Non-parametric Bayesian Inference of Drift and Diffusion Functions for Diffusion Processes Using Data from Stochastic Trajectories

Core Concepts

This paper presents a scalable computational framework for inferring the drift and diffusion functions of diffusion processes from trajectory data using non-parametric Bayesian inference and partial differential equation models.

Abstract

• Bibliographic Information: Kruse, M., & Krumscheid, S. (2024). Non-parametric Inference for Diffusion Processes: A Computational Approach via Bayesian Inversion for PDEs. arXiv preprint arXiv:2411.02324.

• Research Objective: The paper aims to develop a robust and scalable method for inferring the unknown drift and diffusion functions of diffusion processes from potentially noisy trajectory data.

• Methodology: The authors formulate the problem within a Bayesian framework, utilizing the Kolmogorov forward and backward equations to link the unknown parameters (drift and diffusion functions) to observable data (probability density functions or mean first passage times). They employ a Gaussian prior measure for the parameters and incorporate data uncertainty through a Gaussian likelihood function. To solve the resulting Bayesian inverse problem, the authors utilize a combination of optimization and sampling techniques specifically designed for large-scale problems. They determine the maximum a posteriori (MAP) estimate using an inexact Newton-CG method and construct a Laplace approximation of the posterior distribution. Additionally, they employ a dimension-independent Metropolis-Hastings algorithm, specifically the Metropolis-Adjusted Langevin (MALA) sampler, to draw samples from the posterior.

• Key Findings: The authors demonstrate the effectiveness of their approach through numerical experiments involving simulated trajectory data for both single-scale and multi-scale diffusion processes. Their results show that the method can successfully recover the true drift and diffusion functions, even in the presence of noise. Notably, the method can also infer the effective dynamics of a multi-scale process from data generated with a relatively large time-scale separation parameter.

• Main Conclusions: The paper presents a powerful and versatile framework for non-parametric Bayesian inference of diffusion processes. The use of PDE models and scalable computational techniques makes the approach suitable for complex systems with high-dimensional parameter spaces.

• Significance: This research contributes significantly to the field of statistical inference for stochastic processes. The proposed method addresses the challenges of non-parametric inference for diffusion processes, providing a robust and scalable solution for analyzing trajectory data and uncovering the underlying dynamics of complex systems.

• Limitations and Future Research: The authors acknowledge the need for a substantial amount of trajectory data to obtain accurate results, limiting the applicability to scenarios with abundant data. Future research could explore methods for reducing data requirements and extending the framework to non-Markovian processes and PDEs with convolutions.

Stats

The authors use an ensemble of 10^3 trajectories at 51 initial sites each for the single-scale MFPT inference, performing a total of 500,001 steps with a step size of 10^-4.
For the single-scale FP inference, they generate an ensemble of 10^5 trajectories, with initial states distributed as N(0, 1/2), and perform 1,001 EM steps of size 1e-4.
In the multi-scale case, they generate 10^5 trajectories, integrating for 50,001 steps with a step size of 10^-3, and use a time-scale separation parameter of ϵ=0.1.

Quotes

Key Insights Distilled From

Non-parametric Inference for Diffusion Processes: A Computational Approach via Bayesian Inversion for PDEs

by Maximilian K... at arxiv.org 11-05-2024

https://arxiv.org/pdf/2411.02324.pdf

Deeper Inquiries

How could this Bayesian inference framework be extended to incorporate data from multiple sources, such as combining trajectory data with time-series measurements of other system observables?

This Bayesian inference framework can be effectively extended to incorporate data from multiple sources by leveraging the flexibility of the likelihood function. Here's a breakdown of how it can be achieved:

Multiple Likelihoods: Instead of a single likelihood function, define separate likelihood functions for each data source. For example, 𝜋_like_traj for trajectory data based on the Fokker-Planck or MFPT equations, and 𝜋_like_ts for time-series measurements of other observables.

Joint Likelihood: Construct a joint likelihood function by assuming (conditional) independence of the data sources given the parameters m. This is often a reasonable assumption if the measurement noise in each data source is independent. The joint likelihood is then the product of the individual likelihoods:
𝜋_like(𝒚_traj, 𝒚_ts | 𝒎) = 𝜋_like_traj(𝒚_traj | 𝒎) * 𝜋_like_ts(𝒚_ts | 𝒎)

Here, 𝒚_traj represents the trajectory data and 𝒚_ts represents the time-series data.

