confidence that may remain high across a validated OOD boundary; it does not plot measured accuracy or a universal calibration curve.SUPPORT MAY FALLCONFIDENCE MAY NOTDISTANCE FROM SUPPORTEVIDENCESUPPORTSTATEDCONFIDENCETESTED EDGESUPPORTEDUNTESTEDSHIFT CALIBRATIONMUST BE TESTED
  • Low support does not guarantee low expressed confidence
  • Calibration under distribution shift must be measured, not assumed
  • The model may not reliably flag its own operating boundary

Fluent output is not evidence of competence. Confidence can remain high after validated support thins, but there is no universal curve saying where or by how much. That mismatch can help hide the jagged frontier.

ATYPICAL IS NOT NEW

Uncertainty is not an OOD detector

Two OOD-detection failures: an atypical in-distribution case flagged by high uncertainty and an OOD airplane confidently assigned to a familiar class.The figure shows why classifier uncertainty is misaligned with OOD detection: it can create both false alarms and missed distribution shifts.UNCERTAINTY IS NOTAN OOD DETECTORATYPICALIN-DIST.HIGHUNCERTAINTYFALSE OODALARMAIRPLANEOUT OFDISTRIBUTIONCONFIDENTCAT OR DOGFAMILIARFEATURESMISSED OODSHIFTCLASSIFIER ANSWERSTHE CLASS QUESTION
Source: Li et al., “Position: Supervised Classifiers Answer the Wrong Questions for OOD Detection,” ICML/PMLR 267, 2025.

Li, Lu, Kirichenko, Qiu, Rudner, Bruss and Wilson argue that uncertainty- and feature-based detectors answer the wrong questions; more scale or outlier exposure does not remove that objective mismatch.

  • An atypical in-distribution case can trigger high uncertainty — a false alarm
  • An OOD case can confidently activate familiar class features — a missed shift
  • A supervised classifier answers the class question, not 'did this come from my distribution?'

ALL MODELS ARE WRONG

Essentially, all models are wrong, but some are useful.

George E. P. Box, Empirical Model-Building and Response Surfaces, 1987

The statistician's oldest warning, and the cleanest gloss of OOD: a model is a usable approximation of the region it was built on — never a mirror of the world.

A BOUNDED ANALOGY

OOD and Knightian uncertainty can overlap; they are not the same

Two separate axes for distribution membership and availability of a reliable reference class.The comparison shows that OOD status and Knightian no-class uncertainty can overlap without being equivalent.TWO DIFFERENT QUESTIONSOODCASEOUTSIDECHOSENREF. DIST.DISTRIB.QUESTIONNO RELIABLECLASSEVIDENCESUPPORTSA CLASS?ODDS STABLE?EPISTEMICQUESTIONMAY OVERLAPNEITHER IMPLIES THE OTHERKNIGHTIAN ANALOGYIS BOUNDED
  • OOD is defined relative to a reference distribution
  • No reliable class is an epistemic claim about defensible comparison and odds
  • Knightian uncertainty is an analogy for the second problem, not an OOD definition

Use the Knightian analogy only when no stable reference class supports odds. An OOD case may still have useful precedents, and an in-distribution decision may still lack a defensible class.

WHERE THE MAP RUNS OUT

The honest edge is the useful one

  • For shift: test deployment performance and calibration
  • For OOD detection: validate the detector against the chosen reference distribution
  • For no reliable class or check: abstain and retain human ownership

Use a verifiable space where an external check exists; where no defensible class or check exists, abstain.

Read the transcript

01 · THE SURVEYOR'S VALLEY

A surveyor spent his whole life mapping one valley. Every field, every lane, every bend of the river, drawn beautifully, because he walked all of it. Now ask him for the country past the ridge, land he has never seen. A careful man would leave the parchment blank. This one keeps drawing: coastlines, towns, roads, all in the same steady, confident hand. The new detail may be invented. That is the risk this entry is about. Strong performance on covered ground does not certify the map past its validated edge.

