Bayesian Learning in a Nonlinear Multiscale State-Space Model

TL;DR

This work tackles the challenge of inferring hidden states in a nonlinear multiscale system that couples developmental processes within a lifetime to hereditary dynamics across generations. It introduces a Bayesian learning framework that jointly estimates latent trajectories and unknown noise covariances $\Sigma_f$ and $\Sigma_{c,d}$ using a Particle Gibbs with Ancestor Sampling (PGAS) approach, leveraging Gaussian emissions and inverse-Wishart priors. Through simulated experiments, the method demonstrates accurate recovery of both fine- and coarse-scale states and reliable learning of the covariances, with RMSE indicating strong performance across most dimensions and individuals, alongside some variability in select cases. The proposed methodology provides a principled, scalable tool for modeling and inference in complex biological systems exhibiting multiscale feedback, with potential applicability to evolutionary and developmental dynamics where latent states and noise structures are uncertain.

Abstract

The ubiquity of multiscale interactions in complex systems is well-recognized, with development and heredity serving as a prime example of how processes at different temporal scales influence one another. This work introduces a novel multiscale state-space model to explore the dynamic interplay between systems interacting across different time scales, with feedback between each scale. We propose a Bayesian learning framework to estimate unknown states by learning the unknown process noise covariances within this multiscale model. We develop a Particle Gibbs with Ancestor Sampling (PGAS) algorithm for inference and demonstrate through simulations the efficacy of our approach.

Figures (22)

• Figure 1: True vs. estimated fine time scale trajectories at coarse time step $t=11$ for individuals $d=0, d=1, d=2$, and $d=3$.
• Figure 2: True vs. estimated coarse time scale trajectories for individuals $d=0, d=1, d=2$, and $d=3$.
• Figure 3: Trace plots for each coarse scale dimension and each individual $d$. The true process noise covariance matrices are $\Sigma_{c,1}^{\text{true}} = 0.3 \times \mathbf{I}_{M_{\tilde{\textbf{X}}}}, \; \Sigma_{c,2}^{\text{true}} = 0.5 \times \mathbf{I}_{M_{\tilde{\textbf{X}}}}, \; \Sigma_{c,3}^{\text{true}} = 0.7 \times \mathbf{I}_{M_{\tilde{\textbf{X}}}}, \; \Sigma_{c,4}^{\text{true}} = 0.2 \times \mathbf{I}_{M_{\tilde{\textbf{X}}}}$.
• Figure 4: Trace plots for each fine scale dimension. The true covariance is $0.2 \cdot I_{N_{\textbf{x}}}$.
• Figure 5: True vs. estimated fine time scale trajectories at coarse time step $t=1$ for individuals $d=0, d=1, d=2$, and $d=3$.