We introduce a new technique to constrain the line-of-sight integrated electron density of our Galactic halo $text{DM}_text{MW,halo}$ through analysis of the observed dispersion measure distributions of pulsars $text{DM}_text{pulsar}$ and fast radio bursts $text{DM}_text{FRB}$. We model these distributions, correcting for the Galactic interstellar medium, with kernel density estimation---well-suited to the small data regime---to find lower/upper bounds to the corrected $text{DM}_text{pulsar}$/$text{DM}_text{FRB}$ distributions: $max[text{DM}_text{pulsar}] approx 7pm2 text{ (stat)} pm 9 text{ (sys) pc cm}^{-3}$ and $min[text{DM}_text{FRB}] approx 63^{+27}_{-21} text{ (stat)} pm 9 text{ (sys) pc cm}^{-3}$. Using bootstrap resampling to estimate uncertainties, we set conservative limits on the Galactic halo dispersion measure $-2 < text{DM}_text{MW,halo} < 123 text{pc cm}^{-3}$ (95% c.l.). The upper limit is especially conservative because it may include a non-negligible contribution from the FRB host galaxies and a non-zero contribution from the cosmic web. It strongly disfavors models where the Galaxy has retained the majority of its baryons with a density profile tracking the presumed dark matter density profile. Last, we perform Monte Carlo simulations of larger FRB samples to validate our technique and assess the sensitivity of ongoing and future surveys. We recover bounds of several tens $text{pc cm}^{-3}$ which may be sufficient to test whether the Galaxy has retained a majority of its baryonic mass. We estimate that a sample of several thousand FRBs will significantly tighten constraints on $text{DM}_text{MW,halo}$ and offer a valuable complement to other analyses.
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