{"id":"f6ccf720-dfba-40c5-8e8a-5d8ecf6160f8","arxiv_id":"2412.04309","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Tile maps every canonical ranking score for binary classifiers to a point in the unit square, placing common metrics such as accuracy, F-beta, sensitivity, and precision at geometrically meaningful locations.","lead":"This paper introduces the Tile, a two-dimensional map that arranges an infinite family of scores for ranking two-class classifiers, including accuracy, F-beta, sensitivity, and precision, at specific coordinates. It offers a unified way to compare classifiers according to application-specific preferences, going beyond the two-axis ROC and precision-recall spaces.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Tile's universality — every ranking score is order-equivalent to some canonical R_{a,b} — is asserted in Sec. 3.3 without a proof or theorem reference; if the companion paper's characterization does not cover the two-class case, the Tile organizes only a restricted family, not all rankings.","rationale":"The central contribution is the two-dimensional organization, and its advertised scope is universality. The paper gives a clean construction of R_{a,b}, proves canonicalization within that family, and derives correspondences with ROC and no-skill curves. I looked for an independent soft spot. The candidate is the converse: all ranking scores are order-equivalent to some canonical score. That is not established in this paper. Axioms 2 and 3 are plausibility constraints on any performance ordering; they do not obviously force the particular ratio form (3), and the paper contains no theorem to that effect. Since the claims 'infinity of rankings' and 'captures all the rankings' depend on this, the missing proof is load-bearing. The Reader's verdict already conditionalizes on exactly this concern; I agree. My recommended verdict remains unchanged, i.e. CONDITIONAL. If the companion paper contains the missing theorem, the paper is acceptable with a proper reference; if not, the claim should be weakened to the family actually constructed. This is not a challenge to the internal algebraic results, which appear sound and independently checkable.","tokens_in":30916,"tokens_out":5496,"duration_ms":58535,"concrete_test":"Independently re-derive the representation theorem of [28] for the particular space Ω={tn,fp,fn,tp}: prove that every ordering on P(Ω,Σ) satisfying Axioms 1-3 is induced by some R_I(P)=E_P[I S]/E_P[I] with nonnegative I, including degenerate-prior and zero-probability cases. If such a theorem with proof exists in [28] and covers this space, the concern is resolved by adding a pointer at Sec 3.3. If the theorem is absent or requires extra assumptions, replace the universality claim by 'the Tile organizes the family (3) and those orderings shown equivalent to it'.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the completeness claim in Sec. 3.3: 'for any ranking score there exists a canonical ranking score such that the orderings induced by them are equal.' This sentence turns the Tile from a useful map of the two-parameter family R_{a,b} into a complete organization of all admissible performance orderings. In this paper the sentence is unsupported: no proof follows, no theorem number is cited, and the only relevant reference is the companion paper [28], cited in footnote 1 rather than at the point of use. The restated Axioms 2 and 3 are qualitative constraints; they do not, by themselves, imply that every score satisfying them is a ratio of expectations of the form (3) with nonnegative importance I. Property 2 only removes two scaling redundancies inside the already-restricted family (3); it does not show that this family is exhaustive. If the representation theorem in [28] has additional hypotheses, or if its proof fails at boundary cases of two-class classification (degenerate priors, undefined scores, events of probability zero), then the Tile covers only the canonical scores and the advertised 'all rankings' claim is unsupported. The geometric and algebraic content around the ROC pencils, Fβ, κ, and the γπ/γτ curves is internally consistent and useful; the gap is specifically the missing converse that every admissible ordering appears somewhere on the Tile.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the Tile, a two-dimensional map of ranking scores for two-class classification. The authors particularize the framework of their companion paper A to two-class classification, defining ranking scores R_I as the ratio of an importance-weighted satisfaction expectation to the total importance expectation, and canonical scores R_{a,b} after normalization. They establish geometric correspondences between R_{a,b} and the ROC space via pencils of iso-performance lines, place many known scores (accuracy, TPR, TNR, NPV, PPV, F_beta, Jaccard, balanced accuracy, Cohen's kappa) on the Tile, and present applications: ranking classifiers, visualizing rank correlations, analyzing no-skill performances, and interpreting prior shifts. The appendix provides algebraic proofs (Lemmas 1-10) for the placements and for the effect of performance