Generalized Mean-Based Metrics

• Generalized mean-based metrics are a unified framework that redefines centrality, dispersion, and prediction using metric, algebraic, and functional generalizations.
• They leverage transformed metric spaces (via functions like φ) to compute means and variances in non-Euclidean settings, enhancing statistical inference.
• Applications span reinforcement learning, matrix analysis, and statistical prediction, where methods like the Fréchet mean optimize estimation and decision-making.

Generalized mean-based metrics provide a unified, rigorous framework for quantifying central tendency, dispersion, and statistical discrimination in situations where the classical Euclidean definition of the mean and variance is inadequate or insufficient. These constructs extend the notion of the mean through metric, algebraic, and functional generalization, revealing deep connections to geometric prediction, decision theory, group and manifold structures, and statistical divergences.

1. Metric and Geometric Foundations of Generalized Means

Generalized mean-based metrics arise from equipping a set $I \subset \mathbb{R}$ (or more generally, a metric space $(X,d)$) with a metric of the form $d_\phi(x,y) = |\phi(x) - \phi(y)|$, where $\phi$ is a strictly monotone, continuous, and bijective map (typically viewed as a utility-type transformation) (Gzyl, 2020). This induces a metric on $I$, allowing analysis via Euclidean projections in the transformed $\phi$-space.

For multivariate extensions over $I \subset \mathbb{R}^d$, the same definition applies component-wise, and the induced metric becomes the Euclidean norm on vectors of the form $\phi(\mathbf{x})$.

In the broader setting of arbitrary metric spaces, the Fréchet mean (or Karcher mean) is defined as the minimizer of the sum of squared distances:

$m^* = \arg\min_{m \in X} \sum_{i=1}^n d(x_i, m)^2,$

establishing a generalized notion of centrality that relies only on the structure of the metric, not on linearity or vector space properties (Bilisoly, 2014).

2. Algebraic Structures and Functional Equations among Mean Functions

On symmetric domains $D \subset \mathbb{R}^2$, the set $\mathcal{M}_D$ of mean functions admits an abelian group structure. This is achieved via a bijection $\varphi$ between $\mathcal{M}_D$ and the additive group of antisymmetric functions, transporting the group law to $\mathcal{M}_D$:

$$M_1 \oplus M_2 = \varphi^{-1}(\varphi(M_1) + \varphi(M_2)),$$

with the arithmetic mean as the neutral element (Farhi, 2010).

The induced metric on means is

$$d(M, N) = \sup_{(x,y) \in D, x \neq y} \frac{|M(x,y) - N(x,y)|}{|x - y|},$$

rendering $(\mathcal{M}_D, d)$ isometric to a closed ball (radius $1/2$) centered at the arithmetic mean. Notably, the geometric and harmonic means reside on the boundary of this ball due to the unboundedness of their associated $\varphi$ images.

Functional symmetries among mean functions are addressed via equations such as $M_0(M_1(x,y), M_2(x,y)) = M_0(x,y)$, yielding solutions that coincide with group-theoretic symmetries in special cases.

3. Generalized Means and Statistical Prediction

For real-valued random variables $X \in I$ over a probability space $(\Omega, \mathcal{F}, P)$, the generalized $\phi$-mean is defined as the minimizer of the mean square prediction error under the induced metric:

$m = \phi^{-1}(E[\phi(X)]),$

which serves as the best predictor in $\phi$-transformed space (Gzyl, 2020). The corresponding $\phi$-variance is $$\operatorname{Var}_\phi(X) = \operatorname{Var}(\phi(X))$$.

Conditional generalized means extend to sub-$\sigma$-algebras $\mathcal{G} \subset \mathcal{F}$ via

$$E_\phi[X | \mathcal{G}] = \phi^{-1}(E[\phi(X) | \mathcal{G}]),$$

with the property that conditional tower rules apply analogously to the classical setting.

Certainty equivalents and conditional preferences are formulated by associating $\phi$ with utility functions, such that for payoffs $X$ and $Y$,

$$Y \,\, \text{preferred to} \,\, X \iff E[u(Y) | \mathcal{G}] \geq E[u(X) | \mathcal{G}],$$

with certainty equivalents computed as $C = u^{-1}(E[u(X) | \mathcal{G}])$. These conditional certainty equivalents satisfy martingale properties, establishing fair pricing in dynamic environments.

