Probabilistic graphical models have been successfully applied in a lot of different fields, e.g., medical diagnosis and bio-statistics. Multiple specific extensions have been developed to handle, e.g., time-series data or Gaussian distributed random variables. In the case that handles both Gaussian variables and time-series data, downsides are that the models still have a discrete time-scale, evidence needs to be propagated through the graph and the conditional relationships between the variables are bound to be linear. This paper converts two probabilistic graphical models (the Markov chain and the hidden Markov model) into Gaussian processes by constructing covariance and mean functions, that encode the characteristics of the probabilistic graphical models. Our developed Gaussian process based formalism has the advantage of supporting a continuous time scale, direct inference from any time point to the other without propagation of evidence and flexibility to modify the covariance function if needed.