Pattern recognition is becoming an increasingly important tool for making inferences from the massive amounts of data produced in magnetic confinement fusion experiments. However, the measurements obtained from the various plasma diagnostics are typically affected by a considerable statistical uncertainty. In this work, we consider the inherent stochastic nature of the data by modeling the measurements by probability distributions in a metric space. Information geometry permits the calculation of the geodesic distances on such manifolds, which we apply to the important problem of the classification of plasma confinement regimes. We use a distance-based conformal predictor, which we first apply to a synthetic data set. Next, the method yields an excellent classification performance with measurements from an international database. The conformal predictor also returns confidence and credibility measures, which are particularly important for interpretation of pattern recognition results in stochastic fusion data.