We have constructed a comprehensive statistical model for Type Ia supernova (SN Ia) light curves spanning optical through near-infrared (NIR) data. A hierarchical framework coherently models multiple random and uncertain effects, including intrinsic supernova (SN) light curve covariances, dust extinction and reddening, and distances. An improved BayeSN Markov Chain Monte Carlo code computes probabilistic inferences for the hierarchical model by sampling the global probability density of parameters describing individual SNe and the population. We have applied this hierarchical model to optical and NIR data of 127 SNe Ia from PAIRITEL, CfA3, Carnegie Supernova Project, and the literature. We find an apparent population correlation between the host galaxy extinction AV and the ratio of total-to-selective dust absorption RV. For SNe with low dust extinction, A_V_<~0.4, we find R_V_~2.5-2.9, while at high extinctions, A_V_>~1, low values of R_V_<2 are favored. The NIR luminosities are excellent standard candles and are less sensitive to dust extinction. They exhibit low correlation with optical peak luminosities, and thus provide independent information on distances. The combination of NIR and optical data constrains the dust extinction and improves the predictive precision of individual SN Ia distances by about 60%. Using cross-validation, we estimate an rms distance modulus prediction error of 0.11mag for SNe with optical and NIR data versus 0.15mag for SNe with optical data alone. Continued study of SNe Ia in the NIR is important for improving their utility as precise and accurate cosmological distance indicators.