  We review and update constraints on the Early Dark Energy (EDE) model from
cosmological data sets, in particular Planck PR3 and PR4 cosmic microwave
background (CMB) data and large-scale structure (LSS) data sets including
galaxy clustering and weak lensing data from the Dark Energy Survey, Subaru
Hyper Suprime-Cam, and KiDS+VIKING-450, as well as BOSS/eBOSS galaxy clustering
and Lyman-$\alpha$ forest data. We detail the fit to CMB data, and perform the
first analyses of EDE using the CAMSPEC and Hillipop likelihoods for Planck CMB
data, rather than Plik, both of which yield a tighter upper bound on the
allowed EDE fraction than that found with Plik. We then supplement CMB data
with large-scale structure data in a series of new analyses. All these analyses
are concordant in their Bayesian preference for $\Lambda$CDM over EDE, as
indicated by marginalized posterior distributions. We perform a series of tests
of the impact of priors in these results, and compare with frequentist analyses
based on the profile likelihood, finding qualitative agreement with the
Bayesian results. All these tests suggest prior volume effects are not a
determining factor in analyses of EDE. This work provides both a review of
existing constraints and several new analyses.
