Theoretical guarantees for Monte Carlo loss estimation under change of measure in TPPF

Establish whether Monte Carlo estimators of the loss functions for the Twisted-Path Particle Filter—specifically the KL-divergence losses D_KL(P^φ || P^{φ*}), D_KL(P^{φ*} || P^φ), and their sum—together with their gradients, deteriorate when computed via the untwisted change-of-measure representation (equation (eq:untwisted) in Section 4.2) instead of sampling from the twisted Markov chain; rigorously determine conditions under which these estimators are consistent and quantify any bias or variance introduced by the change of measure.

Background

In Section 4.2 the authors propose training the twisting function for their Twisted-Path Particle Filter (TPPF) using KL-divergence losses between path measures. Because sampling from the twisted transition kernel can be difficult, they provide an alternative "untwisted" change-of-measure expression (equation (eq:untwisted)) to compute the loss and its gradient using samples from the original, untwisted Markov chain.

This practical workaround raises a methodological concern: whether Monte Carlo approximation of the loss and its gradient remains reliable after such a change of measure. The authors explicitly note that they lack a theoretical guarantee for the accuracy or stability of this approximation, even though empirical results are encouraging. Formalizing guarantees would strengthen the theoretical foundation of TPPF and inform its use across models and dimensions.