Power divergence is a family of statistical divergences between two probability distributions P and Q, parameterized by a real number α ≠ 0,1. The general form is D_α(P||Q) = (1/(α(α-1))) Σ (p^α q^(1-α) - ...) with specific special cases: α=1 gives Pearson chi-squared divergence, α=0 gives the reverse form, and limits yield KL divergence. Parameters: two probability distributions P, Q over the same support and a real parameter α. The family unifies Pearson, Tsallis, and many other divergence measures under a single parametric framework. Non-negative for appropriate α, zero iff P=Q. Persistence: defined in mathematical statistics literature (Cressie and Read 1984), used in robust estimation, goodness-of-fit testing, and density estimation. [formal: power-divergence | substrate: mind | horizon: a moment | explicit: yes | epoch: 0.01]
Accepted ontology entry
power-divergence
Power divergence is a family of statistical divergences between two probability distributions P and Q, parameterized by a real number α ≠ 0,1. The general form is D_α(P||Q) = (1/(α(α-1))) Σ (p^α q^(1-α) - ...) with specific special cases:…
Definition
Why it is in scope
Power divergence is a family of statistical divergences parameterized by α, including both Rényi and Tsallis divergences as special cases. It is a human-authored mathematical framework for measuring the discrepancy between probability distributions, used in statistics, information theory, and machine learning.
Names and aliases
- power-divergenceen · CANONICAL
Relations from this entry
- f-divergenceINSTANCE_OF →
power-divergence D_alpha = (1/(alpha*(alpha-1))) sum(p^alpha q^(1-alpha) - ...) = sum(q_i * f(p_i/q_i)) with f(t) = (t^alpha - alpha*t + alpha - 1)/(alpha*(alpha-1)). This satisfies the f-divergence form D_F(P||Q) = sum(q_i * F(p_i/q_i)) where F convex, F(1)=0. Specific instance of general f-divergence class.
- cmsdard4403ny3vv3nkbt821lDEPENDS_ON →
Power divergence D_alpha(P||Q) = (1/(alpha(alpha-1))) sum(p^alpha q^(1-alpha) - ...) is a functional of two probability distributions P and Q. Remove probability distributions and the divergence has no operands. Operational cessation per Law 8b.
Relations to this entry
- tsallis-divergence← DERIVED_FROM
Tsallis divergence shares the power-divergence building block Σ p^α q^(1-α): Tsallis applies (1/(α-1))(1 - Σ p^α q^(1-α)) while power divergence uses (1/(α(α-1))) Σ(p^α q^(1-α) - ...). Both are transformations of the same underlying power kernel.
Record identity
- Created
- Sep 4, 2026, 7:16 PM UTC
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- afb9856182e624d864b9f9f93de7b1da3239fb36b56ac6bf36577d330cdb290e