DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Likelihood densities

Definition

Let $E = ((X, A), Θ, (P_{θ})_{θ \in Θ})$ be dominated by the fixed sigma-finite positive measure $μ$ . For each $θ$ , the Radon–Nikodym theorem gives the almost-everywhere class $d P_{θ} / d μ$ ; a sigma-finite exhaustion for $μ$ is also finite for the probability measure $P_{θ}$ . Under AC, select simultaneously a finite-valued measurable nonnegative representative $p_{θ}$ from each class. The nonnegative version exists by A sigma-finite positive measure has a nonnegative Radon–Nikodym density version.

For fixed observed $x \in X$ , the function $L_{x} : Θ \to [0, + \infty), L_{x} (θ) : = p_{θ} (x),$ is the likelihood at $x$ . This is a function on the parameter labels, not a probability distribution on $Θ$ . No sigma-algebra on $Θ$ or measurability of $L_{x}$ is implied. Each density is determined only $μ$ -almost everywhere; there is no canonical pointwise version and, for arbitrary uncountable $Θ$ , no common exceptional null set is asserted.

Depends on

Used by

Expectation-maximization Algorithm
The QMD route uses normalized absolutely continuous likelihoods and a separate uniform LAN premise Corollary
An external dominating law need not be a valid sufficiency reference Counterexample
Identifiability and KL support do not replace testability Counterexample
A dominated differentiable-in-quadratic-mean submodel Definition
Exact inferential targets and approximation chains Definition
Exponential families and full rank Definition
Families with a monotone likelihood ratio Definition
Ideal and finite-simulation synthetic likelihood Definition
KL projection sets and relative KL neighborhoods Definition
Latent-variable joint and observed-data factorization Definition
Likelihood-ratio equivalence on common discrete support Definition
Likelihood-ratio tests with boundary randomization Definition
Marginal likelihoods, proper model evidence, and Bayes factors Definition
Root-L2 Hellinger distance for dominated laws Definition
Score and Fisher information Definition
Variational family and evidence lower bound Definition
A finite identifiable model with posterior consistency Example
Bernoulli likelihood ratio and change of measure Example
Exponential and Weibull hazards with right-censored exposure Example
Gaussian shift LAN and the regular sample mean Example
A QMD score is centered and has finite information Lemma
Countably partitioned sieves yield posterior contraction by weighted entropy Lemma
KL prior mass gives a posterior denominator lower bound in probability Lemma
KL-neighborhood prior mass lower-bounds the likelihood denominator Lemma
Local KL and log-square mass bounds a non-IID posterior denominator Lemma
Normalized product likelihood expansion under QMD Lemma
Observed-data likelihood under a response model Lemma
QMD controls singular and zero-overlap masses Lemma
Tests and a sieve tail imply posterior contraction from a denominator bound Lemma
The common-support likelihood ratio is measurable Lemma
The likelihood-ratio rule is a measurable test of exact size Lemma
Compact models admit tests at a unique KL projection under fractional-moment continuity Proposition
A countable mixture dominates a sigma-finite dominated family Theorem
A DQM score has mean zero and is unique Theorem
A sufficient summary preserves the Bayesian posterior Theorem
Compact-uniform LAN and rate contraction imply a total-variation BvM limit Theorem
Dirichlet normal-mixture posterior contraction Theorem
Equivalent dominating measures give equivalent likelihood ratios Theorem
Finite-model posterior probabilities and Bayes-factor odds Theorem

…and 17 more results.

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §§2.2 and 3.2, PDF pp. 30–33 and 54–55 (standard reference, not scraped)
Joshua M. Tebbs, STAT 512 Mathematical Statistics Lecture Notes, §9.3, PDF pp. 118–123 (standard reference, not scraped)
Charles J. Geyer, Stat 5421 Lecture Notes: Exponential Families, §3, PDF pp. 3–5 (standard reference, not scraped)