Conditional event probability and two-event independence.
2026-10-10 · Checked denominator condition and independence formula.
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Positive-probability event conditioning and product-form independence.
This event ratio does not define conditioning on a zero-probability continuous value.
For square-integrable real X,Y, Var X=E[(X−EX)²] and Cov(X,Y)=E[(X−EX)(Y−EY)]. If both variances are positive, Corr=Cov/√(Var X Var Y) lies in [−1,1]. Independence gives zero covariance; its converse generally fails.
If finite X→Y→Z form a Markov chain, meaning Z is conditionally independent of X given Y, then I(X;Z)≤I(X;Y). Processing the observation cannot increase its mutual information about the original variable.
A chain satisfies P(Xₙ₊₁=j|X₀,…,Xₙ)=Pᵢⱼ when Xₙ=i, where P has nonnegative entries and each row sums to one. With row-vector initial law μ, the law after n steps is μPⁿ.
A Bayesian network uses a directed acyclic graph and local conditional distributions to factor a joint law. d-Separation supplies conditional independences implied by its Markov structure; the converse inference from observed independences needs additional faithfulness assumptions. Variable elimination sums latent variables exactly, but cost depends exponentially on induced width in general. The DAG alone does not encode a causal intervention without additional causal assumptions.
Local conditional laws combine into a joint model.
Specific risk concerns an item given its measurement information. Global consumer/producer risk averages erroneous acceptance/rejection over a defined process population and measurement model. Its value depends on the production distribution, sampling and decision rule; a precise instrument or narrow coverage interval alone does not determine it.
Conditional and joint event probabilities have different denominators.
If E₁,…,Eₖ partition the outcome space, total probability sums P(Eᵢ)P(B|Eᵢ) to obtain P(B). Bayes' theorem then gives P(Eᵢ|B)=P(Eᵢ)P(B|Eᵢ)/P(B). Reversing a conditional probability therefore requires the prior event probabilities.
Each conditional term requires a positive conditioning probability.