4. Parametric Families: Power Means, Divergences, and Inequalities

The power mean family exemplifies generalized means through the parameter $p$:

$$M_p(x_1, ..., x_n) = \left( \frac{1}{n} \sum_{i=1}^n x_i^p \right)^{1/p}.$$

Special cases include the arithmetic $(p = 1)$, geometric $(p \to 0)$, harmonic $(p = -1)$, maximum $(p \to +\infty)$, and minimum $(p \to -\infty)$ means (Dam et al., 2019). These means interpolate between conservative estimation and aggressive maximization strategies in algorithms such as Power-UCT for MCTS, where $p$ dynamically tunes the backup strategy between the average and maximum operators.

Mean-based kernels also generate a rich taxonomy of convex, symmetric, and nonnegative divergence measures. The generalized triangular discrimination and its families form parameterized divergences:

$$L_t(x, y) = \frac{2^t (xy)^{1/2} (x - y)^2}{(x + y)^{t+1}},$$

recovering classical measures (triangular discrimination, Jain–Srivastava, Kumar–Johnson) for specific $t$. Similar families (squared Hellinger, $K_t$, and higher order divergences) possess exponential representations and admit convexity proofs via kernel functions and second derivatives (Tameja, 2012).

The seven classical means, and their chain of inequalities,

$$H(x,y) \leq G(x,y) \leq N(x,y) \leq A(x,y) \leq R(x,y) \leq S(x,y) \leq C(x,y),$$

anchor these constructions, with nonnegative mean differences forming the basis for refined statistical comparisons.

5. Computation, Existence, and Applications

Algorithms for computing generalized means depend on the underlying space:

• In Euclidean or $\phi$-metric settings, closed-form solutions follow from minimization in transformed space (Gzyl, 2020).
• On Riemannian manifolds, the Fréchet mean is computed via Karcher-flow gradient descent, using exponential and logarithmic maps (Bilisoly, 2014).
• For finite or discrete sets, brute-force minimization over candidate points is feasible.

Applications span:

• Quantification of variability in categorical and group-theoretic data via appropriate metrics (e.g., Levenshtein or word metric) (Bilisoly, 2014).
• Algorithmic innovation in reinforcement learning and planning (MCTS), where the power mean modulates exploration and exploitation for improved performance and faster convergence (Dam et al., 2019).
• Statistical inference and information theory through the generation of $f$-divergences and tunable measures, enabling robust estimation and refined hypothesis testing (Tameja, 2012).
• Functional mean construction and operator means in matrix analysis, interpolation between means, and study of inequality stability under perturbations of the metric (Farhi, 2010).

6. Summary of Key Formulas and Properties

Concept	Formula/Definition	Notes/Context
$\phi$-distance	$d_\phi(x,y) = |\phi(x) - \phi(y)|$	Induced by monotone bijection
Generalized $\phi$-mean	$$m = \phi^{-1}\left( \frac{\sum w_k \phi(x_k)}{\sum w_k} \right)$$	Weighted case
Power mean	$M_p(\mathbf{x}) = (\frac{1}{n}\sum x_i^p)^{1/p}$	Interpolates between mean/max/min
Fréchet mean (metric space)	$m^* = \arg\min_{m \in X} \sum d(x_i, m)^2$	Applies for arbitrary $d$
Mean kernel divergence	See $L_t(x,y)$ formula above	Parametric, convex, exponential forms
Metric on mean functions	$d(M,N) = \sup_{(x,y)} |M(x,y) - N(x,y)| / |x-y|$	Closed ball structure
$\phi$-variance	$$\operatorname{Var}_\phi(X) = \operatorname{Var}(\phi(X))$$	Measures residual error

Generalized mean-based metrics, through their geometric, algebraic, and statistical extensions, form a robust theoretical and computational apparatus for centrality, variability, and decision-making in diverse mathematical and applied contexts. Their utility spans metric-based statistics, optimization on manifolds, machine learning algorithms, and advanced statistical inference, enabling the systematic exploitation of underlying symmetries and metric properties for optimal prediction and discrimination.