Amortized Bayesian inference for actigraph time sheet data from mobile devices

1. Problem Statement

The paper addresses the challenge of analyzing high-resolution actigraph data (accelerometer data) collected from wearable devices. While these devices generate massive amounts of data regarding human movement (e.g., magnitude of acceleration, MAG), traditional statistical analysis faces two main hurdles:

• Computational Cost: Exact Bayesian inference for complex hierarchical models (like Dynamic Linear Models) on large, high-frequency time-series datasets is computationally expensive and slow, making it difficult to scale or use for real-time applications.
• Data Sparsity and Irregularity: Raw actigraph data is often sparse, irregular, and contains gaps due to the nature of human movement and device usage. Standard regression models struggle with missing time steps and the need for probabilistic imputation.
• Need for Transfer Learning: There is a growing demand for AI frameworks in health that can perform "amortized inference"—training a model once on a distribution of synthetic data so it can instantly infer parameters for new, unseen datasets without retraining.

The specific goal is to develop a framework that can:

1. Perform probabilistic imputation of missing actigraph time sheets.
2. Learn time-varying impacts of explanatory variables (e.g., age, BMI, GPS location) on physical activity intensity (MAG).
3. Enable fast, reliable inference suitable for health recommender systems.

2. Methodology

The authors propose a framework combining Hierarchical Dynamic Linear Models (DLM) with Amortized Bayesian Inference (ABI) using the BayesFlow architecture.

A. Data Preprocessing and the "Actigraph Timesheet"

• Data Source: The study uses data from the PASTA-LA study (460 individuals in Los Angeles).
• Transformation: Raw 30Hz data is averaged over 20-second intervals. The Magnitude of Acceleration (MAG) is transformed to $\log(\text{MAG})$ to handle non-negativity and allow for negative values in linear models.
• Trajectory Filtering: Trajectories shorter than 5 minutes or longer than 22 minutes are filtered. Gaps are handled by defining "trajectories" as distinct segments separated by >3 minutes.
• Actigraph Timesheet Structure: The data is restructured into a matrix where rows represent (Subject, Date) pairs and columns represent discrete time steps. Missing values (gaps in activity) are treated as unobserved outcomes ($y_{t,u}$) to be imputed.

B. The Statistical Model: Dynamic Linear Model (DLM)

The authors model the log(MAG) using a time-varying regression:
$$y_t = X_t \beta_t + \nu_t, \quad \nu_t \sim N(0, \sigma^2 V_t)$$
$$\beta_t = G_t \beta_{t-1} + \omega_t, \quad \omega_t \sim N(0, \sigma^2 W_t)$$

• $\beta_t$: Time-varying coefficients representing the impact of covariates (e.g., age, distance from home, time of day).
• $\sigma^2$: Variance parameter.
• Priors: A Normal-Gamma prior is placed on $\beta_t$ and $\sigma^2$.
• Benchmark: The authors use the Forward Filter Backward Sampling (FFBS) algorithm to compute exact posterior samples. This serves as the "ground truth" benchmark to evaluate the performance of the amortized method.

C. Amortized Bayesian Inference (ABI) via BayesFlow

Instead of running MCMC or FFBS for every new dataset, the authors train a neural network to approximate the posterior distribution directly.

• Architecture: A Normalizing Flow (specifically a coupling flow) is used as an Invertible Neural Network (INN), denoted as $f_\phi(\theta; y)$.
• Training Process:
  1. Synthetic Data Generation: The DLM is used to generate thousands of synthetic datasets $(\theta^{(m)}, y^{(m)})$ by sampling from the prior and simulating outcomes.
  2. Objective Function: The network is trained to minimize the Kullback-Leibler (KL) divergence between the true posterior and the flow-transformed distribution. The loss function (Equation 6) is:
     $$\mathcal{L} = \frac{1}{M} \sum_{m=1}^M \left( \frac{1}{2} \| f_\phi(\theta^{(m)}; y^{(m)}) \|^2 - \log |\det J_{f_\phi}^{(m)}| \right)$$
  3. Inference: Once trained, for any new observed data $y_{obs}$, the network samples a latent variable $z \sim N(0, I)$ and computes $\theta = f_\phi^{-1}(z; y_{obs})$ to instantly generate posterior samples.
• Handling Time: To manage long time series, the authors split the time horizon into blocks (intervals) and train separate networks for each block, using the posterior of the previous block as the prior for the next (Hierarchical ABI).

D. Imputation Strategy

The framework handles missing data by:

1. Imputing unobserved covariates ($X_{t,u}$) using linear interpolation (for time-based) or spatial averaging (for GPS).
2. Using the trained ABI network to sample from the posterior of parameters $\theta | y_{obs}$.
3. Synthesizing missing outcomes $y_{t,u}$ using the sampled parameters and the model equation.

3. Key Contributions

1. Novel Framework for Wearables: The first application of amortized Bayesian inference specifically tailored for high-resolution actigraph time-series data, enabling rapid analysis of wearable device outputs.
2. Probabilistic Imputation: A robust method for filling gaps in actigraph time sheets while fully propagating uncertainty, rather than using deterministic imputation.
3. Time-Varying Coefficient Learning: Successfully demonstrated the ability to learn how explanatory variables (e.g., time of day, BMI) dynamically influence physical activity intensity over the course of a trajectory.
4. Transfer Learning Capability: Showed that a model trained on synthetic data generated from a DLM can generalize effectively to real-world data and unseen trajectories without retraining.

4. Results

• Accuracy vs. FFBS: In simulations, the ABI method produced posterior estimates (means and credible intervals) that closely matched the exact FFBS algorithm. While ABI intervals were slightly wider (conservative), they captured the true parameter trends effectively.
• Generalization: The trained ABI model successfully predicted synthetic values for trajectories not included in the training set, demonstrating strong generalizability.
• Real Data Application: Applied to the PASTA-LA dataset, the ABI model identified significant time-varying effects. For example, it detected that "Start Time in Day" and "BMI" had significant impacts on activity levels at specific intervals, though some significance patterns differed slightly from the exact FFBS due to the approximation nature of the neural network.
• Computational Efficiency: The training phase (approx. 7.5 hours) was a one-time cost. The inference phase (generating posterior samples for new data) was instantaneous, contrasting sharply with the high computational cost of running FFBS or MCMC for every new subject.

5. Significance and Future Directions

• Scalability: This approach solves the scalability bottleneck in Bayesian analysis of wearable data, making it feasible to deploy in real-time health monitoring systems.
• Health Interventions: By providing a fast, probabilistic model of activity, the framework can power "recommender systems" that suggest optimal exercise paths or intensity targets based on environmental factors and individual history.
• Limitations: The current framework relies on Markovian assumptions (local dependence). The authors note that extending this to non-Markovian systems with long-range dependencies (e.g., seasonal effects spanning weeks) is a future challenge.
• Future Work: The authors plan to explore graph neural networks for spatial-temporal dependencies and investigate supervised training using Bayesian predictive stacking to handle more complex models where exact posterior summaries are unavailable.

In summary, the paper presents a powerful synthesis of hierarchical Bayesian modeling and deep learning (normalizing flows) to create a fast, flexible, and uncertainty-aware tool for analyzing human movement data, bridging the gap between rigorous statistical inference and modern AI scalability.