operations.","tokens_in":31201,"tokens_out":14704,"duration_ms":132121,"significance":"The Tile is a potentially useful visual and conceptual tool: it unifies a broad family of performance orderings on a single diagram and makes explicit a two-parameter structure behind many binary classification scores. The geometric derivations (pencils in ROC space, curves gamma_pi and gamma_tau) are elegant, and the algebraic lemmas are checkable and largely correct. The paper makes concrete, falsifiable statements (e.g., the locations of specific scores and the effect of Cohen's correction), which is a strength. Its central novelty--capturing 'all rankings' on one map--is, however, only as strong as the representability of all admissible orderings by R_I, and this point is not made self-contained.","major_comments":[{"comment":"The sentence 'for any ranking score there exists a canonical ranking score such that the orderings induced by them are equal' is the load-bearing universality claim of the paper. It is neither proved nor accompanied by a theorem reference at the point of use. Within the paper's own definition of ranking scores (Eq. 3), the statement follows from Property 2 and Definition 1 by normalizing I so that I(tn)+I(tp)=1 and I(fp)+I(fn)=1; please include this argument, or state and cite the corresponding representation theorem from paper A [28] with its hypotheses and treatment of boundary cases. As written, a reader cannot tell whether the Tile covers all admissible orderings or only the canonical family R_{a,b}.","section":"Sec. 3.3"},{"comment":"The formula for the location of the ordering induced by R_I is given as (a,b) = (I(tp)/(I(tn)+I(tp)), I(fp)/(I(fn)+I(fp))). By Definition 1, the second coordinate should be I(fn)/(I(fn)+I(fp)); as printed, the formula would place, for example, NPV (I(fn)=1, I(fp)=0) at b=0 instead of b=1, contradicting Table 2 and Lemma 6. This is load-bearing because this formula is the recipe for placing any ranking score on the Tile; please correct it.","section":"Sec. 4.2"}],"minor_comments":[{"comment":"The sentence 'The performance orderings induced by the scores RIa,b are all different' is stated without proof. Since the Tile is advertised as having no redundancy, please provide a short proof or a reference for this injectivity claim.","section":"Sec. 4"},{"comment":"The assertions that the orderings induced by ACP, P4, and VUT are incompatible with the axioms of ranking are not demonstrated; please add a citation or a brief justification.","section":"Sec. 3.2"},{"comment":"The claim that Scott's pi and Fleiss's kappa do not satisfy the axioms of ranking, even for fixed priors, is unsupported; please add a reference or a counterexample.","section":"Sec. 4.4"},{"comment":"In the proof of Lemma 8, the line 'dom(kappa) cap P* = { P in P(Omega,Sigma) } cap P*' appears to have a missing condition; please check the typesetting.","section":"Appendix A.3.3"},{"comment":"The abstract and conclusion say the Tile 'captures all the rankings', while the body defines ranking scores only as in Eq. (3). Please align the wording with the precise scope of the claim, for instance by saying 'all orderings induced by ranking scores of the form (3)'.","section":"Abstract and Sec. 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is the second of a trilogy and depends heavily on the authors' companion paper [28] for the axiomatic basis of 'ranking scores'. The editor may wish to consider whether the journal requires the present paper to be readable without that companion. In addition, the empirical correlation figures (e.g., Figs. 4 and 7) do not include code or data; if the journal values reproducibility, the authors should be encouraged to release the simulation code."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"By way of quick take: this is a useful methodological paper. The Tile gives you a clean 2D parameterization of the canonical ranking scores R_{a,b}, places the standard scores correctly, and ties them to ROC geometry with pencils. The appendix lemmas are checkable and I didn't find an algebraic error. The no-skill curves and the ranking-region algorithm are genuinely handy.\n\nThe thing to press is the completeness claim. Sec 3.3 says \"for any ranking score there exists a canonical ranking score such that the orderings induced by them are equal,\" and the same idea appears in the claim that the Tile \"captures all rankings.\" That is load-bearing if you want to say the Tile organizes all admissible rankings, not just the two-parameter family. In this paper it is asserted without proof and without a theorem reference; the companion paper [28] is cited only in a footnote. It may well be true - the restated Axioms 2 and 3 are qualitative, and I don't see how they alone force the representation - but the reader shouldn't have to guess which theorem in paper A does the job. If the companion paper has a proof, add the precise statement and a proof sketch or a clear pointer. If not, the Tile is still a good map of the family you define, but drop or soften the universal phrasing.