Bayesian parameter estimation for the SWIFT model of eye-movement control during reading

Process-oriented theories of cognition must be evaluated against time-ordered observations. Here we present a representative example for data assimilation of the SWIFT model, a dynamical model of the control of spatial fixation position and fixation duration during reading. First, we develop and test an approximate likelihood function of the model, which is a combination of a pseudo-marginal spatial likelihood and an approximate temporal likelihood function. Second, we use a Bayesian approach to parameter inference using an adapative Markov chain Monte Carlo procedure. Our results indicate that model parameters can be estimated reliably for individual subjects. We conclude that approximative Bayesian inference represents a considerable step forward for the area of eye-movement modeling, where modelling of individual data on the basis of process-based dynamic models has not been possible before.


Background
Reading is characterized by the successful coordination between key cognitive and motor subsystems, e.g., visual information processing, attention, word recognition, and saccade programming. Even during reading of simple texts, there is considerable stochastic variability in fixation positions and fixation durations (Fig. 1). One motivation for the development of mathematical models of eye-movement control during reading is to explain the observed variability.

The SWIFT Model
The SWIFT (saccade generation with inhibition by foveal targets, Engbert, Nuthmann, Richter, & Kliegl, 2005) is a spatially-extended dynamical system that seeks to explain Der gierige Beamte war bei den Einwohnern sehr unbeliebt. Eye trajectory Figure 1: Sequence of fixations during reading. The eye trajectory (red line) is segmented into alternating periods of stationarity (fixations; dotted lines, number indicates order, durations beneath) and quick repositioning (saccades). The sequence contains refixations (3,5), word skipping (8) and regression (9) to a previous word. saccadic selection by the temporal evolution of an activation field. The lexical processing of each word i in a given sentence is represented by an activation variable a i (t). The target selection probability π n (t) for word n at time t is computed from relative activation. As time evolves, relative activations change to produce a continuous-time process that predicts saccadic selection over time, i.e., where γ is a weighting exponent. Fixation durations can be approximated (at first order) by an uncorrelated random process.
To introduce word difficulty effects, however, we modulate fixation duration by a process called foveal inhibition that delays upcoming saccades to prolong ongoing fixations. A simulated trajectory of the model is shown in Figure 2.

Parameter Estimation The Likelihood Function
For parameter estimation, the likelihood of fixation locations (spatial contribution) and fixation durations (temporal contribution) must be calculated incrementally with respect to all previous events in the fixation sequence. We recently showed that  (Seelig et al., 2019). The contributions of the temporal and spatial parts of the likelihood function are shown in Figure 3, where one parameter was varied and the likelihood for simulated data with known parameters was evaluated.

Results
We implemented a fully Bayesian framework for parameter inference (Schütt et al., 2017) and used an adaptive MCMC procedure, the DREAM framework with improvements (ter Braak & Vrugt, 2008). For parameter estimation, we used eye tracking data of 36 participants who read 150 single sentences each. For every participant 70% of the data were used during the estimation. The remaining 30% were then compared with simulated data sets which were based on point estimates of the obtained posterior parameter distributions. We compared typical measures of fixation durations (contingent on saccade programming) and fixation probabilities (relating to oculomotor behavior and target selection). The comparisons indicate a remarkable agreement of artificial and experimental data.

Conclusion
We studied Bayesian parameter inference for a dynamical cognitive model of eye-movement control during reading. Using an adapative MCMC framework, we were able to estimate model parameters on the level of individual readers. Simulation on a test data set indicate that a high correlation between important measures for experimental and simulated data was obtained.