First, the problem is unmodeled, which means, the algorithm is not informed that something has happened and has to infer it from the given observations. The algorithmic approach to learn a Markovian model usually works by defining observations in terms of "observation features" (e.g. observed gene expression value, time point, etc.) and then estimating a conditional distribution
Second, there exists a time lag between transcriptional changes of cells and phenotypic changes of the corresponding cancer cells. Hence, the time-reversibility of the transition process from liver to cancer is not simple to model; according to our observations, changes from non-cancerous cells to cancerous cells should not be taken into account in our transition. Transcriptional changes in cancer cells usually lead to phenotypic effects within minutes
Hence, the used term for such transitions is denoted with "tumor hypoxia" (see Eq. (12) in the original paper) and it is assumed to be caused by a constant influx of oxygen to the cancer cells via macro-vessels within the tumor.
Note, that the algorithm is not designed to give exact results, but rather to calculate an approximation of the true matrix, which enables inference for larger, more complex and even unmodeled networks.
The assumptions that induce the model can also be found in the original article. It is assumed that the transition is time reversible so states in the past can be derived from states in the future. d2c66b5586