We employ the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. This family of problems is indexed by a physical parameter $m$, which characterizes the constitutive relation between density and pressure, and these models converge to a Hele-Shaw type problem as $m$ tends to infinity.
For each model (fix a $m$), we provide well-posedness and stability theories for its Bayesian inversion problem and employ an MCMC method to obtain the posterior distribution in practice. It is worth mentioning that these theoretical and numerical methods work uniformly well for the whole family of models. And we can even prove that for a given observation data, the corresponding posterior distribution converges in the sense of Hellinger distance as $m$ increases. We were able to achieve such results; the theoretical part is mainly due to the forward problems possessing some uniform estimates with respect to $m$. The numerical experiments, however, benefit from the asymptotically preserving (AP) property of our forward problem solver. Finally, we establish some numerical results that validate our theory proof.