This study presents a novel application of a Probabilistic Bayesian Neural Network (PBNN) for estimating vocal function variables and enhancing non-invasive ambulatory voice monitoring by addressing aleatoric and epistemic uncertainties in regression tasks. The proposed PBNN allows for estimating key physiological parameters including subglottal pressure, vocal fold contact pressure, thyroaritenoid and cricothyroid muscle activations, from seven aerodynamic and acoustic features. The PBNN is trained on the Triangular Body-Cover Model (TBCM) of the vocal folds to produce a non-linear inverse mapping between its inputs and outputs. Furthermore, the selected aerodynamic and acoustic features can be obtained in ambulatory settings, thus enhancing the practical applicability of the proposed method. Transfer Learning is then applied to integrate real voice data into the initially synthetic-trained network to refine subglottal pressure estimations. The confidence intervals generated by the PBNN illustrate its ability to identify uncertain estimations, as the results show correlations between prediction errors and the estimated aleatoric and epistemic uncertainties. This correlation is advantageous because it shows that the network can effectively predict potential inaccuracies in its estimations. Increased uncertainty is mainly observed at operating points where the TBCM is likely to exhibit non-linear behaviors, at higher subglottal pressures. This suggests that the selected input features may not be robust enough for capturing the nonlinear effects in the TBCM. These results highlight the potential for future research to assess the viability of incorporating new features and additional measurements that could better capture non-linear responses.