Recent studies have documented that the Madden-Julian Oscillation (MJO) has impacts in extreme dry/wet conditions over tropical regions and in atmospheric state. They are based, however, in correlation analysis, and therefore do not consider non-linear interactions nor they establish cause-effect relationships.In this study we introduce a generalization of the non-linear Granger causality (GC) test to identify causal relations between MJO and hydrological extremes. The method is able to identify causal relations under noisy, nonlinear and non-stationary scenarios. A probabilistic extension is also introduced where the causal test operates directly on the marginal likelihood (also called evidence) of the observations, which is analytic. We apply our proposed method to MJO and satellite-based soil moisture (SM) data, and revisit the global teleconnection patterns induced by MJO events. Since El Ni\~no Southern Oscillation (ENSO) is a modulating factor that can result in abnormal SM global distributions, we also include it in the analysis as a potential driver of SM variability. Including ENSO allows us to differentiate the effect of the MJO and ENSO on the global SM anomalies and to learn the causal graph of their cause-effect relationships, and also the mutual relation between MJO and ENSO extreme events.