Marginal Girsanov Reweighting (MGR)

• Marginal Girsanov Reweighting (MGR) is a method that reweights endpoint transitions to obtain unbiased estimates from biased stochastic simulations.
• It reduces the exponential variance growth seen in classical Girsanov Reweighting by marginalizing over intermediate states and recursively composing short-lag weights.
• MGR is applied in molecular dynamics, Bayesian inference, and high-dimensional graphical models to improve effective sample sizes and inference reliability.

Marginal Girsanov Reweighting (MGR) is an advanced statistical methodology for estimating unbiased quantities from biased or perturbed simulations of stochastic processes. MGR was introduced to address the instability and exponential variance explosion inherent to classical Girsanov Reweighting (GR) when reweighting pathwise observables over long time horizons. Opposed to the trajectory-level Radon–Nikodym derivative utilized in classical GR, MGR operates by marginalizing over the intermediate states of a simulation path and performing density ratio estimation only on the endpoint transition pairs. This approach significantly stabilizes variance and enables scalable reweighting for kinetic and inference tasks in high-dimensional and long-timescale settings (Wang et al., 30 Sep 2025).

1. Foundational Principles and Motivation

In classical GR, the probability ratio (Radon–Nikodym derivative) $w^{\mathrm{GR}}(\mathbf{x}_{t:t+\tau})$ between the target and perturbed measures is computed along full trajectories:

$$w^{\mathrm{GR}}(\mathbf{x}_{t:t+\tau}) = \frac{d\mu}{d\tilde{\mu}}(\mathbf{x}_{t:t+\tau})$$

for a path $\mathbf{x}_{t:t+\tau}$ connecting $x_t$ and $x_{t+\tau}$. As the time horizon $\tau$ increases, the variance of $w^{\mathrm{GR}}$ grows exponentially (a consequence of log-weight statistics aggregating stochastic increments), leading to pronounced sample inefficiency and instability.

MGR addresses this by constructing the weight for endpoints $(x_t, x_{t+\tau})$ as a marginal density ratio:

$$w_t(x_t, x_{t+\tau}) = \frac{\rho_t(x_t, x_{t+\tau})}{\tilde{\rho}_t(x_t, x_{t+\tau})}$$

where $\rho_t$ and $\tilde{\rho}_t$ are the joint transition densities under target and perturbed dynamics, respectively. The key insight is that $w_t(x_t, x_{t+\tau})$ may be expressed as the expectation of $w^{\mathrm{GR}}(\mathbf{x}_{t:t+\tau})$ conditional on the endpoints, thus marginalizing over intermediate states:

$$w_t(x_t, x_{t+\tau}) = \mathbb{E}_{\tilde{\mu}}\left[ w^{\mathrm{GR}}(\mathbf{x}_{t:t+\tau}) \mid X_t = x_t, X_{t+\tau} = x_{t+\tau} \right]$$

Consequently, the variance does not grow with $\tau$, and the computation becomes tractable for longer time intervals (Wang et al., 30 Sep 2025).

2. Iterative Ratio Estimation and Neural Implementation

MGR employs an iterative density ratio estimation across increasing lags $k\tau$. Given a simulation at the perturbed (reference) dynamics, the procedure is as follows:

1. Short lag bootstrapping: For small lag $\tau$, the classical GR weight $w^{\mathrm{GR}}_{\tau}$ is well-behaved and directly estimable.
2. Composite weight construction: For longer lags $k\tau$, composite weights $c_t$ for pair $(x_t, x_{t+k\tau})$ are constructed by multiplying previously estimated marginal weights and current short-lag GR weights:

$$c_t = w_{(k-1)\tau}(x_t, x_{t+(k-1)\tau}) \cdot w^{\mathrm{GR}}_\tau(x_{t+(k-1)\tau}, x_{t+k\tau})$$

1. Classifier-based ratio estimation: A binary classifier (neural network) $h_\theta(x_t, x_{t+k\tau})$ is trained using $c_t$ as importance weights, yielding for each endpoint pair:

$$w_{k\tau}(x_t, x_{t+k\tau}) = \frac{h_\theta(x_t, x_{t+k\tau})}{1 - h_\theta(x_t, x_{t+k\tau})}$$

The classifier is trained to distinguish (possibly weighted) samples from the target and reference joint transition distributions.

This recursive construction controls variance and leverages neural ratio estimation, enabling "sequential learning" of marginal weights over long time intervals (Wang et al., 30 Sep 2025).

