DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Conditional law given a random element

Definition

Let $X : (Ω, F, P) \to (E, S)$ and $Y : (Ω, F, P) \to (T, T)$ be measurable random elements. A conditional law of X given Y is a probability kernel $K : (T, T) \to (E, S)$ such that $ω \mapsto K (Y (ω), \cdot)$ is a regular conditional distribution of X given $σ (Y)$ in Regular conditional distribution. Thus for every $A \in S$ and $H \in σ (Y)$ , $\int_{H} K (Y (ω), A) d P = P (H \cap {X \in A}) .$

Composition with Y makes each evaluation $σ (Y)$ -measurable, and every section remains a probability by Measure kernel and probability kernel. The notation $P (X \in A ∣ Y = y) = K (y, A)$ refers to a chosen kernel; it is not a ratio involving the possibly zero probability $P (Y = y)$ .

Write $P_{Y} (B) = P (Y^{- 1} (B))$ for the law in Law or distribution of a random element. If a specified countable family $C \subseteq S$ determines probability measures, two conditional laws K,L agree as measures outside a single $T$ -measurable $P_{Y}$ -null set. Indeed for each $A \in C$ the measurable discrepancy $D_{A} = {y : K (y, A) \neq L (y, A)}$ has null inverse image under Y by Conditional expectation is unique almost surely. Hence $P_{Y} (D_{A}) = 0$ by the law definition; the countable union $D$ is null by Finite and countable subadditivity of measures, and off D the determining property gives equality of measures. This uses a supplied determining family and makes no AC assertion about obtaining one.

Values on a measurable $P_{Y}$ -null subset may be replaced by a specified fixed probability on E without changing these identities. The inverse image of that subset is measurable null, and all event evaluations are bounded, so the modified event integrals agree. Empty-event evaluations remain zero and whole-target evaluations remain one. Existence on standard-Borel spaces is proved separately; the definition alone does not assert a conditional kernel exists.

Depends on

Used by

Conditional flow matching for inverse posteriors Algorithm
Positive covariance does not imply PRDS Counterexample
Assumptions for identifying natural effects Definition
Bayesian posterior as a regular conditional law Definition
Conditional mutual information by a same-marginal product law Definition
Event-time observation, censoring, and truncation schemes Definition
Filtering, predictive, and smoothing laws in a finite-horizon state-space model Definition
Finite-alphabet conditional entropy given an observation Definition
Location-shift pattern-mixture sensitivity function Definition
MCAR, MAR, and MNAR relative to an observation pattern Definition
Misfit evidence and the potential posterior law Definition
Poisson and Cox point processes Definition
Positive regression dependence on true-null coordinates (PRDS) Definition
Randomized conditional upper-tail p-values Definition
Sufficient statistics Definition
Temporal point-process models and conditional intensity Definition
A posterior mean can lie outside a nonconvex signal class Example
A proper Gaussian likelihood gives a proper posterior Example
A zero-evidence fiber has no normalized fiberwise update Example
An exact posterior sample need not minimize point risk Example
BY and BH thresholds for independent valid p-values Example
Poisson and Cox counts with the same mean on a finite window Example
A PRDS bound for each BH true-null contribution Lemma
Independent noise extends and pushes forward a conditional law Lemma
Population regression and exact-flow meaning of conditional flow matching Lemma
Regular conditional kernels factor through a standard borel conditioning variable Lemma
The same-marginal conditional-product law is well-defined Lemma
The posterior mean is the unique squared-loss Bayes action in a Hilbert target Proposition
A sufficient summary preserves the Bayesian posterior Theorem
Campbell, Laplace, Mecke, and Cox moment formulas Theorem
Dirichlet-process posterior update Theorem
Discrete-treatment potential-outcome means are identifiable Theorem
Filtering and finite-horizon smoothing laws exist Theorem
Finite-alphabet conditional entropy is version-invariant and obeys its chain rule Theorem
Likelihood contributions under censoring, truncation, and delayed entry Theorem
Randomized conditional p-values are uniform Theorem
Standard-Borel Bayesian posterior kernel exists Theorem
Sufficiency makes the conditional rejection probability nuisance-free Theorem
The finite-horizon linear-Gaussian Kalman filter is exact Theorem
The natural-effect mediation formula under independent errors Theorem

Dependency tree · two levels

16 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

Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)
Varadhan, Probability Theory, Chapter 4 (standard reference, not scraped)