02 · OUTSIDE THE REGION

Separate two terms. An input is out of distribution only relative to a specified reference distribution: the case does not look like draws from that reference under the features and test being used. Distribution shift compares populations: the deployment distribution differs from the training or validation reference. One concerns a case relative to a distribution; the other concerns a change between distributions. Neither term by itself proves the case is novel, absent from training, difficult, or destined to fail. Those are separate empirical questions.

03 · GEOMETRY, NOT A VERDICT

Now separate a third concept. In the geometry used by Balestriero, Pesenti and LeCun, interpolation means a point lies inside the convex hull of a finite dataset, and extrapolation means it lies outside. They show that in more than one hundred dimensions, geometric interpolation almost surely never occurs under their conditions. Extrapolation here is neutral. It does not identify a model mechanism, imply failure, establish probabilistic out-of-distribution status, or prove distribution shift. It only states where a point lies relative to the dataset's convex hull.

04 · NO RELIABLE REFERENCE CLASS

A fourth concept is epistemic rather than geometric. No reliable reference class means the available evidence does not support a stable comparison set or defensible odds for the question. That is not another definition of out of distribution. An input can be out of one reference distribution while useful evidence exists elsewhere. An input can also look in distribution while the decision being asked has no validated class. Distribution membership and evidential support must be assessed separately.

05 · SUREST WHERE IT IS BLINDEST

Here is the part that turns a limitation into a trap. You might hope expressed confidence falls whenever evidence support does. That relationship cannot be assumed. Under distribution shift, a system may remain confident on some unsupported inputs, and its calibration may differ from what was measured in distribution. There is no universal curve saying hallucination rises or calibration degrades by a fixed amount at one exact line. The operating requirement is narrower: test calibration under relevant shifts, and do not use fluent confidence as the boundary detector.

06 · ATYPICAL IS NOT NEW

And it gets subtler. You might hope classifier uncertainty can raise a hand and say, this one came from somewhere else. Li, Lu, Kirichenko, Qiu, Rudner, Bruss and Wilson show why that hope is misaligned. An atypical case that still belongs to the training distribution can produce high uncertainty, creating a false alarm. An out-of-distribution case can carry features that make the classifier confidently choose a familiar class, creating a missed shift. Their example is plain: a cat-dog classifier may confidently label an airplane if the airplane contains features useful for separating cats from dogs. The classifier answered the class question it was trained to answer. It was never trained to answer whether this input came from its distribution.

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08 · ALL MODELS ARE WRONG

The statistician George Box put all of this in a line, decades before any of it was built. Essentially, all models are wrong, but some are useful. That is not a shrug. It is a discipline. A model is a working approximation of the slice of reality it was built from. Inside that slice it is useful, sometimes astonishingly so. But it is still a map, and the map is not the country. The mistake is not using the model. The mistake is forgetting it has an edge, walking past that edge, and reading the same confidence you trusted inside as if it still meant something outside. Useful is not the same as true. Only one of them survives the boundary.

09 · A BOUNDED ANALOGY

Frank Knight's distinction offers a bounded analogy, not an equivalence. Knightian uncertainty concerns decisions for which no stable reference class supports odds. Out-of-distribution concerns a case relative to a chosen reference distribution. The two can overlap, but neither implies the other. An out-of-distribution case may have documented precedents in another evidence base. An apparently in-distribution case may still support no defensible odds for the decision. Use the Knightian label only for that no-class limit, not as a synonym for OOD.

10 · WHERE THE MAP RUNS OUT

Match the control to the problem. For distribution shift, measure deployment performance and calibration against the relevant shifted population. For OOD detection, validate the detector against an explicitly chosen reference distribution and alternatives. Convex-hull extrapolation needs no moral reading; it is geometry. Where no reliable reference class or external check exists, the answer is not more confidence but abstention and human ownership. Keep those boundaries separate, and each term becomes useful instead of rhetorical.

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