Simulation-based inference for rapid Bayesian parameter estimation in epidemiological models: a comparison with MCMC

Technical Summary: Simulation-Based Inference for Rapid Bayesian Parameter Estimation in Epidemiological Models

Problem Statement
Mechanistic epidemiological models are essential for forecasting infectious disease dynamics and guiding public health decisions. However, calibrating these models to surveillance data using standard Bayesian inference, specifically Markov chain Monte Carlo (MCMC), is computationally prohibitive for high-dimensional, nonlinear systems. MCMC requires thousands to millions of likelihood evaluations, often taking days or months for detailed analyses. This latency prevents the repeated, near-real-time recalibration necessary when epidemic conditions change rapidly, such as during the COVID-19 pandemic. The authors investigate whether Simulation-Based Inference (SBI), specifically Neural Posterior Estimation (NPE), can serve as a scalable, computationally efficient alternative for Bayesian calibration without sacrificing the fidelity of the posterior distribution.

Methodology
The study utilizes a mechanistic SECIR (Susceptible-Exposed-Infectious-Symptomatic-Severe-Critical-Recovered-Dead) compartmental model implemented in the MEmilio framework to simulate COVID-19 dynamics in Germany during 2020. The model is fitted to Intensive Care Unit (ICU) occupancy data, chosen for its direct relevance to healthcare burden and lower sensitivity to testing variability compared to case counts.

• Inference Targets: The Bayesian model estimates five epidemiological disease progression parameters (transition times and probabilities) and a time-varying effective contact rate represented by a grid of reduction factors ($r_k$).
• Reference Method (MCMC): A differential-evolution Metropolis-Z (DE-MZ) algorithm is used to generate reference posterior distributions. This method runs entirely on CPU hardware and relies on explicit likelihood evaluations using a Student's t-distribution for observation errors.
• Proposed Method (SBI/NPE): The authors employ Amortized Neural Posterior Estimation.
  1. Training: A neural density estimator (Masked Autoregressive Flow, MAF) is trained on a large dataset of simulated parameter-trajectory pairs $(\theta, \hat{X})$. Parameters are sampled from prior distributions, and the SECIR simulator generates synthetic ICU trajectories.
  2. Architecture: A Convolutional Neural Network (CNN) embeds the time-series trajectory into a lower-dimensional feature representation, which conditions the MAF to approximate the posterior $p(\theta | X)$.
  3. Inference: Once trained, the network rapidly generates posterior samples for observed data without further likelihood evaluations.
• Evaluation: The study compares SBI and MCMC across three 31-day inference windows (representing distinct epidemic phases) and one challenging 201-day reconstruction window involving multiple transmission change points. Agreement is quantified using 1-Wasserstein distance and Kullback–Leibler (KL) divergence between marginal posteriors, alongside posterior predictive checks (RMSE and credible intervals).

Key Contributions

1. Comparative Framework: The paper provides a rigorous, head-to-head comparison of SBI and MCMC for a complex, high-dimensional epidemiological model, moving beyond simple predictive accuracy to assess the structural agreement of the inferred posterior distributions.
2. Scalability Demonstration: The authors demonstrate that SBI can reduce inference runtime by approximately one order of magnitude for short windows and more than two orders of magnitude for long, high-dimensional windows compared to CPU-based MCMC.
3. Prior Sensitivity Analysis: The study highlights the critical importance of prior specification in SBI. For the 201-day problem, broad priors on time-varying reduction factors led to implausible epidemic dynamics in the training data, impairing learning. The authors show that tightening prior bounds based on prior predictive checks is essential for successful SBI in high-dimensional settings.
4. Hyperparameter Optimization: The work identifies that optimal SBI configurations (simulation budget and batch size) differ based on the inference horizon and dimensionality, suggesting that hyperparameter selection cannot be blindly transferred between short and long inference windows.

Results

• Posterior Agreement: Across all settings, SBI recovered posterior distributions broadly consistent with the MCMC reference. While SBI marginals were generally smoother and slightly broader (indicating higher uncertainty), the dominant posterior modes and parameter trends were preserved.
  • For 31-day windows, the Wasserstein distance between SBI and MCMC posteriors ranged from 0.63 to 2.47, with symmetric KL divergences generally under 100.
  • For the 201-day window, despite increased dimensionality and uncertainty, SBI preserved the main posterior structure, with a Wasserstein distance of 0.35 and symmetric KL of 72.14.
• Predictive Performance: SBI accurately reproduced observed ICU trajectories. For 31-day windows, RMSE values were comparable to MCMC (e.g., 66.41 vs. 31.8 for the offset 0 window). In the 201-day setting, SBI RMSE was 325.72 compared to MCMC's 23.1; however, both methods captured the dominant temporal dynamics, including the post-first-wave decline and the autumn surge. The higher error in the long window is attributed to the difficulty of reconstructing multiple change points simultaneously.
• Runtime Efficiency:
  • 31-day windows: MCMC required ~1,077 seconds per run. SBI required 60–71 seconds on a single NVIDIA A100 GPU.
  • 201-day window: MCMC required ~19,102 seconds and failed to achieve convergence for all parameters (maximum $\hat{R} = 1.605$). SBI required ~157 seconds and maintained broad agreement with the MCMC posterior structure.
• Convergence: MCMC struggled to converge in the high-dimensional 201-day setting, whereas SBI provided stable posterior estimates in a fraction of the time.

Significance and Claims
The paper claims that SBI with NPE provides a rapid and computationally efficient framework for the Bayesian calibration of mechanistic epidemiological models. The primary significance lies in enabling repeated near-real-time inference and rapid outbreak analysis, which are critical for public health decision-making but are currently bottlenecked by the computational cost of MCMC.

The authors modestly note that while SBI preserves the dominant posterior structure, it may produce broader uncertainty estimates than MCMC, potentially due to the lack of explicit observation noise modeling in the training phase. They emphasize that SBI is an amortized method: its advantages are most pronounced in settings requiring repeated inference under the same model and prior specification, where the initial simulation and training costs are distributed across multiple analyses. The study concludes that SBI is a promising approach for rapid calibration, provided that rigorous prior predictive checks and posterior validation are integrated into the workflow to ensure parameter identifiability and the plausibility of simulated trajectories.