Many cellular effectors of pluripotency are dynamically regulated. In principle, regulatory mechanisms can be inferred from single-cell observations of effector activity across time. However, rigorous inference techniques suitable for noisy, incomplete, and heterogeneous data are lacking. Here, we introduce stochastic inference on lineage trees (STILT), an algorithm capable of identifying stochastic models that accurately describe the quantitative behavior of cell fate markers observed using time-lapse microscopy data collected from proliferating cell populations. STILT performs exact Bayesian parameter inference and stochastic model selection using a particle-filter-based algorithm. We use STILT to investigate the autoregulation of Nanog, a heterogeneously expressed core pluripotency factor, in mouse embryonic stem cells. STILT rejects the possibility of positive Nanog autoregulation with high confidence; instead, model predictions indicate weak negative feedback. We use STILT for rational experimental design and validate model predictions using novel experimental data.