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114THE SURVEYOR'S VALLEY

Out of Distribution: A Map of Where the Data Lived

Out of distribution describes a case relative to a named reference. Model reliability still needs evidence for the task.

A surveyor's valley map continues with invented marks beyond its ridge.Solid river, field and settlement marks occupy the surveyed valley. Dashed accent marks add an unsupported coastline and settlements beyond the ridge. This is an analogy, not a depiction of every OOD output.SurveyedInvented
An illustrative survey: solid marks come from walked ground; dashed marks beyond the ridge are invented.

OUTSIDE THE REGION

Name the reference before calling a case OOD

An individual case lies outside a chosen reference region around a small sample.Sixteen illustrative observations stay fixed. A dashed ellipse identifies a chosen reference test region; a separate ring lies outside it. The region is not a confidence interval, fitted density or reliability boundary.Reference regionCase
Illustrative two-dimensional sample and chosen reference contour. The contour is a test region, not a measured boundary of reliability.
  • OOD detection assesses a case against a specified reference distribution.
  • Distribution shift compares the deployment and reference populations.

The chosen features and test determine what counts as unusual. A distributional label alone establishes neither novelty nor failure, and does not reveal every example a model encountered in training. Li et al., §2, distinguish label-preserving changes from inputs belonging to new classes.

GEOMETRY, NOT A VERDICT

A convex hull describes the sample geometry

A computed convex hull contains the observations and an interior midpoint; another case lies outside.The polygon is the convex hull of all sixteen displayed observations. The interior ring is the exact midpoint of two connected observations. The exterior ring does not move to a neighbour or receive a correctness verdict. This is a two-dimensional example, not a projection of the paper's high-dimensional datasets.HullInsideOutside
Illustrative geometry, not study data. The hull connects the outermost plotted observations; the interior case is a midpoint of two observations. Source: Balestriero et al. (2021), definition 1, theorem 1 and §2.1.

A point inside or on the convex hull is an interpolation; a point outside is an extrapolation under Balestriero, Pesenti and LeCun's definition. Their theorem for independent uniform samples from a high-dimensional ball requires exponentially more observations to maintain interpolation as dimension grows. The relevant dimension is the smallest flat (affine) space containing the data. There is no universal cutoff at 100 coordinates, and geometric extrapolation alone gives no performance verdict.

Familiar inputs can still lack a defensible comparison

The available evidence may support no stable odds for the decision being asked. That is a limit on inference, separate from an input's distributional status. An OOD case may have useful evidence elsewhere; a familiar-looking case may lack relevant validation.

CLASS CONFIDENCE

A confident class choice leaves another question open

An airplane is assigned confidently to a familiar cat label while its source remains unresolved.A classifier has only cat and dog branches. The airplane's highlighted branch selects cat, while a separate source question remains open. The emphasis is schematic and carries no numerical confidence or measured accuracy.CatDog?Source?
Illustrative cat–dog classifier from Li et al.'s motivating example. Branch emphasis represents a confident choice, not a measured probability. Source: Li et al. (2025), abstract and §4.2.

A classifier can strongly prefer one of its familiar labels for an unfamiliar input. That preference does not establish where the input came from. Test performance and calibration on the relevant deployment population; neither confident wording nor uncertainty alone maps the jagged frontier.

TWO DETECTION ERRORS

Uncertainty can raise the wrong alarm

An ambiguous familiar-class image raises a false alarm; an OOD airplane passes without an alarm.The upper example contains a cat and a dog, creating label ambiguity within the illustrative reference. Both class branches remain plausible and an alert appears. The lower airplane gets one strong class branch while its alarm indicator stays empty. These illustrate case-level false and missed OOD detections, not measured rates or a population shift.!False alarmMissed case
Illustrations of the two error types, not measured classifier outputs. The upper image contains both familiar classes; the airplane belongs to neither. An empty indicator means no OOD alarm.
  • An ambiguous in-distribution image can trigger a false OOD alarm.
  • An OOD airplane can receive a confident familiar label and escape detection.

Li et al., §4.2, separate uncertainty about the class label from uncertainty about distribution membership. Their position paper critiques detectors built from supervised classifiers' uncertainty or features. Some interventions improve particular benchmarks; that does not remove the mismatch between the two questions.

ALL MODELS ARE WRONG

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

George E. P. Box and Norman R. Draper, Empirical Model-Building and Response Surfaces (1987), p. 424

Box and Draper's passage concerns approximating a response with a polynomial. It supports judging an approximation by its use. It does not define OOD or guarantee usefulness inside a training region.

A BOUNDED ANALOGY

Distribution membership and defensible odds are separate

Two cases separate distribution membership from support for decision odds.Inside and outside occupy separate columns; defensible and unresolved odds occupy separate rows. An outside case connects to comparable records in the upper half. An inside case has an open comparison bracket in the lower half. The drawing illustrates that neither assessment determines the other; unresolved odds still permit human judgment.InsideOutsideDefensibleoddsUnresolvedodds?
Two illustrative decisions: an outside case with comparable records and an inside case with unresolved odds. The axes separate questions; they are not a measured risk scale.

An OOD case can have documented precedents in another evidence base. An in-distribution case can still lack a defensible basis for forecasting the outcome of a decision. Knightian uncertainty concerns that second question. Knight describes gradations of comparison and judgment, so missing stable odds does not mean that all evidence or human judgment disappears.

