The Global History of Reionisation
  Using a Bayesian framework, we quantify what current observations imply about
the history of the epoch of reionisation (EoR). We use a popular,
three-parameter EoR model, flexible enough to accommodate a wide range of
physically-plausible reionisation histories. We study the impact of various EoR
observations: (i) the optical depth to the CMB measured by Planck 2016; (ii)
the dark fraction in the Lyman $\alpha$ and $\beta$ forests; (iii) the redshift
evolution of galactic Ly$\alpha$ emission (so-called "Ly$\alpha$ fraction");
(iv) the clustering of Ly$\alpha$ emitters; (v) the IGM damping wing imprint in
the spectrum of QSO ULASJ1120+0641; (vi) and the patchy kinetic
Sunyaev-Zel'dovich signal. Combined, (i) and (ii) already place interesting
constraints on the reionisation history, with the epochs corresponding to an
average neutral fraction of (75, 50, 25) per cent, constrained at 1$\sigma$ to
$z= (9.21\substack{+1.22 -1.15}, 8.14\substack{+1.08 -1.00},
7.26\substack{+1.13 -0.96})$. Folding-in more model-dependent EoR observations
[(iii--vi)], strengthens these constraints by tens of per cent, at the cost of
a decrease in the likelihood of the best-fit model, driven mostly by (iii). The
tightest constraints come from (v). Unfortunately, no current observational set
is sufficient to break degeneracies and constrain the astrophysical EoR
parameters. However, model-dependent priors on the EoR parameters themselves
can be used to set tight limits by excluding regions of parameter space with
strong degeneracies. Motivated by recent observations of $z\sim7$ faint, lensed
galaxies, we show how a conservative upper limit on the virial temperature of
haloes which host reionising galaxies can constrain the escape fraction of
ionising photons to $f_{\rm esc} = 0.14\substack{+0.26 -0.09}$
