 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, soSet $t_0=0$ 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.