Uniform-interval encoding of conditioning and a binary query ($\mathbb{P}(A,Y=1\mid \theta )=L(\theta )h(\theta )$)

Let a model have parameter prior density $\rho$, observation probability $L(\theta )\in [0,1]$ and conditional binary-query probability $h(\theta )\in [0,1]$ after observation. Draw $\theta \sim \rho$ and an independent uniform random variable $U$. Accept exactly when $U<L(\theta )$, then output whether $U<L(\theta )h(\theta )$. At fixed $\theta$, the accepted true and false intervals have lengths $Lh$ and $L(1-h)$. Integrating against $\rho$ and dividing by the positive evidence proves equality of the joint posterior law of parameters and answer. This can replace many latent choices by one auxiliary coordinate. It preserves that joint law, not unreported latent traces. More generally the bounded factor must be a probability or first be divided by a valid common rejection envelope.