Formulate the Posterior: The posterior is then proportional to the product of the joint likelihood and the prior:
𝑑𝜇_post(𝒎 | 𝒚_traj, 𝒚_ts) ∝ 𝜋_like(𝒚_traj, 𝒚_ts | 𝒎) * 𝑑𝜇_prior(𝒎)

Computational Considerations: The solution methods, such as the inexact Newton-CG for MAP estimation and dimension-independent MCMC, can be adapted to handle the joint likelihood. The gradient and Hessian computations would involve contributions from all likelihood terms.

Benefits of Multi-Source Inference:

Improved Parameter Estimates: Combining information from different sources can significantly reduce uncertainty in the inferred parameters, leading to more robust and reliable estimates for the drift and diffusion functions.
Enhanced Model Validation: Agreement between the posterior predictive and multiple data sources provides stronger evidence for the validity of the inferred model.
Deeper System Understanding: Incorporating diverse data can reveal complex relationships and dependencies within the system that might not be apparent from a single data source.

While the Bayesian framework offers a principled way to incorporate prior information, could an overly informative prior potentially bias the inference towards the prior belief and mask important features in the data?

Yes, an overly informative prior can indeed bias the inference and potentially mask important features present in the data. This is a general concern in Bayesian inference, often referred to as prior dominance.
Here's how it can happen:

Strong Prior Influence: If the prior assigns very low probability to certain regions of the parameter space, the posterior will also tend to have low probability in those regions, even if the data provide evidence to the contrary.
Suppressed Data Features: When the prior strongly favors specific parameter values or shapes of the drift and diffusion functions, it can overshadow subtle but important features in the data that might suggest deviations from the prior belief.
Mitigating Prior Dominance:

Prior Sensitivity Analysis: Perform a sensitivity analysis by varying the prior parameters (e.g., mean, covariance) and observing the impact on the posterior. This helps assess the influence of the prior on the inference results.
Weakly Informative Priors: If possible, use weakly informative priors that encode general knowledge about the parameters without being overly restrictive. For example, instead of a narrow Gaussian prior, consider a wider Gaussian or a less informative prior like a Student-t distribution.
Data-Driven Prior Selection: If sufficient data are available, explore hierarchical Bayesian modeling, where hyperparameters governing the prior distribution are themselves learned from the data. This allows the data to inform the prior to some extent.
Prior Relaxation: If there's a strong suspicion that the prior might be biasing the inference, consider relaxing the prior during the inference process. This can be done by gradually decreasing the prior's influence or by using annealing techniques in MCMC sampling.
Balancing Act:
Finding the right balance between prior information and data is crucial in Bayesian inference. While the prior should guide the inference in the absence of strong data evidence, it's essential to ensure that the prior does not overwhelm the information contained in the data.

How can the insights gained from inferring the drift and diffusion functions of complex systems be used to design more efficient control strategies or predict the system's long-term behavior?

Inferring the drift and diffusion functions provides a probabilistic model of the system's dynamics, which can be invaluable for designing control strategies and predicting long-term behavior:
Control Strategies:

Optimal Control: The inferred drift and diffusion functions can be used to formulate and solve optimal control problems. By defining a cost function that penalizes undesirable states and control efforts, techniques like dynamic programming or stochastic optimal control can determine the control inputs that steer the system towards desired states while minimizing costs.

Feedback Control: Knowledge of the drift and diffusion functions enables the design of feedback control laws. These laws continuously adjust the control inputs based on the system's current state, as predicted by the inferred model, to drive it towards a target state or trajectory.

Robust Control: The uncertainty quantification provided by the Bayesian framework (e.g., posterior variance of the drift and diffusion functions) can be incorporated into robust control design. This allows for the development of controllers that maintain performance even in the presence of uncertainties in the system dynamics.

Long-Term Behavior Prediction:

Forward Simulation: With the inferred drift and diffusion functions, one can perform forward simulations of the stochastic differential equation (SDE) using techniques like the Euler-Maruyama method. This allows for predicting the system's likely future states and their associated probabilities.

Stationary Distribution: For ergodic systems, the inferred drift and diffusion functions can be used to characterize the system's stationary distribution, which describes the long-term probabilities of the system occupying different states.

First Passage Time Analysis: The inferred model enables the computation of Mean First Passage Times (MFPTs), which estimate the average time it takes for the system to reach a particular target state. This is valuable for understanding long-term transition behaviors.

Example Applications:

Molecular Dynamics: Inferred models can guide the design of molecules with desired properties by predicting their dynamics and interactions.
Finance: Inferred models of financial markets can inform trading strategies and risk management by predicting asset price movements.
Climate Science: Inferred models of climate systems can improve long-term climate projections and inform mitigation strategies.