Modeling the arousal potential of epistemic emotions using Bayesian information gain: a framework for inquiry cycles driven by free energy fluctuations.

Epistemic emotions, such as curiosity and interest, drive the inquiry process. This study proposes a novel formulation of these emotions using two types of information gain derived from the principle of free energy minimization: Kullback-Leibler divergence (KLD), representing free energy reduction through recognition, and Bayesian surprise (BS), representing free energy reduction via Bayesian updating. Conventional Gaussian models predict an infinite divergence in information gain (KLD and BS) as prediction error increases, which contradicts the known limits of human cognitive resources. The key novelty of this study lies in a simple yet impactful modification: incorporating a uniform distribution into the Gaussian likelihood function to model neural activity under conditions of large prediction error. This modification yields an inverted U-shaped relationship between prediction error and both KLD and BS, producing a finite peak in information gain that better reflects cognitive realism. Based on this convexity, we propose that alternating the maximization of BS and KLD generates an ideal inquiry cycle that fluctuates around an optimal arousal level, with curiosity and interest driving this process. We further analyze how prediction uncertainty (prior variance) and observation uncertainty (likelihood variance) affect the peak of information gain. The results suggest that greater prediction uncertainty (reflecting open-mindedness) and lower observation uncertainty (indicating focused observation) promote higher information gains through broader exploration. This mathematical framework integrates the brain's free energy principle with arousal potential theory, providing a unified explanation of the Wundt curve as an information gain function and proposing an ideal inquiry process driven by epistemic emotions.