= Poisson change-point posterior
{c}
{title2=$p(\theta\mid t_1,\ldots,t_n)$}
For an <Inhomogeneous Poisson process> whose known positive intensity changes from $\lambda_1$ to $\lambda_2$ at an unknown time $\theta\in(0,T)$, the ordered arrival-time <likelihood function> is
$$
L(\theta)=\lambda_1^{j(\theta)}\lambda_2^{n-j(\theta)}e^{-\lambda_1\theta-\lambda_2(T-\theta)}.
$$
Under a <uniform prior>, each interval between successive arrivals has an exponential <posterior density> with rate in the exponent $\lambda_2-\lambda_1$. Its integrated weights allow exact interval selection followed by a truncated exponential draw; when the two rates coincide, the <Bayesian posterior> is uniform and the data contain no change-time information.
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