In this paper, we provide a Doob-style consistency theorem for stationary models. Many applications involving Bayesian inference deal with non independent and identically distributed data, in particular, with stationary data. However, for such models, there is still a theoretical gap to be filled regarding the asymptotic properties of Bayesian procedures. The primary goal to be achieved is establishing consistency of the sequence of posterior distributions. Here we provide an answer to the problem. Bayesian methods have recently gained growing popularity in economic modeling, thus implying the timeliness of the present paper. Indeed, we secure Bayesian procedures against possible inconsistencies. No results of such a generality are known up to now.