\n\nMinor things: the rank-correlation tiles (Figs 4, 7) are Monte Carlo estimates without error bars or code; they're illustrative, so this is not serious, but easy to fix. Also the \"all orderings are distinct\" point in Sec 3.3 is argued from distinct pencil vertices; I believe it, but a one-line proof would tighten it. There is a typo \"tartget\" in the appendix; not worth mentioning to the authors unless they ask.\n\nWho this is for: people who design benchmarks, choose evaluation metrics, or teach classification evaluation. It deserves a serious referee. Send it to review, but require that the universality claim be made traceable - either proved in this paper or exactly cited from the companion work.","headline":"The Tile is a well-executed map of an infinite family of ranking scores with sound algebra; the only real gap is the unproved completeness claim in Sec 3.3, which should be traceable to the companion paper or proved here.","tokens_in":31801,"tokens_out":4627,"would_cite":true,"duration_ms":46116,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces the Tile, a two-dimensional map that gives every ranking score for two-class classifiers a definite location, with nearby locations inducing similar classifier rankings.","keywords":["two-class classification","ranking scores","performance evaluation","ROC space","F-beta scores","canonical ranking scores","prior shift","no-skill performances"],"falsifier":"Take any score satisfying Axioms 2 and 3 and plot its iso-performance lines for fixed priors; if these lines do not form a pencil whose vertex lies in the gray regions outside the ROC square, or if no $(a,b)$ reproduces its ordering, the Tile's completeness fails.","tokens_in":30703,"feed_emoji":"🗺️","tokens_out":6854,"duration_ms":64615,"temperature":0.7,"pith_summary":"The paper's aim is to establish that the variety of scores used to rank two-class classifiers is not arbitrary: up to ordering, every admissible ranking score is a member of one two-parameter family. The Tile is the map $(a,b) \\mapsto R_{a,b}$ of the canonical members of that family, so each score, from accuracy and $F_\\beta$ to exotic similarity coefficients, lands at a point whose neighbors are scores that rank classifiers similarly. The authors argue that with this map one can read off which classifier is best under any preference, see how prior class frequencies move rankings, characterize any new score by comparing it to the whole map, and understand why balanced accuracy and Cohen's kappa are not new ranking principles. If the completeness claim holds, the Tile is to classifier evaluation what a periodic table is to chemical elements: a single organizing picture for an apparently endless list.","feed_headline":"One 2D map places every two-class ranking score","feed_subtitle":"Accuracy, F1, precision, recall, and infinitely many more scores each get a point whose nearby rankings are similar.","key_machinery":"The load-bearing object is the canonical ranking score $R_{a,b}$, obtained by normalizing the general ranking score $R_I$ so that the importances on the two correct cells and on the two incorrect cells are each balanced. Explicitly, $R_{a,b}(P)=((1-a)P(\\{tn\\})+aP(\\{tp\\}))/((1-a)P(\\{tn\\})+(1-b)P(\\{fp\\})+bP(\\{fn\\})+aP(\\{tp\\}))$. The two parameters $a$ and $b$ separate two choices: which of the two correct outcomes matters more, and which of the two errors matters more. The whole argument is carried by the fact that ordering depends only on ratios of importances, so a score's ordering is a point in the unit square; the paper's geometry then reads that point through the pencil of iso-performance lines in ROC space, whose vertex lies outside the ROC square for every admissible score.","core_discovery":"Working within the axiomatic ranking theory of the companion paper [28], the authors particularize to two-class crisp classification a family of ranking scores $R_I(P)=\\mathbb{E}_P[I S]/\\mathbb{E}_P[I]$, where $I$ is a nonnegative importance on the four cells of the confusion matrix and $S$ is the satisfaction indicator. Because rescaling the importances on the two correct cells and on the two incorrect cells does not change the induced ordering, every ordering has a canonical representative $R_{a,b}$ with $a,b\\in[0,1]$: $a$ balances true positives against true negatives and $b$ balances false positives against false negatives. The Tile is the map $(a,b)\\mapsto R_{a,b}$; the paper claims this map is complete, in the sense that for any ranking score there is a canonical score inducing exactly the same ordering, and that the orderings at different points of the Tile are all distinct. It then charts the coordinates of familiar scores—the four corners are $NPV$, $TPR$, $TNR$, and $PPV$, accuracy sits at the center, and the $F_\\beta$ family runs along the right edge—and proves geometric correspondences: iso-performance lines in ROC space form a pencil whose vertex encodes $(a,b)$, and operations such as changing the predicted class, swapping classes, or shifting priors act as symmetries or deformations of the square. It also identifies curves $\\gamma_\\pi$ and $\\gamma_\\tau$ of scores that tie all no-skill performances, which explains where chance-corrected scores live.","pith_inferences":["If the inherited completeness theorem is made fully explicit, the