This work has been submitted to the IEEE for possible publication. Copyright may be transfered without notice, after which this version may no longer be accessible. We introduce the Physiologically-informed Gaussian Process (PhGP) model, a novel Bayesian probabilistic approach to integrate and interpret prior physiological knowledge in machine learning models. Existing recognition algorithms often consider either end-end-to models and/or alongside feature extraction techniques. Conversely, our model based on Gaussian Processes (GP) which are grounded on Bayesian statistics proposes a principled and interpretable integration of these techniques for recognition problems in biomedical engineering realm. This paper builds upon but significantly extends our previous conference paper presented at EMBC 2020 [1]. In this paper, we develop a new model that considers both the raw physiological signals and the prior expert knowledge for training GP models. Moreover, unlike our previous paper by relying on the explicit formula of the GP inference equations, we developed an interpretability framework for our proposed recognition model. Finally, we thoroughly validated our new model on two different public datasets. Our paper brings novelties in the field of biomedical engineering, affective computing , machine learning and computer sicence. [1] Ghiasi, S., Patane, A., Greco, A., Laurenti, L., Scilingo, E.P. and Kwiatkowska, M., 2020, July. Gaussian Processes with Physiologically-Inspired Priors for Physical Arousal Recognition. In 2020 42nd Annual International Conference of the IEEE Engineering in Medicine & Biology Society (EMBC) (pp. 54-57). IEEE.