We present observational constraints on the initial-final mass relation (IFMR) using wide double white dwarfs (DWDs). We identify 65 new candidate wide DWDs within the Sloan Digital Sky Survey, bringing the number of candidate wide DWDs to 142. We then engage in a spectroscopic follow-up campaign and collect existing spectra for these objects; using these spectra, we derive masses and cooling ages for 54 hydrogen (DA) WDs in DWDs. We also identify one new DA/DB pair, four candidate DA/DC pairs, four candidate DA/DAH pairs, and one new candidate triple degenerate system. Because wide DWDs are co-eval and evolve independently, the difference in the pre-WD lifetimes should equal the difference in the WD cooling ages. We use this to develop a Bayesian hierarchical framework and construct a likelihood function to determine the probability that any particular IFMR fits a sample of wide DWDs. We then define a parametric model for the IFMR and find the best parameters indicated by our sample of DWDs. We place robust constraints on the IFMR for initial masses of 2-4M_{sun}_. The WD masses produced by our model for stars within this mass range differ from those predicted by semi-empirical fits to open cluster WDs. Within this mass range, where there are few constraining open cluster WDs and disagreements in the cluster ages, wide DWDs may provide more reliable constraints on the IFMR. Expanding this method to the many wide DWDs expected to be discovered by Gaia may transform our understanding of the IFMR.