  In many fields, including biology, medicine, physics, chemistry, economy and
actuarial science, data can be represented as the time-to-event of a finite
state Markov chain model. The distribution of time intervals between successive
recorded events is known as a phase-type distribution. We demonstrate that in
cases where the eigenvalues of the Markov chain transition matrix are distinct
(non-degenerate), the phase-type distribution is multi-exponential. We then
pose and solve an inverse problem: given knowledge of the phase-type
distribution, can we determine the transition rate parameters of the underlying
Markov chain? To tackle this challenge, we initially convert the inverse
problem into a computer algebraic task, involving the solution of a system of
polynomial equations. These equations are symmetrized with respect to the
parameters of the phase-type distribution. For a specific subset of Markov
models that we refer to as "solvable," the inverse problem yields a unique
solution, up to transformations by finite symmetries. We outline a recursive
approach to compute these solutions for specific families of models, regardless
of their number of states. Additionally, we use the Thomas decomposition
technique to calculate solutions for models possessing 2, 3, or 4 states.
Interestingly, we show that models having the same number of states but
different transition graphs can yield identical phase-distributions. In such
"Rashomon effect" scenarios, time-to-event data permits the formulation of
multiple models and interpretations, all of which are consistent with the
observed experimental data. To differentiate among these models, we propose
additional distinguishing properties beyond just the time until the next event.
In order to prove its applicability, we discuss how this method can be used to
infer models of transcription regulation from transcriptional bursting data.
