  Context: Fast radio bursts are transient radio pulses of extragalactic
origin. Their dispersion measure is indicative of the baryon content in the
ionized intergalactic medium between the source and the observer. However,
inference using unlocalized fast radio bursts is degenerate to the distribution
of redshifts of host galaxies. Method: We perform a joint inference of the
intergalactic baryon content and the fast radio burst redshift distribution
with the use of Bayesian statistics by comparing the likelihood of different
models to reproduce the observed statistics in order to infer the most likely
models. In addition to two models of the intergalactic medium, we consider
contributions from the local environment of the source, assumed to be a
magnetar, as well as a representative ensemble of host and intervening
galaxies. Results: Assuming that the missing baryons reside in the ionized
intergalactic medium, our results suggest that the redshift distribution of
observed fast radio bursts peaks at $z \lesssim 0.6$. However, conclusions from
different instruments regarding the intergalactic baryon content diverge and
thus require additional changes to the observed distribution of host redshifts,
beyond those caused by telescope selection effects.
