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F-Divergences, Rényi Information, and Hypothesis Testing — Examples

1 · Prerequisites

2 · Summary

The examples calculate binary KL, Rényi, and total-variation divergences, then check how zero coordinates determine singular divergence values. The testing example uses P as the null, Q as the alternative, and a rejection function whose value is the probability of rejecting P. It computes the type-I budget, the alternative acceptance error, and the likelihood-ratio boundary randomization exactly.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

A binary Neyman–Pearson boundary randomization

Example

Let the null law be $P = (2 / 3, 1 / 3)$ and the alternative law be $Q = (1 / 3, 2 / 3)$ on $X = {a, b}$ . For the type-I bound $ϵ = 1 / 2$ , the optimal test rejects with probability $1 / 4$ at $a$ and probability $1$ at $b$ .

Facts & Assumptions

Given: The displayed null $P$ , alternative $Q$ , and $ϵ = 1 / 2$ .

[F1]

Randomized tests: $ϕ (x) \in [0, 1]$ is the probability of rejecting the null after observing $x$ ; $1 - ϕ (x)$ is the acceptance probability.

[F2]

Test size and power: for a simple null $P$ the type-I error is $E_{P} ϕ$ , and under the alternative $Q$ the rejection probability is the power $E_{Q} ϕ$ .

[F3]

Finite type-II testing function with explicit orientation: $β_{ϵ} (P, Q)$ is the infimum of $E_{Q} [1 - ϕ]$ over all rejection functions with $E_{P} ϕ \leq ϵ$ .

[F4]

Finite probability vectors: $P, Q$ are probability vectors on ${a, b}$ with the displayed coordinates.

[F5]

Finite probability spaces, outcome weights, events, and event probabilities and Expectation of a real random variable on a finite probability space: the displayed vectors give finite laws, with $E_{P} h = \sum_{x} p_{x} h (x)$ and $E_{Q} h = \sum_{x} q_{x} h (x)$ .

Verification

Given: $P = (2 / 3, 1 / 3)$ is the null, $Q = (1 / 3, 2 / 3)$ is the alternative, and the test function is a rejection probability.

1.1F1F2F3F4F5constructalgebra

Define $ϕ (a) = 1 / 4$ and $ϕ (b) = 1$ . By [F1], this is a valid randomized test. Its type-I error is $(2 / 3) (1 / 4) + (1 / 3) (1) = 1 / 2$ , and its type-II error is $(1 / 3) (3 / 4) + (2 / 3) (0) = 1 / 4$ by [F2, F3, F5].

2.1F1F2F3F4F5step 1.1algebra

Consider any feasible test and put $u = ϕ (a)$ , $v = ϕ (b)$ . If $u \leq 1 / 4$ , then its alternative rejection probability is $E_{Q} ϕ = u / 3 + 2 v / 3 \leq 1 / 12 + 2 / 3 = 3 / 4$ by [F5]. If $u > 1 / 4$ , the constraint $2 u / 3 + v / 3 \leq 1 / 2$ gives $v \leq 3 / 2 - 2 u$ , hence $E_{Q} ϕ \leq u / 3 + (2 / 3) (3 / 2 - 2 u) = 1 - u < 3 / 4$ by [F5]. Thus no feasible test has smaller type-II error than the candidate in step 1.1, proving $β_{1 / 2} (P, Q) = 1 / 4$ .

3.1F1F2step 1.1algebra∎

The alternative-to-null likelihood ratios are $q_{a} / p_{a} = 1 / 2$ and $q_{b} / p_{b} = 2$ . The test of step 1.1 rejects fully at the larger ratio $b$ and randomizes at the smaller ratio $a$ to exhaust the remaining type-I budget. This verifies the threshold direction for $H_{0} : P$ against $H_{1} : Q$ .

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

Binary f-, KL-, and Rényi divergences

Example

On $X = {0, 1}$ , take $P = (3 / 4, 1 / 4)$ and $Q = (1 / 4, 3 / 4)$ . Their supports agree and all coordinates are positive.

Facts & Assumptions

Given: The displayed $P, Q$ and the definitions of f-, KL-, and Rényi divergence below.

[F1]

Finite probability vectors: a finite probability vector has nonnegative coordinates summing to one.

[F2]

Finite Kullback–Leibler divergence: $D (P ∥ Q) = \sum_{x : p_{x} > 0} p_{x} \log (p_{x} / q_{x})$ with natural logarithms.

[F3]

Finite Rényi divergence: for $α \neq 1$ , $D_{α} (P ∥ Q) = \log S_{α} (P, Q) / (α - 1)$ with the support conventions stated in its Definition.

[F4]

Finite f-divergence with support conventions: the generator must be finite and convex on $(0, \infty)$ and satisfy $f (1) = 0$ ; its coordinate terms here are $q_{x} f (p_{x} / q_{x})$ .

[F5]

Absolute value in an ordered field and Basic properties of the absolute value: absolute value is real-valued, $∣ c ∣ = c$ for $c \geq 0$ , and $∣ c z ∣ = ∣ c ∣ ∣ z ∣$ .

[F6]

The triangle inequality: $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ .

[F7]

Convex and strictly convex functions on Euclidean convex sets: a function is convex when it lies below each two-point chord.

Verification

Given: $P = (3 / 4, 1 / 4)$ and $Q = (1 / 4, 3 / 4)$ .