Tile becomes a design tool: one could invent a new evaluation score by choosing an $(a,b)$ point rather than deriving a formula, and know in advance how it ranks classifiers.","The same correlation portrait could be turned into a robustness measure: the area or diameter of the region where a benchmark's ranking does not change would quantify how stable a leaderboard is to score choice.","The two-class construction suggests extending the idea to multiclass or soft classifiers by replacing the four events with a continuous satisfaction variable, though the paper does not do that.","The no-skill curves $\\gamma_\\pi$ and $\\gamma_\\tau$ imply that chance-correction is not a special trick for accuracy: any point on the Tile can be corrected the way Cohen corrected accuracy, producing a score whose ranking is again on the Tile."],"forward_implications":["A practitioner can read from one picture which classifier wins under any preference: changing $(a,b)$ changes the winner, and the boundaries between winners are convex polygons when priors are balanced.","Scores that are known to be order-equivalent, such as $F_1$ and the positive Jaccard index, or balanced accuracy and Youden's index, occupy the same point on the Tile, so the map makes ranking equivalences visible at a glance.","Any new score can be characterized by its Kendall rank correlation against all $R_{a,b}$, producing a correlation portrait of that score.","With fixed priors, chance-corrected variants such as Cohen's kappa correspond to a single point on the Tile; applying Cohen's correction to any $R_{a,b}$ collapses an entire horizontal line to a point, showing a large loss of ranking diversity."],"supporting_citations":[{"why":"Supplies the axiomatic ranking framework and the theorem, used without proof here, that every admissible ranking score is order-equivalent to a canonical $R_{a,b}$.","marker":"[28]"},{"why":"Provides the ROC isometrics geometry that the paper generalizes to a full pencil-of-lines construction for all canonical scores.","marker":"[14]"},{"why":"Defines the target/prior shift operation whose effect on the Tile is used to handle fixed non-uniform priors.","marker":"[33]"},{"why":"Characterizes the PABDC families, connecting rational-importance ranking scores to existing presence/absence dissimilarity coefficients.","marker":"[5]"},{"why":"Supplies the experimental-comparison context for structuring performance measures, which the Tile reorganizes into one map.","marker":"[13]"}],"fun_headline_variants":["One Tile to rank them all","All two-class ranking scores, one Tile","A single 2D map for every classifier ranking","The Tile: your complete ranking-score map","Every ranking score gets a Tile spot"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The completeness of the Tile rests on the companion paper's claim that every score satisfying Axioms 2 and 3 is order-equivalent to some canonical $R_{a,b}$, a claim restated here without proof.","fun_headline_variants_meta":{"raw":{"variants":["One Tile to rank them all","All two-class ranking scores, one Tile","A single 2D map for every classifier ranking","The Tile: your complete ranking-score map","Every ranking score gets a Tile spot"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001114,"raw_usage":{"total_tokens":4722,"prompt_tokens":1113,"completion_tokens":3609,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":729,"completion_tokens_details":{"reasoning_tokens":3544}},"tokens_in":729,"tokens_out":3609,"duration_ms":26717,"temperature":1.0,"reasoning_tokens":3544,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:32:58.864066+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any score satisfying Axioms 2 and 3 and plot its iso-performance lines for fixed priors; if these lines do not form a pencil whose vertex lies in the gray regions outside the ROC square, or if no $(a,b)$ reproduces its ordering, the Tile's completeness fails.","supporting_citations":[{"cited_title":"Foundations of the theory of performance-based ranking","cited_arxiv_id":null,"evidence_quote":"Supplies the axiomatic ranking framework and the theorem, used without proof here, that every admissible ranking score is order-equivalent to a canonical $R_{a,b}$."},{"cited_title":"The geometry of ROC space: Understanding machine learning metrics through ROC isometrics","cited_arxiv_id":null,"evidence_quote":"Provides the ROC isometrics geometry that the paper generalizes to a full pencil-of-lines construction for all canonical scores."},{"cited_title":"The hitchhiker’s guide to prior-shift adaptation","cited_arxiv_id":null,"evidence_quote":"Defines the target/prior shift operation whose effect on the Tile is used to handle fixed non-uniform priors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Characterizes the PABDC families, connecting rational-importance ranking scores to existing presence/absence dissimilarity coefficients."},{"cited_title":"An experimental comparison of performance measures for classification","cited_arxiv_id":null,"evidence_quote":"Supplies the experimental-comparison context for structuring performance measures, which the Tile reorganizes into one map."}],"review_version":1}