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Efficient Bayesian Estimation of Dynamic Structural Equation Models via State Space Marginalization

Øystein Sørensen

Abstract

Dynamic structural equation models (DSEMs) combine time-series modeling of within-person processes with hierarchical modeling of between-person differences and differences between timepoints, and have become very popular for the analysis of intensive longitudinal data in the social sciences. An important computational bottleneck has, however, still not been resolved: whenever the underlying process is assumed to be latent and measured by one or more indicators per timepoint, currently published algorithms rely on inefficient brute-force Markov chain Monte Carlo sampling which scales poorly as the number of timepoints and participants increases and results in highly correlated samples. The main result of this paper shows that the within-level part of any DSEM can be reformulated as a linear Gaussian state space model. Consequently, the latent states can be analytically marginalized using a Kalman filter, allowing for highly efficient estimation via Hamiltonian Monte Carlo. This makes estimation of DSEMs computationally tractable for much larger datasets -- both in terms of timepoints and participants -- than what has been previously possible. We demonstrate the proposed algorithm in several simulation experiments, showing it can be orders of magnitude more efficient than standard Metropolis-within-Gibbs approaches.

Efficient Bayesian Estimation of Dynamic Structural Equation Models via State Space Marginalization

Abstract

Dynamic structural equation models (DSEMs) combine time-series modeling of within-person processes with hierarchical modeling of between-person differences and differences between timepoints, and have become very popular for the analysis of intensive longitudinal data in the social sciences. An important computational bottleneck has, however, still not been resolved: whenever the underlying process is assumed to be latent and measured by one or more indicators per timepoint, currently published algorithms rely on inefficient brute-force Markov chain Monte Carlo sampling which scales poorly as the number of timepoints and participants increases and results in highly correlated samples. The main result of this paper shows that the within-level part of any DSEM can be reformulated as a linear Gaussian state space model. Consequently, the latent states can be analytically marginalized using a Kalman filter, allowing for highly efficient estimation via Hamiltonian Monte Carlo. This makes estimation of DSEMs computationally tractable for much larger datasets -- both in terms of timepoints and participants -- than what has been previously possible. We demonstrate the proposed algorithm in several simulation experiments, showing it can be orders of magnitude more efficient than standard Metropolis-within-Gibbs approaches.
Paper Structure (18 sections, 1 theorem, 34 equations, 5 figures, 2 tables)

This paper contains 18 sections, 1 theorem, 34 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Dynamic Structural Equation Models
  4. State Space Models and Kalman Filters
  5. A State Space Formulation of DSEM
  6. Augmented State Space Formulation
  7. Bayesian Estimation via Marginalization
  8. Simulation Experiments
  9. Multilevel Scalar Latent AR(1) Model
  10. Multilevel One-Factor Multi-Indicator AR(1) Model
  11. Multilevel Trivariate VAR(1) Model
  12. Multilevel Latent AR(p) Model
  13. Cross-Classified VAR(1) Model
  14. Sensitivity Analyses
  15. Discussion
  16. ...and 3 more sections

Key Result

Theorem 1

For maximum lag $L \ge 1$, the within-level model eq:WithinLevelModel is exactly equivalent to the state space model where the augmented state vector is $\tilde{\bm{\eta}}_{1,it} = [\bm{\eta}_{1,i,t}^{T}, \dots, \bm{\eta}_{1,i,t-L+1},\bm{y}_{1,i,t}, \dots, \bm{y}_{1,i,t-L+1}]^{T}$ and the measurement matrix extracts the contemporaneous observation, $\bm{Z}_{it} = [\bm{0}_{U \times L V_1} ~ \bm{I}

Figures (5)

  • Figure 1: Box plots of efficiency for the three algorithms compared for the multilevel scalar latent AR(1) model
  • Figure 2: Histograms of efficiency for each parameter of interest across 100 Monte Carlo simulations in the multilevel one-factor three-indicator latent AR(1) model
  • Figure 3: The left box plot shows efficiency plotted against indicators per latent variable and the right plot shows $\hat{R}$ plotted against indicators per latent variable for the trivariate VAR(1) model
  • Figure 4: The left box plot shows efficiency plotted against lag, and the right plot shows $\hat{R}$ plotted against lag for the AR(p) model
  • Figure 5: Efficiency measured in bulk-ESS per minute (left) and tail-ESS per minute (right) for the cross-classified VAR(1) model

Theorems & Definitions (6)

  • Definition 1: Lag Operator
  • Definition 2: Polynomial Matrices
  • Definition 3: Coefficient Extraction
  • Theorem 1
  • proof
  • Definition 4: Strictly Lagged Polynomial Matrices