This paper presents a Bayesian framework for estimating a Probabilistic Linear Discriminant Analysis (PLDA) model in the presence of noisy labels. True class labels are interpreted as latent random variables, which are transmitted through a noisy channel, and received as observed speaker labels. The labeling process is modeled as a Discrete Memoryless Channel (DMC). PLDA hyperparameters are interpreted as random variables, and their joint posterior distribution is derived using meanfield Variational Bayes, allowing maximum a posteriori (MAP) estimates of the PLDA model parameters to be determined. The proposed solution, referred to as VB-MAP, is presented as a general framework, but is studied in the context of speaker verification, and a variety of use cases are discussed. Specifically, VB-MAP can be used for PLDA estimation with unreliable labels, unsupervised PLDA estimation, and to infer the reliability of a PLDA training set. Experimental results show the proposed approach to provide significant performance improvements on a variety of NIST Speaker Recognition Evaluation (SRE) tasks, both for data sets with simulated mislabels, and for data sets with naturally occurring missing or unreliable labels.