  The part played by stars in the ionization of the intergalactic medium
remains an open question. A key issue is the proportion of the stellar ionizing
radiation that escapes the galaxies in which it is produced. Spectroscopy of
gamma-ray burst afterglows can be used to determine the neutral hydrogen
column-density in their host galaxies and hence the opacity to extreme
ultra-violet radiation along the lines-of-sight to the bursts. Thus, making the
reasonable assumption that long-duration GRB locations are representative of
the sites of massive stars that dominate EUV production, one can calculate an
average escape fraction of ionizing radiation in a way that is independent of
galaxy size, luminosity or underlying spectrum. Here we present a sample of NH
measures for 138 GRBs in the range 1.6<z<6.7 and use it to establish an average
escape fraction at the Lyman limit of <fesc>~0.005, with a 98% confidence upper
limit of ~0.015. This analysis suggests that stars provide a small contribution
to the ionizing radiation budget of the IGM at z<5, where the bulk of the
bursts lie. At higher redshifts, z>5, firm conclusions are limited by the small
size of the GRB sample, but any decline in average HI column-density seems to
be modest. We also find no indication of a significant correlation of NH with
galaxy UV luminosity or host stellar mass, for the subset of events for which
these are available. We discuss in some detail a number of selection effects
and potential biases. Drawing on a range of evidence we argue that such
effects, while not negligible, are unlikely to produce systematic errors of
more than a factor ~2, and so would not affect the primary conclusions. Given
that many GRB hosts are low metallicity, high specific star-formation rate,
dwarf galaxies, these results present a particular problem for the hypothesis
that such galaxies dominated the reionization of the universe.
