= Solution
For an <Inhomogeneous Poisson process>, the <likelihood function> is the product of the intensities at arrivals times the exponential of minus the integrated intensity. The integrated intensity is $\theta+2(T-\theta)=2T-\theta$. There are $j(\theta)$ arrivals at rate one and $n-j(\theta)$ at rate two, so
$$
\boxed{L(\theta)=e^{\theta-2T}2^{n-j(\theta)}\ \propto\ e^\theta2^{-j(\theta)},\qquad 0<\theta<T.}
$$
Set $t_0=0$ and $t_{n+1}=T$ to include changes before the first or after the last arrival. An arrival exactly at the change has probability zero, so the convention for $j$ at that point does not affect the <Bayesian posterior>.
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