Default Bayesian analysis for the AR(1) model is considered and an improper prior is suggested,the choice being motivated by consideration of the functional relationship between the autoregressive parameter p and the variance of the process in a time serie framework. We discuss the initialization issue under both stationarity and nonstationarity,and we propose a procedure that avoids discontinuities in statistical inference. We present numerical evidence,based on frequentist criteria,on the performance of the suggested prior and of the fractional Bayes factor when comparing stationary versus nonstationary models