= Solution
Independence of the experiments gives
$$
\mathcal P_{AB}(\theta,\boldsymbol\alpha,\boldsymbol\beta)
=\frac{L_A(\theta,\boldsymbol\alpha)L_B(\theta,\boldsymbol\beta)
\pi(\theta)\pi(\boldsymbol\alpha)\pi(\boldsymbol\beta)}
{\mathcal Z_{AB}},
$$
where
$$
\mathcal Z_{AB}=\iiint
L_AL_B\pi(\theta)\pi(\boldsymbol\alpha)\pi(\boldsymbol\beta)
\,d\theta\,d\boldsymbol\alpha\,d\boldsymbol\beta.
$$
The marginal posterior is obtained by integrating the displayed joint posterior over both nuisance-parameter vectors.
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