Version-Robust Methods for Identifying Minimal Sufficient Statistics
Pith reviewed 2026-05-15 13:51 UTC · model grok-4.3
The pith
The standard criterion for minimal sufficient statistics fails in general because densities can have multiple versions.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
A frequently used criterion asserts that a statistic T(X) is minimal sufficient if, for any sample points x and y, T(x) = T(y) exactly when there exists a finite constant h_xy > 0, independent of theta, such that f_theta(y) = f_theta(x) h_xy for all theta. We show that this criterion is false in general via a counterexample exploiting the non-uniqueness of versions of Radon-Nikodym derivatives. We introduce a version-robust method applicable whenever sufficiency is known and generalize it from Euclidean settings to arbitrary analytic Borel sample spaces and separable measurable statistic spaces. We also obtain a method for exponential-family densities under verifiable hypotheses and show aPf
What carries the argument
Version-robust identification procedure for minimal sufficient statistics that conditions on already-known sufficiency and applies to analytic Borel sample spaces with separable measurable statistic spaces.
If this is right
- Minimal sufficient statistics can be identified reliably even when multiple versions of the densities exist.
- The method extends prior regularity conditions to arbitrary analytic Borel sample spaces.
- Verifiable conditions now exist for identifying minimal sufficient statistics in exponential families.
- Likelihood-ratio arguments for minimal sufficiency are sound only when the version-robust checks are used.
- A separate criterion for minimal sufficiency due to Pfanzagl also fails without extra conditions.
Where Pith is reading between the lines
- Statisticians may need to confirm sufficiency by other means before applying ratio-based tests.
- Numerical routines that search for minimal sufficient statistics could be updated to avoid version-dependent errors.
- Similar version ambiguities may affect other density-based arguments in measure-theoretic statistics.
- The approach could be tested on specific non-Euclidean models such as those on manifolds or function spaces.
Load-bearing premise
The procedure requires that the statistic is already known to be sufficient and that the sample space is analytic Borel while the statistic space is separable measurable.
What would settle it
Find a minimal sufficient statistic T together with points x and y where T(x) equals T(y) but the ratio f_theta(y) over f_theta(x) changes with theta for some theta.
read the original abstract
Let $f_\theta$ be the joint density of a random sample $X$. A frequently used criterion asserts that a statistic $T(X)$ is minimal sufficient if, for any sample points $x$ and $y$, $T(x) = T(y)$ exactly when there exists a finite constant $h_{xy} > 0$, independent of $\theta$, such that $f_\theta(y) = f_\theta(x)h_{xy}$ for all $\theta$. We show that this criterion is false in general via a counterexample exploiting the non-uniqueness of versions of Radon--Nikodym derivatives. Although Sato (1996) established sufficient regularity conditions for the validity of this criterion, these conditions are frequently intractable to verify in practice. We resolve this limitation by introducing a version-robust method applicable whenever sufficiency is known. Moreover, our method allows us to generalize Sato's approach from Euclidean settings to arbitrary analytic Borel sample spaces and separable measurable statistic spaces. We also obtain a method for exponential-family densities under verifiable hypotheses. Taken together, these results clarify when pointwise likelihood-ratio arguments for minimal sufficiency are mathematically sound in irregular settings. Finally, we construct a counterexample demonstrating that a distinct criterion for minimal sufficiency due to Pfanzagl (1994, 2017) similarly fails in the absence of supplementary hypotheses. Identifying minimal sufficient statistics is important not only for parsimonious data reduction but also because, in models admitting complete sufficiency, such statistics provide a practical route to the complete sufficient structure underlying optimal estimation and prediction.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript demonstrates that the standard pointwise likelihood-ratio criterion for minimal sufficiency—T(x)=T(y) iff f_θ(y)=f_θ(x) h_xy for h_xy>0 independent of θ—is false in general, via an explicit counterexample that exploits non-uniqueness of Radon-Nikodym derivative versions. It supplies a version-robust replacement method that applies once sufficiency is already known, generalizes Sato (1996) from Euclidean to arbitrary analytic Borel sample spaces with separable measurable statistic spaces, gives a method for exponential-family densities under verifiable hypotheses, and constructs a counterexample showing that Pfanzagl's (1994, 2017) distinct criterion likewise fails without supplementary conditions.
Significance. If the results hold, the work clarifies the precise scope in which likelihood-ratio arguments for minimal sufficiency are mathematically sound, especially in irregular or non-Euclidean settings. The explicit counterexamples constructed from classical facts about RN versions on analytic Borel spaces, together with the theorem whose hypotheses are checkable once sufficiency is granted, constitute a concrete advance for data reduction and for identifying complete sufficient statistics used in optimal estimation and prediction.
major comments (2)
- [§2] The counterexample establishing failure of the standard criterion (abstract and §2) is load-bearing for the central claim; please confirm in the proof that the two versions of the density induce the same probability measures for all θ while the pointwise ratio condition fails for the chosen T.
- [Theorem 4.1] Theorem 4.1 (version-robust method) presupposes that sufficiency of T is already known; the manuscript should state explicitly whether this hypothesis can be verified in the same analytic-Borel setting without circular appeal to the very likelihood-ratio test being replaced.
minor comments (2)
- [§4] The statement that the sample space is analytic Borel while the statistic space is separable measurable appears in the abstract and §4; a short remark on why separability of the statistic space is needed for the version-robust construction would aid readability.
- [§5] In the exponential-family section, the verifiable hypotheses on the natural parameter space should be cross-referenced to the precise measurability conditions used in the general theorem.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate the requested clarifications in the revised version.
read point-by-point responses
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Referee: [§2] The counterexample establishing failure of the standard criterion (abstract and §2) is load-bearing for the central claim; please confirm in the proof that the two versions of the density induce the same probability measures for all θ while the pointwise ratio condition fails for the chosen T.
Authors: In the counterexample of §2, the two versions of the Radon-Nikodym derivative are constructed to agree almost everywhere with respect to the dominating measure, ensuring they induce identical probability measures for every θ. The pointwise ratio condition nevertheless fails for the chosen T because the versions differ on a null set in a way that violates the required equality at points where T(x) ≠ T(y). We will revise the proof to include an explicit statement confirming these properties. revision: yes
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Referee: [Theorem 4.1] Theorem 4.1 (version-robust method) presupposes that sufficiency of T is already known; the manuscript should state explicitly whether this hypothesis can be verified in the same analytic-Borel setting without circular appeal to the very likelihood-ratio test being replaced.
Authors: Sufficiency of T is to be verified independently of the likelihood-ratio criterion, for instance via the factorization theorem (when the density factors appropriately) or by direct verification that the conditional distribution of X given T is independent of θ. Both approaches are valid in the analytic Borel setting without circularity. Theorem 4.1 is then applied to establish minimality. We will add an explicit remark clarifying this non-circular verification procedure. revision: yes
Circularity Check
No significant circularity; derivation relies on independent measure-theoretic facts
full rationale
The paper's central results rest on the non-uniqueness of Radon-Nikodym derivative versions, a standard fact in measure theory on analytic Borel spaces. The version-robust method is explicitly conditional on prior knowledge of sufficiency, without circular redefinition. Counterexamples for the common criterion and Pfanzagl's criterion are constructed directly without reducing to self-defined quantities. No self-citations are load-bearing for the main claims, and no parameters are fitted and then called predictions. The argument is self-contained against external benchmarks like Sato (1996).
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Radon-Nikodym theorem holds for sigma-finite measures on measurable spaces
- domain assumption Sample space is an analytic Borel space and statistic space is separable measurable
discussion (0)
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