I stumbled upon the following quantity and I'm wondering if anyone knows of anywhere it has appeared in the stats literature previously. Here's the setting:
Suppose you will observe data $y\in \mathcal{Y}$ which is distributed according to the density $p(y\,\vert\,\theta)$ for some parameter $\theta$ having a prior density $p(\theta).$ The marginal likelihood of $y$ is $p(y) = \int p(y\,\vert\,\theta)\,p(\theta)\,\text{d}\theta.$ Let $\theta_\text{ml}(y)$ be the maximum likelihood estimator of $\theta$ based on $y.$
The quantity of interest is $$c=\sup_{y\in \mathcal{Y}} \frac{p(y\,\vert\,\theta_\text{ml}(y))}{p(y)}.$$
I know that $c > 1,$ but I don't know much more than that. It certainly appears impossible to compute except perhaps in the simplest settings. I can't say immediately whether or when it is finite. I'm sure that it's also hard to bound above, which is what I'd be most interested in doing.