Objective Statistical Dissimilarity Measure

• Objective statistical dissimilarity measures are mathematically grounded functions that quantify differences between entities while ensuring strong invariance and interpretability.
• They enable robust applications in clustering, hypothesis testing, and pattern recognition by preserving key properties such as symmetry, reflexivity, and transitivity.
• These measures transform formulations through equivalence functions and monotonic bijections, ensuring reliable and reproducible comparisons across diverse domains.

An objective statistical dissimilarity measure is a rigorously defined function that quantifies the inexactness or divergence between two entities, such as samples, distributions, objects, or data structures, in a manner that is mathematically grounded, interpretable, and as independent as possible from arbitrary choices, subjective heuristics, or application-specific conventions. These measures serve as core quantitative tools across disciplines in artificial intelligence, statistics, data mining, and related fields, where tasks such as classification, clustering, hypothesis testing, and information retrieval require reproducible and theoretically transparent quantification of differences.

1. Formal Foundations: Definitions and Duality

The foundational formalism for objective dissimilarity measures is built on a dual definition over a set $X$, as articulated in (Belanche, 2012). A similarity measure is cast as an upper-bounded, exhaustive, and total mapping $s: X \times X \to I_s \subset \mathbb{R}$ with $\lvert I_s \rvert > 1$ and $s(x, x) = S_{\max} = \sup I_s$. Its dual, the dissimilarity measure, is a lower-bounded, exhaustive, and total mapping $d: X \times X \to I_d \subset \mathbb{R}$ with $\lvert I_d \rvert > 1$ and $d(x, x) = d_{\min} = \inf I_d$. Every property of a similarity measure (reflexivity, boundedness, symmetry, etc.) is echoed in the dissimilarity case, establishing a symmetry (duality) that allows systematic translation between frameworks designed for similarity and those for dissimilarity.

The critical axiomatic properties underlying objectivity include:

• Strong Reflexivity: $d(x, y) = d_{\min}$ if and only if $x = y$.
• Symmetry: $d(x, y) = d(y, x)$ for all $x, y \in X$.
• Boundedness and Closedness: There must exist upper (for $d$) and lower (for $s$) bounds, and these bounds must be achieved for some pairs.
• Complementarity: The extremal value $d_{\max}$ (if it exists) is attained for some unique complement(s).
• Transitivity: Captured by an operator $T_d$, usually linked to the triangle inequality or its generalizations.

These criteria define a class of objective measures against which operational or ad hoc scores can be contrasted.

2. Fundamental Invariance and Transformations

Preservation of ordering and structure under function transformations is a central feature of objective measures. Two classes of transformations play major roles:

• Equivalence Functions: Monotonic increasing and invertible functions $f$ that, when composed with an existing measure, preserve the induced preorder ($d_1 \prec d_2$ if and only if $f(d_1) \prec f(d_2)$). Equivalence functions enable rescalings and monotonic reparameterizations without altering the relative rankings, ensuring that applications are not sensitive to arbitrary changes of metric scale.
• Transformation Functions: More generally, one can combine equivalence functions with monotonic bijections on $[0,1]$ to create measure dualities, mapping similarities to dissimilarities and vice versa, while maintaining critical structural properties (e.g., strong reflexivity, symmetry, transitivity). An example is $f = f^* \circ n \circ f^z$; for instance, $n(z) = 1 - z$ in $[0,1]$.

This formalism ensures objectivity by detaching the measure’s ordering from its numerical representation, making the measure invariant under a wide class of lawful modifications, which is essential for robust cross-domain applicability.

3. Axiomatic and Statistical Characterizations

Several important dissimilarity measures are characterized axiomatically, guaranteeing objectivity through strictly defined behaviors:

Measure	Canonical Formula	Axiomatic Characterization
$\chi^2$-dissimilarity	$$\chi^2_1(x, \pi) = \sum_i \frac{(s(x)\pi_i - x_i)^2}{s(x)\pi_i}$$	Uniquely characterized by homogeneity, deviations balancedness, and inverse effects (Bouyssou et al., 2021)
Latent-Observed Dissimilarity (LOD)	$LOD(X, Y) = D_{KL}(\tilde{p}(X)\,\|\,q(X))$ with $q(X) = \exp(-f(X))/C$	Measures divergence between true and virtual posteriors, objective in evaluating model fit (Terazono, 2016)
KS distance ($c'$)	$c' = D_{F, F'}\sqrt{n n' / (n + n')}$	Inherits nonparametric objectivity, robust for two-sample testing (Fabbri et al., 2017)
Kernel Multi-sample Dissimilarity (KMD)	Graph-based functional of k-NN relationships	Achieves objectivity via data-processing, invariance, and nonparametric consistency (Huang et al., 2022)

Axiomatization ensures that these measures cannot be arbitrarily chosen, but must emerge from rigorous constraints, sharply limiting subjectivity.