WHERE THE MAP RUNS OUT

Choose a control for the evidence you have

A supported claim and a withheld automated answer both reach the person who decides.The case divides into two possible evidence paths. A completed check supports a bounded claim. An unsupported automated answer is crossed out while the unresolved case continues to the owner. The person remains responsible for the decision in both paths; an OOD label does not select either branch.CheckWithholdDecide
A checked claim and a withheld answer both leave the decision with a person. Neither branch is selected by distribution membership alone.
  • For population shift, evaluate performance and calibration on relevant deployment cases.
  • For OOD detection, validate the reference, alternatives and decision threshold.

Use a verifiable space when an external check can settle the relevant claim. If the automated answer lacks adequate support, withhold it and pass the unresolved case to its owner. An OOD label alone does not require abstention; the person can seek evidence, commission a check or decide under uncertainty.

Read the transcript

01 · THE SURVEYOR'S VALLEY

A surveyor has mapped one valley. The river bends between fields and a small settlement; he has walked the ground those marks describe. Past the ridge, his survey ends. He keeps drawing an invented coastline and another town, using the same steady hand. A fluent continuation can carry detail unsupported by the evidence used to validate it. The map beyond the ridge might contain a correct guess. To rely on it, the traveller needs a reason that goes beyond the quality of the penmanship.

02 · OUTSIDE THE REGION

An input is out of distribution only relative to a named reference. Choose the reference distribution, the features that matter, and the test used to compare a case with it. A case can then be flagged as outside that reference. Distribution shift compares populations: the distribution encountered in deployment differs from the training or validation reference. The two questions are related, but one flagged input does not by itself establish a population change. Neither label establishes novelty, difficulty, inevitable failure, or everything a model has encountered during training.

03 · GEOMETRY, NOT A VERDICT

A convex hull asks a geometric question about a finite sample. Stretch a boundary around its outermost observations, leaving every observation inside or on the edge. A midpoint between two observations lies inside that hull; under this definition, it is an interpolation. A point beyond the hull is an extrapolation. Balestriero, Pesenti and LeCun show why extrapolation can be routine in high dimensions. Their theorem for independent uniform samples from a ball requires sample size to grow exponentially with dimension to maintain interpolation. The dimension of the smallest flat, or affine, space containing the data matters, not simply the number of coordinates. One hundred is not a universal cutoff. This geometry does not tell us whether a prediction will work.

04 · NO RELIABLE REFERENCE CLASS

A different problem arises when the available evidence supports no stable odds for the decision. That is a limit on inference, rather than another definition of out of distribution. Useful evidence may exist outside the chosen reference: another collection may contain comparable cases, or a direct check may settle the claim. Conversely, a familiar-looking input may ask for a forecast that has never been validated. Distribution membership and evidence for the particular decision need separate assessments.

05 · CLASS CONFIDENCE

Consider a classifier trained to distinguish cats from dogs. An airplane can activate features that favour one of those labels, so the classifier confidently chooses cat. The class choice leaves the source question unanswered: did this input come from the reference distribution? High confidence over familiar labels does not answer that question. In deployment, performance and calibration need evaluation on the relevant population. Calibration concerns how confidence relates to observed correctness across cases. It cannot be established by one fluent answer, and there is no universal curve that deteriorates at a single distributional boundary.

06 · TWO DETECTION ERRORS

Li and colleagues describe two errors that follow from confusing label uncertainty with distribution membership. First, an image from the reference distribution can contain both a cat and a dog. A classifier may be uncertain which label to choose. If uncertainty triggers an OOD alert, this becomes a false alarm. Second, an airplane outside that reference can receive a confident cat prediction, leaving the OOD case undetected. These are illustrative cases, not reported error rates. The paper examines detectors based on supervised classifiers' uncertainty and features. Better results on a particular benchmark do not erase the difference between deciding which familiar class fits best and detecting whether the input comes from a different distribution.

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08 · ALL MODELS ARE WRONG

Box and Draper wrote: Essentially, all models are wrong, but some are useful. The passage concerns a polynomial that approximates a response over a limited region of interest. Its usefulness depends on the purpose and on whether its errors matter there. This is a reason to examine an approximation's limits. It gives neither a guarantee of usefulness inside the training region nor a ban on useful predictions outside it.

09 · A BOUNDED ANALOGY

Knight's distinction adds a question about decision evidence. An out-of-distribution case may have comparable records in another evidence base, supporting a defensible estimate for the decision. An in-distribution case may still leave the relevant odds unresolved. Knight discusses the gradations between close comparisons and largely unique judgments; missing stable odds does not make all comparison impossible. Calling the latter problem Knightian uncertainty can be useful, provided it remains separate from OOD. The distributional test concerns a reference population. The judgment concerns what the evidence permits us to infer about this decision.

10 · WHERE THE MAP RUNS OUT

Match the response to the question. Test deployment performance under the population changes that matter, and evaluate OOD detectors against a named reference and meaningful alternatives. When an external check can settle the claim, use it and preserve its limits. When support for the automated answer is inadequate, withhold that answer and pass the unresolved case to its owner. The person can seek better evidence, commission a check or decide under uncertainty. An OOD label alone does not require abstention. Some changes preserve the task, such as a noisier image; useful generalization may be the aim. Membership helps describe the case. Evidence and the decision's consequences determine what to do with it.

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