Hierarchical Bayesian parameter estimation for models of decision under uncertainty.

•  Cognitive modeling has long been the key tool in research on decision under uncertainty. Cognitive modeling has the benefits of generating exact model-predictions and providing the opportunity of rigorous model comparison tests. Cognitive modeling is not, however, without problems. This paper focuses on one of these, namely the problem of how to generate reliable estimates of a model´s free parameters.Traditionally, models are fitted to data at the individual level and parameter estimates are retrieved using maximum likelihood (ML) methods. A downside of this procedure is that it, due to the fact that it is susceptible to noise, tends to exaggerate individual differences. An alternative approach is to use hierarchical Bayesian (HB) parameter estimation. The approach is hierarchical because it uses models with several levels. In the simplest case (the case studied in this paper), the value of parameter x for individual i (individual level) is assumed to be sampled from a normally distributed hyper distribution with parameters m and s (the hyper level). The individual x-values and parameters m and s are estimated from individual data simultaneously. Importantly, the HB approach is less susceptible to noise because individual x-values are constrained by the hyper distribution.The present paper: The aim was to explore the benefits of the HB approach. The strategy was to estimate the parameters of cumulative prospect theory (CPT) with both the ML and the HB approach and compare the results. For clarity, CPT has five free parameters. The three that are most important here are α (quantifies the curvature of the value function for gains), β (quantifies the curvature of the value function for losses) and λ (quantifies the amount of loss aversion).Study 1: Study 1 was a parameter recovery study. Synthetic data was created as follows. A deterministic version of CPT, equipped with the parameter estimates from Tversky and Kahneman (1992; Journal of Risk and Uncertainty), generated choice-predictions for the 180 gamble-pairs used in Rieskamp (2008; JEP:LMC). Data sets with choices from 30 synthetic participants were created by adding noise to these predictions. The goal of Study 1 was to explore if the two approaches would be equally good at recovering the underlying parameters. Study 1 provided three key findings. (1) Overall, the medians for the ML-estimates corresponded well with the medians of the HB-estimates. (2) The HB approach was superior at filtering out noise. (3) Regardless of fitting approach, loss aversion was mainly captured by separating α and β so that α < β. As a result, λ was systematically underestimated. When CPT was constrained so that α = β, the underestimation was strongly reduced for ML-estimates and neutralized for HB-estimates.Study 2. CPT was fitted to the behavioral data from Rieskamp (2008). Study 2 replicated the main findings from Study 1. In addition, study 2 provided two finding regarding λ. (1) Neither the median ML-estimate nor the median HB-estimate indicated systematic loss aversion. (2) Extreme ML-estimates for λ can be caused by two very different factors, systematic loss aversion and noise. This was shown by the fact the 6 participants that received the most extreme ML-estimates all received HB-estimates very close to the mean of the hyper distribution for λ.Summary. Parameter λ is the weak-spot of CPT. In particular, it is systematically underestimated if α and β are allowed to vary freely and noisy data tend to attract extreme ML-estimates. On the general level, as it generates both more information and more stable parameter estimates, we conclude that HB parameter estimation has the potential of vitalizing the literature on decisions under uncertainty.

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