Let $X_1,\ldots,X_n$ be <independent random variables>, let $f=f(X_1,\ldots,X_n)$, and let each $f_i$ depend on every coordinate except $X_i$. The <modified logarithmic Sobolev inequality> states that, for every real $\lambda$ for which the expectations exist,
$$
\operatorname{Ent}(e^{\lambda f})
\leq\mathbb E\left[e^{\lambda f}\sum_{i=1}^n
\phi\bigl(-\lambda(f-f_i)\bigr)\right],
\qquad
\phi(u)=e^u-u-1.
$$

To prove it, first apply <tensorization of entropy>:
$$
\operatorname{Ent}(e^{\lambda f})
\leq\sum_{i=1}^n\mathbb E\!\left[
\operatorname{Ent}_{X_i}(e^{\lambda f})
\right].
$$
Condition on all coordinates except $X_i$ and use the stated variational formula with the admissible constant $u=e^{\lambda f_i}$. The $i$th conditional entropy is at most
$$
\begin{aligned}
\mathbb E_{X_i}\!\left[
e^{\lambda f}(\lambda f-\lambda f_i)
-(e^{\lambda f}-e^{\lambda f_i})
\right]
&=\mathbb E_{X_i}\!\left[
e^{\lambda f}\phi\bigl(-\lambda(f-f_i)\bigr)
\right].
\end{aligned}
$$
Summing and taking the remaining expectations proves the inequality.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 208 / 3 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-208/3/a/solution