The ceiling imposed by two disagreeing annotators

Lax303562.AnnotatorCeiling · concepts/Lax303562/AnnotatorCeiling.lean · lax-303562

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    Natural Language Statement

    Theorem

    When the same image is annotated twice, with aa and bb instances, a prediction with mm instances can be scored against both. Its true positives against the first annotator are at most aa, and against the second at most mm. If the annotators disagree in their counts, a<ba < b, the total number of true positives stays strictly below the combined denominator m+(a+b)/2m + (a + b)/2, by a margin of (ba)/2(b − a)/2: half the disagreement.

    Since each matched IoU is at most 11, the combined numerator is at most the number of true positives, so the combined Panoptic Quality is strictly below 11 for every prediction. The ceiling is set by the annotators, not by the model.

    Concept map
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    Lean source view on GitHub

    1import Lax303562.PanopticQuality
    2
    3/-!
    4---
    5title: The ceiling imposed by two disagreeing annotators
    6type: theorem
    7---
    8When the same image is annotated twice, with `a` and `b` instances, a prediction
    9with `m` instances can be scored against both. Its true positives against the
    10first annotator are at most `a`, and against the second at most `m`. If the
    11annotators disagree in their counts, `a < b`, the total number of true positives
    12stays strictly below the combined denominator `m + (a + b)/2`, by a margin of
    13`(b − a)/2`: half the disagreement.
    14
    15Since each matched IoU is at most `1`, the combined numerator is at most the
    16number of true positives, so the combined Panoptic Quality is strictly below `1`
    17for every prediction. The ceiling is set by the annotators, not by the model.
    18-/
    19
    20namespace Lax303562.AnnotatorCeiling
    21
    22/-- With `a < b` annotated instances and `m` predicted ones, the true positives
    23against both annotators are strictly fewer than the combined denominator. -/
    24axiom ceiling_two_annotators (m a b tpA tpB : ℕ)
    25 (hab : a < b) (hA : tpA ≤ a) (hB : tpB ≤ m) :
    26 ((tpA : ℝ) + tpB) < (m : ℝ) + ((a : ℝ) + b) / 2
    27
    28/-- The combined Panoptic Quality against two disagreeing annotators is strictly
    29below `1`, whatever the prediction. -/
    30axiom pq_lt_one_of_disagreement (sA sB : ℝ) (m a b tpA tpB : ℕ)
    31 (hab : a < b) (hA : tpA ≤ a) (hB : tpB ≤ m)
    32 (hsA : sA ≤ tpA) (hsB : sB ≤ tpB) (hm : 0 < m) :
    33 (sA + sB) / ((m : ℝ) + ((a : ℝ) + b) / 2) < 1
    34
    35end Lax303562.AnnotatorCeiling
    36
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