  In this paper, 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. The models contain uncertain parameters and are
indexed by a physical parameter $m$, which characterizes the constitutive
relation between density and pressure. Based on these models, we employ the
Bayesian inversion framework to infer parametric and nonparametric unknowns
that affect tumor growth from noisy observations of tumor cell density. We
establish the well-posedness and the stability theories for the Bayesian
inversion problem and further prove the convergence of the posterior
distribution in the so-called incompressible limit, $m \rightarrow \infty$.
Since the posterior distribution across the index regime $m\in[2,\infty)$ can
thus be treated in a unified manner, such theoretical results also guide the
design of the numerical inference for the unknown. We propose a generic
computational framework for such inverse problems, which consists of a typical
sampling algorithm and an asymptotic preserving solver for the forward problem.
With extensive numerical tests, we demonstrate that the proposed method
achieves satisfactory accuracy in the Bayesian inference of the tumor growth
models, which is uniform with respect to the constitutive relation.