4. Representative Classes, Examples, and Extensions

The general formalism admits classical and modern instances:

• Metric-based Dissimilarities: Euclidean, Manhattan, Minkowski, Mahalanobis, and Wasserstein distances fit the above framework, with positive definiteness and symmetry linking directly to objective criteria.
• Structural and Distributional Dissimilarities: KL divergence and $f$-divergences for distributions, Hotelling's $T^2$ and Bhattacharyya distances for multivariate and Lie group-valued data, and spectral/tensor-based distances for graphs and hypergraphs are all fit for objective comparison, provided their defining properties conform to the above axioms (Hanik et al., 20 Feb 2024, Surana et al., 2021).
• Combinatorial Measures for Categorical Data: The dissimilarity for unordered categorical draws, such as $D(G_1, G_2) = 1 - \langle g_1, g_2 \rangle/K^2$, is shown to yield an expectation $E[(p, q)] = 1 - \langle p, q \rangle$, independent of the ploidy $K$, reinforcing objectivity and interpretability in applications such as genetics (Ahsan et al., 30 Sep 2024).

Across these domains, the guiding principle is structure-preserving quantification, matching the intended notion of (dis)similarity with sharp mathematical definitions.

5. Behavior under Transformations and Invariance

A key insight is that objective dissimilarity measures are constructed to be robust under a class of transformations that includes:

• Monotonic rescalings: Ordering is preserved so all valid comparisons and statistical tests remain meaningful.
• Symmetries and invariances: For example, bi-invariance under group actions in Lie groups (Hanik et al., 20 Feb 2024), invariance under bijective measurable transformations for KMD (Huang et al., 2022), and invariance under diffeomorphisms for function-based signals (Cantelobre et al., 2022).
• Transitivity preservation: Ensures that the measure retains compatibility with essential properties like the triangle inequality, or their generalizations through t-norms for similarities and their duals for dissimilarities.

Such behaviors are essential for meaningful use in knowledge transfer, benchmarking, and generalization: objectivity equates with invariance to non-informative choices.

6. Practical Examples and AI Applications

The abstract framework yields direct operational tools as demonstrated by:

• Pattern Recognition: Objectively defined dissimilarity is central to case-based reasoning, clustering, classification, and anomaly detection (Belanche, 2012).
• Generative Models and Representation Learning: LOD and mutual information-based measures underpin advances in probabilistic model evaluation and the design of representations that are information-preserving (Terazono, 2016).
• Non-Euclidean Data Analysis: In shape and motion analysis, bi-invariant and Lie group-based measures enable unbiased statistical testing; in hypergraph comparison, both tensor-based and expansion-based measures enable the comparison of complex multi-way relations (Surana et al., 2021, Hanik et al., 20 Feb 2024).
• Hypothesis Testing and Robust Statistics: Robust two-sample testing using $c'$ or KMD provides scale-invariant, nonparametric alternatives to parametric divergence-based tests, critical in bioinformatics, signal processing, and scientific research (Fabbri et al., 2017, Huang et al., 2022).

These practical applications depend critically on foundational properties—robustness, consistency, invariance, and interpretability—established by the objective dissimilarity framework.

7. Challenges, Limitations, and Future Directions

Challenges inherent to the construction and deployment of objective statistical dissimilarity measures include:

• Transformation Selection: Choosing the right transformation (e.g., in duality or for ensuring metric properties) can be nontrivial, especially when empirical goals must align with theoretical invariance.
• Computational Overhead: Structural and distributional measures that preserve objectivity can be computationally intensive, particularly for large sets, structured data, or high dimensions (e.g., tensor-based measures on hypergraphs, Lie group computations).
• Data Heterogeneity and Missing Data: Measures such as CDM and its weighted and compound variants have been developed to ensure objectivity in the context of incomplete data, but the tuning of regularization and weighting can introduce modeling choices that must be carefully calibrated (Zhou et al., 2018, Zhou et al., 2019).
• Combinatorial Complexity: For measures defined over unordered samples and identity states, enumeration and probability assignments must correctly account for relabeling and ordering invariances, a nontrivial group-theoretic problem (Ahsan et al., 30 Sep 2024).

Future directions include unifying frameworks for non-Euclidean and structured data, further axiomatic studies to characterize new dissimilarities, computational advances for efficient estimation under the objective paradigm, and broadening applications in scientific analysis, machine learning, and hypothesis testing.

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Objective statistical dissimilarity measures are thus defined and understood as rigorously constructed, property-preserving, and context-invariant quantifiers of difference, built upon a duality-theoretic, axiomatic, and transformation-resilient foundation. Their development and use enable robust, interpretable, and transferable comparison and analysis in statistical inference, artificial intelligence, and data science.