Estimating the Trial-by-Trial Learning Curve in Perceptual Learning with Hierarchical Bayesian Modeling

The learning curve serves as a crucial metric for assessing human performance in perceptual learning. It may encompass various component processes, including general learning, between-session forgetting or consolidation, and within-session rapid relearning and adaptation or deterioration. Typically, empirical learning curves are constructed by aggregating tens or hundreds of trials of data in blocks or sessions. Here, we devised three inference procedures for estimating the trial-by-trial learning curve based on the multi-component functional form identified in Zhao et al. (submitted): general learning, between-session forgetting, and within-session rapid relearning and adaptation. These procedures include a Bayesian inference procedure (BIP) estimating the posterior distribution of parameters for each learner independently, and two hierarchical Bayesian models (HBMv and HBMc) computing the joint posterior distribution of parameters and hyperparameters at the population, subject, and test levels. The HBMv and HBMc incorporate variance and covariance hyperparameters, respectively, between and within subjects. We applied these procedures to data from two studies investigating the interaction between feedback and training accuracy in Gabor orientation identification across about 2000 trials spanning six sessions (Liu et al., 2010, 2012) and estimated the trial-by-trial learning curves at both the subject and population levels. The HBMc generated best fits to the data and the smallest half width of 68.2% credible interval of the learning curves compared to the BIP and HBMv. The parametric HBMc with the multi-component functional form provides a general framework for trial-by-trial analysis of the component processes in perceptual learning and for predicting the learning curve in unmeasured time points.