1.1F1F2algebra

By [F2], $D (P ∥ Q) = \frac{3}{4} \log 3 + \frac{1}{4} \log (1 / 3) = \frac{1}{2} \log 3$ . The direction is $P$ to $Q$ ; reversing the vectors gives the same value here by symmetry.

1.2F3algebra

For order $1 / 2$ , each affinity term is $\sqrt{p_{x} q_{x}}$ , so $S_{1 / 2} = 2 \sqrt{3 / 16} = \sqrt{3} / 2$ by [F3]. Therefore $D_{1 / 2} = - 2 \log (\sqrt{3} / 2) = \log (4 / 3)$ .

1.3F3algebra

For order $2$ , [F3] gives $S_{2} = (9 / 16) / (1 / 4) + (1 / 16) / (3 / 4) = 9 / 4 + 1 / 12 = 7 / 3$ , so $D_{2} = \log (7 / 3)$ .

1.4F4F5F6F7algebra

Put $f (t) = \frac{1}{2} ∣ t - 1 ∣$ for $t > 0$ . It is finite-valued and $f (1) = 0$ . For $x, y > 0$ and $s \in [0, 1]$ , $(1 - s) x + s y > 0$ , so [F5] and [F6] give $f ((1 - s) x + s y) = \frac{1}{2} ∣ (1 - s) (x - 1) + s (y - 1) ∣ \leq (1 - s) f (x) + s f (y) .$ Thus [F7] proves that $f$ is convex, so it is an admissible generator in [F4].

2.1F1F4step 1.4algebra∎

The generator in [F4] now gives $D_{f} (P ∥ Q) = \frac{1}{2} \sum_{x} ∣ p_{x} - q_{x} ∣ = \frac{1}{2} (1 / 2 + 1 / 2) = 1 / 2$ .

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passOpen item page →

Disjoint supports and singular divergence values

Example

Let $X = {a, b}$ , $P = (1, 0)$ , and $Q = (0, 1)$ . The two laws have disjoint supports.

Facts & Assumptions

Given: The displayed $P, Q$ , the total-variation generator $f (t) = \frac{1}{2} ∣ t - 1 ∣$ , and the KL and Rényi conventions.

[F1]

Finite probability vectors: coordinates may be zero, and $supp P = {x : p_{x} > 0}$ .

[F2]

Finite f-divergence with support conventions: the generator must be finite and convex on $(0, \infty)$ and satisfy $f (1) = 0$ ; a coordinate with $q_{x} > 0 = p_{x}$ contributes $q_{x} f (0)$ , while one with $p_{x} > 0 = q_{x}$ contributes $p_{x} f_{\infty}$ .

[F3]

Finite Kullback–Leibler divergence: any coordinate with $p_{x} > 0 = q_{x}$ makes $D (P ∥ Q) = + \infty$ ; zero $P$ -mass contributes zero.

[F4]

Finite Rényi divergence: below order one a $p_{x} > 0 = q_{x}$ coordinate contributes zero to the affinity; above order one it makes the affinity infinite.

[F5]

Absolute value in an ordered field and Basic properties of the absolute value: absolute value is real-valued, $∣ c ∣ = c$ for $c \geq 0$ , and $∣ c z ∣ = ∣ c ∣ ∣ z ∣$ .

[F6]

The triangle inequality: $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ .

[F7]

Convex and strictly convex functions on Euclidean convex sets: a function is convex when it lies below each two-point chord.

Verification

Given: $P = (1, 0)$ and $Q = (0, 1)$ on ${a, b}$ .

1.1F2F5F6F7algebra

1.2F3algebra

Since $p_{a} = 1 > 0 = q_{a}$ , [F3] gives $D (P ∥ Q) = + \infty$ . This records the singular direction; $D (Q ∥ P)$ is also infinite by the same support calculation with $a, b$ exchanged.

2.1F1F2step 1.1algebra

For $0 < t < 1$ , $f (t) = \frac{1}{2} (1 - t)$ , so $f (0) = \lim_{t ↓ 0} f (t) = 1 / 2$ . For $t > 1$ , $f (t) / t = \frac{1}{2} (1 - 1 / t)$ , so $f_{\infty} = \lim_{t \to \infty} f (t) / t = 1 / 2$ . The coordinate $a$ contributes $p_{a} f_{\infty} = 1 / 2$ by [F2], and coordinate $b$ contributes $q_{b} f (0) = 1 / 2$ . Thus $D_{f} (P ∥ Q) = 1$ .

3.1F3F4step 1.2algebra∎

If $0 < α < 1$ , the only coordinate with $q_{x} > 0$ is $b$ , where $p_{b} = 0$ , so $S_{α} (P, Q) = 0$ and $D_{α} (P ∥ Q) = + \infty$ by [F4] and the definition. If $α > 1$ , coordinate $a$ gives infinite affinity by [F4]. At $α = 1$ , KL is infinite by step 1.2.

F-Divergences, Rényi Information, and Hypothesis Testing — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

A binary Neyman–Pearson boundary randomization

Example

Facts & Assumptions

Verification

Binary f-, KL-, and Rényi divergences

Example

Facts & Assumptions

Verification

Disjoint supports and singular divergence values

Example

Facts & Assumptions

Verification

Sources