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Suppose that $\mu(dx) = \exp(-\psi(x)) \, \mathrm{dx}$ is a probability measure on $\mathbf{R}^d$. For suitable functions $g \geqslant 0$, define $$\text{Ent}(g) = \int \mu(dx) g(x) \log \left( \frac{ …

Let $W: \mathbf{R}^d \to \mathbf{R}$ be a convex function such that $\int \exp(-W) = 1$, and define probability measures $\mu_y$ by $$\mu_y (dx) = \exp( - W (x - y)) \,dx,$$ i.e. each $\mu_y$ is a tra …

Suppose I have a well-behaved, strictly convex function $f : \mathbf{R}^d \to [0, \infty)$, and assume that $f$ has its unique minimiser at $x = 0$, with $f(0) = 0$. For $y > 0$, I define the level …