When studying some concentration inequalities, it became relevant to consider the following discrepancy between two probability measures $\pi$ and $\nu$ (treating $\sigma \in \left( 0, \frac{1}{2} \right)$ as a free parameter):
\begin{align} \mathsf{d}_{\sigma} \left(\nu,\pi\right) := \sup \left\{ \nu \left(f\right) - \pi \left(f\right) : \mathrm{osc} \left(f\right) \leq 1, \mathrm{var} \left(f ;\pi \right) \leq \sigma^{2} \right\} \end{align}
i.e. when considering bounded functions $f$ of a given variance under $\pi$, how much can $\nu$ and $\pi$ disagree about their expectations?
By inclusion,
- one can readily see that $\mathsf{d}_{\sigma} (\nu, \pi)$ is upper-bounded by a multiple of the total variation distance (since all of the $f$ which we consider are bounded),
- and also by a multiple of the chi-squared divergence in a suitable direction (since all of the $f$ which we consider have finite variance under $\pi$).
So, some control of $\mathsf{d}_{\sigma}$ is at least possible. Perhaps (hopefully!) more can be said.
My question is then whether this discrepancy functional is already known, well-understood, or otherwise. If there is a more explicit characterisation of the functional (as e.g. an explicit integral rather than as an optimisation problem), that would also be a great resolution.
