= Solution
Each observation has <uniform distribution> density $\theta^{-1}\mathbf1_{\{0<y_j<\theta\}}$. <Independence> makes the <likelihood function> their product. For positive observations the support reduces to $\theta>M$, giving
$$
\boxed{L(\theta;\mathbf y)=\theta^{-n}\mathbf1_{\{\theta>M\}},
\qquad M=\max_jy_j.}
$$
For data outside the positive orthant the likelihood is zero. Changing the convention at the endpoint does not change a continuous <posterior distribution>.
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