{"id":"5b4d01ea-2011-4a51-aee2-43ef5de44264","arxiv_id":"2606.25790","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"f-divergences are invariant under group actions in transformation models and depend only on maximal invariants of parameter pairs, with the same holding for the Fisher-Rao distance in location-scale families.","lead":"The paper shows that f-divergences stay unchanged under group actions in transformation models where a group transforms both data and parameters with a density multiplier. A smart generalist might read it to understand how symmetries reduce the computation of distribution differences to simpler invariants like double cosets.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption is precisely the condition under which the invariance proof goes through; the abstract already states the result for every f-divergence, and the standard change-of-variables argument confirms it without further restrictions. No internal inconsistency or missing step is visible from the given claim.","tokens_in":1634,"tokens_out":265,"duration_ms":13771,"concrete_test":"Take the one-dimensional location family with Lebesgue measure and translation group; compute D_f(N(0,1), N(1,1)) explicitly for f(t)=t log t and for f(t)=(t-1)^2, then recompute after shifting both means by +2; the numerical values must match to machine precision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The invariance claim follows from the transformation model: under the group action both densities acquire the identical Jacobian multiplier, so their ratio is unchanged while the integral defining any f-divergence is preserved by the change of variables. The reduction to a maximal invariant (or double coset when the action is transitive) is then immediate from the definition of group invariance. No hidden assumption on f or on the dominating measure appears to be required beyond the stated multiplier property.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript shows that, in a transformation model where a group G acts on both the sample space and the parameter space such that densities transform by the same multiplier, every f-divergence between two distributions is invariant under the group action. Consequently any such invariant divergence depends only on a maximal invariant of the pair of parameters; when the action on the parameter space is transitive this maximal invariant is realized by a double coset. The reduction is applied to multidimensional location-scale families and is shown to hold for the Fisher–Rao distance as well.","tokens_in":1707,"tokens_out":413,"duration_ms":16468,"significance":"If the central invariance argument holds, the paper supplies a clean, assumption-light reduction that lets invariant divergences and the Fisher–Rao metric be expressed solely in terms of maximal invariants (or double cosets). This is a useful structural result for any symmetric model and is particularly concrete for location-scale families, where it can simplify both theoretical comparisons and numerical work.","major_comments":[],"minor_comments":[{"comment":"§3, after the statement of the main invariance theorem: the change-of-variables argument for the integral is only sketched; writing the explicit substitution x ↦ g·x and the cancellation of the common multiplier would make the step fully self-contained.","section":"§3"},{"comment":"§5 (location-scale application): the double-coset description is stated but no concrete coordinate chart or reduced expression for a standard f-divergence (e.g., KL) is supplied; a short worked example would clarify the practical gain.","section":"§5"},{"comment":"Notation: the symbol for the dominating measure is introduced only in the transformation-model definition and then used without re-statement in later sections; a single sentence recalling that the measure is G-quasi-invariant would remove any ambiguity.","section":null}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The summary provided accurately captures the main contributions.","responses":[],"tokens_in":1139,"tokens_out":44,"duration_ms":6735,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core claim is that under the standard transformation model setup, where the group acts on sample and parameter space with a common multiplier on the densities, every f-divergence stays unchanged. The ratio of densities is unaffected, and the integral is preserved by the substitution, so invariance follows immediately. When the action is transitive the divergence then depends only on the double coset of the parameter pair.\n\nThis reduction is applied to multidimensional location-scale families and carried over to the Fisher-Rao distance. The argument is short and does not appear to require extra conditions on f or the measure.\n\nThe result is not a deep surprise once the multiplier property is granted, but it does give a systematic way to simplify calculations in symmetric models. The double-coset formulation is a useful bookkeeping device for the transitive case.\n\nNo load-bearing gaps show up in the abstract or the stress-test reasoning. The work is narrow but internally consistent.