**Multiple solutions to bifurcation inference**

*Starting with three cell states, we would like to infer a bifurcation process from one to the other two. If a single gene is up-regulated in one of the states, yet down-regulated in the other two, then clearly any state may act as the beginning of the trajectory. For example, if we start in state 1 then the gene is up-regulated along state 2 and stays constant in state 3; if we start in state 2 then the gene is down-regulated in states 1 & 3; if we start in state 3 then the gene is up-regulated in state 2 and remains down-regulated in state 1. However, due to the non-identifiability this is true if we add additional genes that are up-regulated in one or two of the cell states. The equivalent geometric argument is that we can build the transcriptomic profiles across all genes by spinning the figure about B (with possible inversion) and “adding” that gene. No matter how many additional genes we add, any one of the three states can act as the root state or beginning of pseudotime. Therefore, in the absence of any additional information there are always three equally valid solutions to bifurcation inference from gene expression data alone.*

Campbell KR, Yau C. (2017) **Probabilistic modeling of bifurcations in single-cell gene expression data using a Bayesian mixture of factor analyzers**. *Wellcome Open Res* 2:19. [article]