3. Mathematical Formulation and Pathwise Weights

The GR weight for discretized SDEs is typically computed as

$$\log w^{\mathrm{GR}}_t(\mathbf{x}_{t:t+\tau}) \approx \sum_{k} [f(x^k, t^k) - \tilde{f}(x^k, t^k)]^T g(t^k) \sqrt{\Delta t} \, \xi^k - \frac{\Delta t}{2} \left\| \frac{f(x^k, t^k) - \tilde{f}(x^k, t^k)}{g(t^k)} \right\|^2$$

where the sum runs over time-steps between $t$ and $t+\tau$, $f$ and $\tilde{f}$ are drift terms, $g$ is the diffusion, and $\xi^k$ are the underlying Gaussian increments. MGR sidesteps the direct accumulation of these weights over long trajectories by focusing on the marginalized, endpoint conditional expectation.

The composition property of transition densities ensures that MGR's marginal ratio for lag $k\tau$ may be iteratively obtained from short-lag ratios and GR weights, thus propagating stable weights through sequential "blocks" of the time axis.

4. Practical Applications

MGR has been shown to be effective in several high-impact contexts:

• Molecular Dynamics (MD): In umbrella sampling and other enhanced sampling schemes, biased simulations accelerate rare events but distort kinetic estimates. MGR provides stable weights for kinetic observables, ensuring correct recovery of equilibrium and rate matrices for Markov State Model (MSM) construction. Empirical results in one-dimensional multi-well systems and biomolecules (alanine dipeptide) demonstrate stable eigenfunctions and timescales, with improved effective sample sizes relative to GR (Wang et al., 30 Sep 2025).
• Bayesian Parameter Inference in SDEs: When parameters of a stochastic process are inferred from sparsely observed data, MGR is used to estimate the likelihood ratio between candidate parameter sets without re-simulation. The marginal weights allow efficient calculation of the posterior over Bayesian parameters, for example in Ornstein–Uhlenbeck processes and Lotka–Volterra models. Experiments demonstrate that MGR yields more stable and concentrated posteriors compared to particle marginal Metropolis–Hastings and variational inference baselines.
• Markov Random Fields and High-Dimensional Graphical Models: Papers on interacting SDEs indexed by graphs show that post-Girsanov change of measure, the law of the system can be factorized over cliques, facilitating marginal reweighting strategies amenable to parallel and local computation in large systems (Hu et al., 14 May 2024).

5. Comparison to Classical Girsanov Reweighting

Classical GR directly reweights observables along full trajectories, which is reliable only for short time intervals or strongly overlapping path distributions. The variance of GR weights for long $\tau$ is prohibitively large, with effective sample sizes decaying rapidly. MGR circumvents this by focusing on marginal endpoint distributions:

• For MSM construction, MGR allows lag times $k\tau$ comparable to slow processes while controlling variance.
• For Bayesian inference, likelihood ratios over wide observation gaps can be computed efficiently.

A plausible implication is that MGR is suitable for rare-event simulation, parameter inference in sparsely sampled or expensive-to-simulate models, and reweighting in enhanced sampling contexts where classical GR fails due to numerical instability.

6. Limitations and Sensitivity

MGR remains sensitive to severe drift mismatch: if the reference process is sufficiently different from the target, even short-lag GR weights can become unstable, impeding recursive estimation. Furthermore, MGR assumes that endpoint pairs (reaction coordinates) adequately capture the relevant transitions; poor coordinate choices can lead to ineffective reweighting.

Model choices—such as the neural classifier architecture—affect estimation accuracy, and further exploration of density ratio estimators (e.g., normalizing flows, score matching) is proposed.

7. Outlook and Extensions

The MGR methodology is amenable to further enhancement:

• Extension to more complex neural estimation schemes may improve robustness and accuracy.
• Integration with biased simulation frameworks (umbrella sampling, metadynamics) is natural, potentially supporting online reweighting during enhanced sampling runs.
• Theoretical analysis of error propagation across recursive compositions will inform better scaling and reliability.

Given its variance control, flexibility, and compatibility with modern ML approaches for density ratio estimation, MGR represents a rigorously grounded and scalable strategy for unbiased reweighting in high-dimensional, long-timescale simulations and inference tasks (Wang et al., 30 Sep 2025).

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References

• Marginal Girsanov Reweighting: Stable Variance Reduction via Neural Ratio Estimation (Wang et al., 30 Sep 2025).
• The fundamental martingale with applications to Markov Random Fields (Hu et al., 14 May 2024).
• Girsanov reweighting for path ensembles and Markov state models (Donati et al., 2017).
• A review of Girsanov Reweighting and of Square Root Approximation for building molecular Markov State Models (Donati et al., 2022).
• Girsanov reweighting for simulations of underdamped Langevin dynamics. Theory (Kieninger et al., 2023).