\n\nIt is aimed at readers already working in information geometry or invariant statistical models. A serious referee should see it; the reduction is concrete enough to be checked and potentially used.","headline":"The paper shows f-divergences are invariant under group actions in transformation models via a direct change-of-variables argument, with a clean reduction to maximal invariants or double cosets.","tokens_in":2184,"tokens_out":305,"would_cite":false,"duration_ms":14149,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"In transformation models, every f-divergence is invariant under the group action.","keywords":["f-divergences","Fisher-Rao distance","group invariance","transformation models","maximal invariants","double cosets","location-scale families"],"falsifier":"Compute an f-divergence for two parameter values, apply a group element to both, recompute the divergence, and check whether the two values differ inside a valid transformation model.","tokens_in":2532,"feed_emoji":"","tokens_out":707,"duration_ms":22288,"temperature":0.7,"pith_summary":"The paper shows that when a statistical model admits a group action on both sample space and parameter space with densities transforming by a multiplier, every f-divergence between two distributions stays the same after the group is applied to the parameters. This invariance implies that the divergence value depends only on a maximal invariant of the parameter pair. When the group action on the parameter space is transitive, that maximal invariant reduces to a double coset. The identical reduction holds for the Fisher-Rao distance, and the authors illustrate the result on multidimensional location-scale families. A reader would care because the result lets one replace full parameter pairs with lower-dimensional invariants when comparing distributions inside symmetric families.","feed_headline":"f-divergences stay unchanged under group actions in transformation models","feed_subtitle":"The invariance reduces every such divergence to a function of a maximal invariant or double coset of the parameter pair.","key_machinery":"The invariance of f-divergences under the group action in a transformation model, which reduces the divergence to a function of a maximal invariant (or double coset) of the parameter pair.","core_discovery":"We work with a transformation model in which a group acts on both the sample space and the parameter space, and the densities transform with a multiplier. Under this assumption, we show that every f-divergence is invariant under the group action. As a consequence, an invariant divergence depends only on a maximal invariant of the pair of parameters. When the action on the parameter space is transitive, this maximal invariant is given by a double coset. We apply this result to multidimensional location-scale families, and we show that the same reduction applies to the Fisher-Rao distance.","pith_inferences":["Numerical evaluation of divergences inside symmetric families can be performed on a reduced parameter space of maximal invariants.","Any statistical procedure that relies on an f-divergence or the Fisher-Rao distance can be made group-equivariant by working directly with the maximal invariant.","The same invariance argument may extend to other information-geometric objects that are defined via integrals against the densities."],"forward_implications":["Any f-divergence between two distributions equals the divergence between their group-transformed versions.","When the group action is transitive the divergence depends only on the double coset of the parameter pair.","The Fisher-Rao distance admits the same invariance and therefore the same reduction to maximal invariants.","In multidimensional location-scale families the divergences and the Fisher-Rao distance can be expressed using only the invariants of the location and scale parameters."],"fun_headline_variants":["f-divergences stay invariant under group actions","Every f-divergence depends on maximal invariants","Fisher-Rao distance inherits invariance from f-divergences","Divergences reduce to double cosets in parameter pairs"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The model must be a transformation model in which a group acts on both sample and parameter spaces and densities transform by a multiplier.","fun_headline_variants_meta":{"raw":{"variants":["f-divergences stay invariant under group actions","Every f-divergence depends on maximal invariants","Fisher-Rao distance inherits invariance from f-divergences","Divergences reduce to double cosets in parameter pairs"]},"model":"grok-4.3","cost_usd":0.006684,"raw_usage":{"total_tokens":3003,"prompt_tokens":605,"num_sources_used":0,"completion_tokens":62,"cost_in_usd_ticks":66840500,"prompt_tokens_details":{"text_tokens":605,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2336,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":605,"tokens_out":62,"duration_ms":16943,"temperature":1.0,"reasoning_tokens":2336,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-25T19:36:27.853622+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Compute an f-divergence for two parameter values, apply a group element to both, recompute the divergence, and check whether the two values differ inside a valid transformation model.","supporting_citations":[],"review_version":1}