The Chernoff partition function Z(λ) := ∑_a P₁(a)^(1-λ) · P₂(a)^λ.

The Chernoff information C(P₁, P₂) := -min_{λ ∈ [0,1]} log Z(λ), stated as -sInf of the image; chernoffInfo_attained shows the minimum is realized.

Z(0) = ∑ P₁(a) = 1 for pmf P₁.

Z(1) = ∑ P₂(a) = 1 for pmf P₂.

Each summand (P₁ a)^(1-λ) · (P₂ a)^λ is strictly positive under full support.

Z(λ) > 0 under full support (any λ ∈ ℝ).

λ ↦ Z(λ) is continuous on ℝ.

λ ↦ log Z(λ) is continuous on ℝ (under full-support strict positivity of Z).

Chernoff information is attained: ∃ λ* ∈ Icc 0 1, chernoffInfo = -log Z(λ*).

chernoffInfo P₁ P₂ ≥ 0.

Symmetry of chernoffZSum under λ ↔ 1 - λ: Z_{P₁,P₂}(λ) = Z_{P₂,P₁}(1 - λ).

klDivPmf P P = 0: the Kullback–Leibler divergence of a positive pmf with itself is zero.

Hoeffding tradeoff exponent at Type I level alpha: E₂(α) := min { klDivPmf Q P₂ | Q ∈ stdSimplex ∧ klDivPmf Q P₁ ≤ α }.

The Hoeffding constraint set K := {Q ∈ stdSimplex | klDivPmf Q P₁ ≤ α}.

The Hoeffding constraint set is non-empty when α ≥ 0 and P₁ is a positive pmf: P₁ itself satisfies klDivPmf P₁ P₁ = 0 ≤ α.

The Hoeffding constraint set is a subset of the simplex.

The Hoeffding constraint set is closed (intersection of the closed simplex with the closed sublevel set of the continuous function Q ↦ klDivPmf Q P₁).

The Hoeffding infimum is attained: there exists Q* ∈ K with hoeffdingE2 P₁ P₂ alpha = klDivPmf Q* P₂.

hoeffdingE2 P₁ P₂ alpha ≥ 0.

Hölder multiplicative form for the Chernoff partition function: Z(αλ₁ + βλ₂) ≤ Z(λ₁)^α · Z(λ₂)^β for α, β ∈ (0, 1), α + β = 1.

Stated under full-support P₁, P₂ > 0 to avoid 0^x corner cases.

λ ↦ log Z(λ) is convex on Icc 0 1.

The Chernoff mediator pmf T_λ(a) := P₁(a)^(1-λ) · P₂(a)^λ / Z(λ).

T_λ(a) > 0 under full-support P₁, P₂ > 0.

T_λ is a pmf: ∑ a, T_λ(a) = 1 (by construction).

Mediator pmf bound: at endpoints lam = 0 (resp lam = 1), T_λ = P₁ (resp P₂) under full support + probability conditions.

The Hoeffding minimizer is unique: klDivPmf · P₂ is strictly convex on the convex constraint set K, so the minimizer of hoeffdingE2 is unique.

n-IID Bayes error in pmf form: bayesErrorMinPmf P₁ P₂ n := (1/2) · ∑_{x : Fin n → α} min(∏ P₁(x_i), ∏ P₂(x_i)).

This is the optimal Bayes error for the 2-class hypothesis test with equal priors 1/2 : 1/2 on n IID samples (Bayes-optimal rule decides i := argmax_i P_i^n(x), giving error contribution (1/2) · min(P₁^n(x), P₂^n(x)) per x).

Geometric mean inequality (degenerate Hölder form): min(a, b) ≤ a^{1-λ} · b^λ for a, b ≥ 0, λ ∈ [0, 1].

Auxiliary: n-IID per-point factorization: ∏ i, (P₁ (x i)) ^ (1-lam) * (P₂ (x i)) ^ lam = (∏ i, P₁ (x i)) ^ (1-lam) * (∏ i, P₂ (x i)) ^ lam under P₁, P₂ ≥ 0.

n-IID Chernoff partition function via product factorization: ∑_{x : Fin n → α} (∏ i, P₁ (x i))^(1-lam) · (∏ i, P₂ (x i))^lam = Z(λ)^n.

The Chernoff bound bayesErrorMinPmf ≤ (1/2) · Z(λ)^n for each λ ∈ Icc 0 1.

bayesErrorMinPmf > 0 under full support P₁, P₂ > 0.

For each fixed λ ∈ Icc 0 1, -(1/n) log bayesErrorMinPmf ≥ -log Z(λ) + (log 2)/n (eventually for n ≥ 1).

The + log 2 / n slack term vanishes as n → ∞, leaving -log Z(λ) (and after min over λ, chernoffInfo).

For each fixed λ* ∈ Icc 0 1 attaining chernoffInfo (= -log Z(λ*)), -(1/n) log bayesErrorMinPmf ≥ chernoffInfo + (log 2)/n (eventually for n ≥ 1).

Chernoff achievability (rate-side lower bound): liminf_n -(1/n) log bayesErrorMinPmf ≥ chernoffInfo P₁ P₂.

Chernoff bound (achievability half): the exponential convergence rate of bayesErrorMinPmf is at least chernoffInfo P₁ P₂.

See also chernoff_achievability.