Suppose I am given

- A probability distribution on $ \mathbf R^d$ , with density $ \pi (x)$ .
- A family of transition kernels $ \{ q^0 (x \to \cdot) \}_{x \in \mathbf R^d}$ on $ \mathbf R^d$ , with densities $ q^0 (x \to y)$ .

I now want to find a *new* family of transition kernels $ \{ q^1 (x \to \cdot) \}_{x \in \mathbf R^d}$ such that

- $ q^1$ is in detailed balance with respect to $ \pi$ , i.e.

\begin{align} \pi (x) q^1 ( x \to y ) = \pi (y) q^1 ( y \to x) \end{align}

- $ q^1$ are as close as possible to $ q^0$ in KL divergence, averaged across $ x \sim \pi$ .

To this end, I introduce the following functional

\begin{align} \mathbb{F} [ q^1 ] &= \int_{x \in \mathbf R^d} \pi (x) \cdot KL \left( q^1 (x \to \cdot ) || q^0 (x \to \cdot ) \right) dx\ &= \int_{x \in \mathbf R^d} \int_{y \in \mathbf R^d} \pi (x) \cdot q^1 (x \to y) \log \frac{ q^1 (x \to y) }{ q^0 (x \to y) } \, dx dy \ &= \int_{x \in \mathbf R^d} \int_{y \in \mathbf R^d} \pi (x) \cdot \left\{ q^1 (x \to y) \log \frac{ q^1 (x \to y) }{ q^0 (x \to y) } – q^1 (x \to y) + q^0 (x \to y) \right \} \, dx dy. \end{align}

I thus wish to minimise $ \mathbb{F}$ over all transition kernels $ \{ q^1 (x \to \cdot) \}_{x \in \mathbf R^d}$ satisfying the desired detailed balance condition. Implicitly, there are also the constraints that the $ q^1$ are nonnegative and normalised.

I would like to solve this minimisation problem, or at least derive e.g. an Euler-Lagrange equation for it.

Currently, I can show that the first variation of $ \mathbb{F}$ is given by

\begin{align} \left( \frac{d}{dt} \vert_{t = 0} \right) \mathbb{F} [q^1 + t h] = \int_{x \in \mathbf R^d} \int_{y \in \mathbf R^d} \pi (x) \cdot \log \frac{ q^1 (x \to y) }{ q^0 (x \to y) } \cdot h (x, y) \, dx dy. \end{align}

Moreover, my constraints stipulate that any admissible variation $ h$ must satisfy the following two conditions:

- $ \forall x, y, \quad \pi (x) h (x, y) = \pi (y) h (y, x)$
- $ \forall x, \quad \int_{y \in \mathbf R^d} h (x, y) dy = 0$

I have not been able to translate these conditions into an Euler-Lagrange equation. I acknowledge that since the functional involves no derivatives of $ q^1$ , the calculus of variations approach may be ill-suited. If readers are able to recommend alternative approaches, this would also be appreciated. Anything which would allow for a more concrete characterisation of the optimal $ q^1$ would be ideal.