We have developed a Bayesian statistical framework for quantitative geosteering in real time. Two types of contemporary geosteering approaches, model based and stratification based, are introduced. The latter is formulated as a Bayesian optimization procedure: The log from a pilot reference well is used as a stratigraphic signature of the geologic structure in a given region; the observed log sequence acquired along the wellbore is projected into the stratigraphic domain given a proposed earth model and directional survey; the pattern similarity between the converted log and the signature is measured by a correlation coefficient; then stochastic searching is performed on the space of all possible earth models to maximize the similarity under constraints of the prior understanding of the drilling process and target formation; finally, an inference is made based on the samples simulated from the posterior distribution using stochastic approximation Monte Carlo in which we extract the most likely earth model and the associated credible intervals as a quantified confidence indicator. We extensively test our method using synthetic and real geosteering data sets. Our method consistently achieves good performance on synthetic data sets with high correlations between the interpreted and the reference logs and provides similar interpretations as the geosteering geologists on four real wells. We also conduct a reliability performance test of the method on a benchmark set of 200 horizontal wells randomly sampled from the Permian Basin. Our Bayesian framework informs geologists with key drilling decisions in real time and helps them navigate the drilling bit into the target formation with confidence.