Acceptance of novelty depends on the receiver's emotional state. This paper proposes a novel mathematical model for predicting emotions elicited by the novelty of an event under different conditions. It models two emotion dimensions, arousal and valence, and considers different uncertainty levels. A state transition from before experiencing an event to afterwards is assumed, and a Bayesian model estimates a posterior distribution as being proportional to the product of a prior distribution and a likelihood function. Our model uses Kullback-Leibler divergence of the posterior from the prior, which we termed information gain, to represent arousal levels because it corresponds to surprise, a high-arousal emotion, upon experiencing a novel event. Based on Berlyne's hedonic function, we formalized valence as a summation of reward and aversion systems that are modeled as sigmoid functions of information gain. We derived information gain as a function of prediction errors (i.e., differences between the mean of the posterior and the peak likelihood), uncertainty (i.e., variance of the prior that is proportional to prior entropy), and noise (i.e., variance of the likelihood function). This functional model predicted an interaction effect of prediction errors and uncertainty on information gain, which we termed the arousal crossover effect. This effect means that the greater the uncertainty, the greater the information gain for a small prediction error. However, for large prediction errors, greater uncertainty means a smaller information gain. To verify this effect, we conducted an experiment with participants who watched short videos in which different percussion instruments were played. We varied uncertainty levels by using familiar and unfamiliar instruments, and we varied prediction error magnitudes by including congruent or incongruent percussive sounds in the videos. Event-related potential P300 amplitudes and subjective reports of surprise in response to the percussive sounds were used as measures of arousal levels, and the findings supported the hypothesized arousal crossover effect. The concordance between our model's predictions and our experimental results suggests that Bayesian information gain can be decomposed into uncertainty and prediction errors and is a valid measure of emotional arousal. Our model's predictions of arousal may help identify positively accepted novelty.

Appropriate levels of arousal potential induce hedonic responses (i.e., emotional valence). However, the relationship between arousal potential and its factors (e.g., novelty, complexity, and uncertainty) have not been formalized. This paper proposes a mathematical model that explains emotional arousal using minimized free energy to represent information content processed in the brain after sensory stimuli are perceived and recognized (i.e., sensory surprisal). This work mathematically demonstrates that sensory surprisal represents the summation of information from novelty and uncertainty, and that the uncertainty converges to perceived complexity with sufficient sampling from a stimulus source. Novelty, uncertainty, and complexity all act as collative properties that form arousal potential. Analysis using a Gaussian generative model shows that the free energy is formed as a quadratic function of prediction errors based on the difference between prior expectation and peak of likelihood. The model predicts two interaction effects on free energy: that between prediction error and prior uncertainty (i.e., prior variance) and that between prediction error and sensory variance. A discussion on the potential of free energy as a mathematical principle is presented to explain emotion initiators. The model provides a general mathematical framework for understanding and predicting the emotions caused by novelty, uncertainty, and complexity. The mathematical model of arousal can help predict acceptable novelty and complexity based on a target population under different uncertainty levels mitigated by prior knowledge and experience.