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Suppose

$q^*(\theta) = \underset{q \in \mathcal{Q}}{argmin} \,\, KL[q(\theta)||p(\theta)] = \min_{q\in\mathcal{Q}} \int_{\theta\in\Theta}q(\theta)\ln\Big( \frac{q(\theta)}{p(\theta)} \Big)d\theta$

is an approximation to $p(\theta)$ where $Q$ is a family of densities. Does a solution imply boundedness of $q^*/p$ (if such a $q^*$ is allowed within the family $Q$)?

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