SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\inf  _ {p} \mathbb {E} _ {x \sim p (x)} \left[ f (x) + \ln \frac {p (x)}{p _ {0} (x)} \right] \geq - \ln \mathbb {E} _ {x \sim p _ {0}} \exp (- f (x)).
$$
Proof of Lemma G.3. Define the density function
$$
q (x) = \frac {p _ {0} (x) \exp (- f (x))}{Z (f)} \text {w h e r e} Z (f) := \mathbb {E} _ {x \sim p _ {0} (x)} \exp (- f (x)).
$$
$$
\begin{array}{l} \mathbb {E} _ {x \sim p (x)} \left[ f (x) + \ln \frac {p (x)}{p _ {0} (x)} \right] = \mathbb {E} _ {x \sim p (x)} \ln \frac {p (x)}{q (x)} - \ln Z (f) \\ = K L [ p \| q ] - \ln Z (f) \\ \geq - \ln Z (f). \\ \end{array}
$$