Fraction of ionized hydrogen as a function of redshift, obtained by solving Equation (6) with our model luminosity density. We plot our results from integrating the model UV LFs to two magnitude limits of ﹩{M}_{{\rm{AB}}}=-17﹩ (green) and ﹩{M}_{{\rm{AB}}}=-12﹩ (purple), with 1σ confidence regions as shaded regions. We also plot constraints derived from observations of: Lyα emission from galaxies (open circles, Ouchi et al. 2010; Faisst et al. 2014; Pentericci et al. 2014; Schenker et al. 2014; Tilvi et al. 2014); the Lyα forest (filled circles, Fan et al. 2006); the clustering of Lyα emitting galaxies (square, Ouchi et al. 2010); GRB spectra damping wings (diamond, McQuinn et al. 2008); dark gaps in the Lyα forest (upper triangles, McGreer et al. 2015); quasar near zones (star, Venemans et al. 2015); and quasar spectra damping wings (lower triangle, Schroeder et al. 2013). We also plot the Planck Collaboration et al. (2015) redshift of instantaneous reionization. We note that the conversion from the Lyα escape fraction to the global ionized hydrogen fraction is uncertain and relies on several model assumptions (Mesinger et al. 2015).