We consider the inverse problem of parameter estimation in a diffuse interface model for tumor growth. The model consists of a fourth-order Cahn--Hilliard system and contains three phenomenological parameters: the tumor proliferation rate, the nutrient consumption rate, and the chemotactic sensitivity. We study the inverse problem within the Bayesian framework and construct the likelihood and noise for two typical observation settings. One setting involves an infinite-dimensional data space where we observe the full tumor. In the second setting we observe only the tumor volume; hence the data space is finite-dimensional. We show the well-posedness of the posterior measure for both settings, building upon and improving the analytical results in [C. Kahle and K. F. Lam, Appl. Math. Optim., (2018)]. A numerical example involving synthetic data is presented in which the posterior measure is numerically approximated by the sequential Monte Carlo approach with tempering.