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Bernoulli likelihood ratio and change of measure

Example

Let $X = {0, 1}$ , $A = 2^{X}$ , and let $μ$ be counting measure. Set $Θ = {0, 1}$ and define the experiment's probability measures by

P_{0} ({1}) = \frac{1}{3}, P_{0} ({0}) = \frac{2}{3}, P_{1} ({1}) = \frac{2}{3}, P_{1} ({0}) = \frac{1}{3} .

Both are dominated by $μ$ . Their likelihood-ratio density is $r (0) = 1 / 2$ and $r (1) = 2$ .

Facts & Assumptions

Given: AC (for the linked general theorem), the two-point measurable space, counting measure, and probability mass assignments in the example.

[F1]

Sigma-algebras: a sigma-algebra is an algebra of subsets closed under countable unions; the full power set of ${0, 1}$ is such an algebra.

[F2]

Measurable spaces and measurable sets: a measurable space is a set with a sigma-algebra.

[F3]

Counting measure on an arbitrary set, Counting measure is a measure, and Finite, sigma-finite, and semifinite measures: counting measure assigns mass $1$ to each singleton and is a measure; here its total mass is $2$ , so it is finite and sigma-finite.

[F4]

Probability measures and probability spaces: a measure of total mass $1$ is a probability measure.

[F5]

Statistical experiments: an experiment is a parameter-indexed family of probability measures on one measurable sample space.

[F6]

Dominated statistical experiments: domination means each sampling law is absolutely continuous with respect to the supplied sigma-finite positive measure.

[F7]

Likelihood densities: for a fixed observation $x$ , likelihood is the selected density value $p_{θ} (x)$ .

[F8]

Likelihood ratio as a Radon–Nikodym derivative: for $P_{1} ≪ P_{0}$ , the full likelihood-ratio class is $d P_{1} / d P_{0}$ .

[F9]

Likelihood-ratio change of measure: when $P_{1} ≪ P_{0}$ , a nonnegative likelihood-ratio representative satisfies $E_{P_{0}} [g r] = E_{P_{1}} [g]$ for every nonnegative measurable $g$ and every $P_{1}$ -integrable real $g$ .

[A1]

The Axiom of Choice: AC states that every family of nonempty sets has a choice function; the finite example itself uses explicit mass functions.

Verification

Given: $X = {0, 1}$ , the four masses above, and counting measure $μ$ .

1.1F1F2F3F4F5F6givenalgebra

The full power set is a sigma-algebra by [F1], so [F2] gives a measurable space. The counting measure is finite with $μ (X) = 2$ by [F3], hence sigma-finite. Each displayed mass is nonnegative and the two masses for each $P_{i}$ sum to $1$ . For any disjoint sequence of events, each point belongs to at most one event, so the mass formula is countably additive; each $P_{i}$ is therefore a probability measure by [F4]. Since counting measure assigns positive mass to every nonempty subset, $P_{i} ≪ μ$ , and the family is a dominated experiment by [F5]–[F6].

1.2A1F7F8givenalgebra

Put $p_{i} (x) : = P_{i} ({x})$ for $i \in {0, 1}$ . For every $A \subseteq X$ , $\int_{A} p_{i} d μ = \sum_{x \in A} p_{i} (x) = P_{i} (A)$ , so these are density representatives as in [F7]. Since $p_{0}$ is positive at both points, $P_{1} ≪ P_{0}$ ; define $r (x) = p_{1} (x) / p_{0} (x)$ . For every $A \subseteq X$ , $\int_{A} r d P_{0} = \sum_{x \in A} r (x) p_{0} (x) = \sum_{x \in A} p_{1} (x) = P_{1} (A)$ ; thus $r$ represents the likelihood-ratio derivative [F8], with $r (0) = (1 / 3) / (2 / 3) = 1 / 2$ and $r (1) = (2 / 3) / (1 / 3) = 2$ .

2.1A1F9givenalgebra∎

For any real-valued $g$ on $X$ , $E_{P_{0}} [g r] = \frac{2}{3} g (0) \frac{1}{2} + \frac{1}{3} g (1) 2 = \frac{1}{3} g (0) + \frac{2}{3} g (1) = E_{P_{1}} [g]$ . Thus the change-of-measure identity holds directly, in agreement with [A1, F9].

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