Gaussian Process Amplitude Modulation (GPAM) is a probabilistic model that assigns Gaussian Process priors to the modulator and the carrier and allows us to solve the amplitude demodulation (AD) problem by using inference methods in probability theory. Inference in GPAM results in Gaussian Process Probabilistic Amplitude Demodulation (GP-PAD). However, the mostly used inference technique for GP-PAD is maximum a posteriori (MAP), a point estimate method that is not entirely representative of Bayesian methods in general. In this paper, we provide a full Bayesian inference approach to GP-PAD model. More specifically, we represent the GP-PAD model as a factor graph and use message-passing rules, namely Belief Propagation (BP) and Expectation Propagation (EP), to infer the marginal posteriors of the modulator and the carrier. Furthermore, we employ the Kalman smoothing solution to temporal GP regression models to achieve fast inference for GP models. We compare our approach to the baseline, popular demodulation methods in synthetic and real data experiments. The result shows that our method outperforms the baseline methods and converges