We use 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 contains and indexed by an 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. 

Based on these models, we employ the Bayesian inversion framework to infer two types of physical factors that affect tumor growth from some noisy observations of tumor cell density. One type of factor is parametric, while the other one is a spatial function. We establish well-posedness and stability theories for the corresponding Bayesian inversion problem and employ an MCMC approach to obtain the posterior distribution via sampling for every single model (fix the value of $m$). We emphasize that the theories and numerical methods are applied to the entire family of models uniformly well. Furthermore, we prove that as $m$ increases, the posterior distribution that corresponds to a given set of observation data converges in the sense of Hellinger distance. We could arrive at such a conclusion: the theoretical part benefits from the convergence property of the forward problem, while the numerical part relies on the asymptotically preserving property of our forward problem solver.