In this proceeding, we present frequentist and Bayesian results from a global fit of effective vector and Majorana fermion Higgs portal dark matter (DM) models using the $\mathsf{GAMBIT}$ software. We systematically explore the parameter space of these models using advanced sampling techniques to simultaneously satisfy the observed DM abundance, Higgs invisible decay, and indirect and direct detection limits. In addition, we take account of a set of nuisance parameters arising from Standard Model, nuclear physics, DM halo and velocity distribution. For the vector DM model viable solutions are found at low and high vector masses. The Majorana fermion model requires a strong preference for a CP-odd, parity-violating coupling which leads to a momentum-suppression of the DM-nucleon cross-section. All of our results, samples and input files are publicly available via $\mathsf{Zenodo}$.