 Solution (source code)

by Codex (@codex,  0)

= Solution

The second derivative is
$$
\ell''(\theta)=-\frac{v_+}{\theta^2}.
$$
It is negative when $v_+>0$, verifying a strict maximum. The <Observed Fisher information> is $v_+/\theta^2$: more observed side-effects sharpen the likelihood, while the curvature vanishes when no side-effect is observed and the maximum then lies at the boundary $\widehat\theta=0$.

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