Summary: Bayesian estimation of the unknown parameters of a non-homogeneous Gaussian hidden Markov model is described. The hidden Markov chain presents time-varying transition probabilities, depending on exogenous variables through a logistic function. Bayesian model choice is also proposed to select the unknown number of states of the hidden non-homogeneous Markov chain. Both the analyses are developed by using Markov chain Monte Carlo algorithms. Model selection and parameter estimation are performed after making the model identifiable, by selecting suitable constraints through a data-driven procedure. The methodology is illustrated by an empirical analysis of ozone data.