The concept of cell states is increasingly used to classify cellular behaviour in development, regeneration, and cancer. This is driven in part by a deluge of data comprising snapshots of cell populations at single-cell resolution. Yet quantitative predictive models of cell states and their transitions remain lacking. Such models could help, for example, to optimise differentiation protocols in vitro. Here, starting with a tractable immunostaining dataset of transcription factor expression we explore systematically if cell state transition rates can be inferred quantitatively and what information is required to do this. We investigate early cell fate decisions in primitive streak-like populations derived from epiblast stem cells (Tsakiridis et al., 2014). A particular challenge of the existing data is that labelling of cell states can be incomplete, i.e., not all of the markers that define a cell state are read out simultaneously in a given experiment. Using a top-down approach, we enumerate all possible cell states from known lineage markers, and build a minimal mathematical model for the transitions between these states in a growing colony. We adopt a Bayesian inference approach to quantify cell state transition rates and their uncertainties.