The 2004-2009 uplift episode is the largest recorded episode of unrest at Yellowstone caldera. We use GPS and InSAR time series spanning 2004-2015, with a focus in the aforementioned event to understand the mechanisms of unrest. InSAR data recorded ~25 and ~20 cm of uplift at the Sour Creek (SCD) and Mallard Lake (MLD) resurgent domes during 2004-2009, and ~8 cm of subsidence at the Norris Geyser Basin (NGB). The SCD/MLD uplift was followed by subsidence across the caldera floor with a maximum at MLD of ~1.5-2.5 cm/yr and no deformation at NGB. The best-fit source models are two horizontal sills at depths of ~8.7 and 10.7 km for the caldera source and NGB respectively, with volume changes of 0.354 and -0.121 km3, and an overpressure of ~0.1 MPa. The InSAR and GPS time series record an exponential increase followed by exponential decrease in the uplift, which is indicative of magma injection into the caldera reservoir, with no need for other mechanisms. However, magma extraction from NGB to the caldera is unable to explain the subsidence coeval with the caldera uplift. The GPS time series of the 2014-2015 episode of caldera uplift can also be explained by a magma injection model. Distributed sill opening models show that magma is stored across the caldera source with no clear boundary between MLD and SCD. Since the magma overpressure is orders below the tensile strength of the encasing rock, historical episodes of unrest like these are very unlikely to trigger an eruption.