  Event Horizon Telescope (EHT) observations of the core of the galaxy M87
suggest an observational appearance dominated by a ring of approximately
40$\mu$as in diameter. The thickness of the ring is less certain: imaging
efforts constrained it to be less than half the diameter (consistent with an
imaging resolution of 20$\mu$as), while visibility-domain modeling suggested a
variety of fractional widths, including as low as $10\%$ on some days. The
fractional width is very interesting as it has the potential to discriminate
between different astrophysical scenarios for the source; in fact, the
$10$--$20\%$ range is so narrow as to be in tension with theoretical
expectations. In the first of a series of papers on the width of the observed
ring, we reproduce a subset of EHT visibility-domain modeling results and we
explore whether alternative data analysis methods might favor thicker rings. We
point out that the closure phase (and closure amplitude) likelihood function is
not independent of residual station gain amplitudes, even at high
signal-to-noise, and explore two approximations of practical interest: one
standard in the field (and employed by the EHT collaboration), and a new one
that we propose. Analyzing the public data, we find that the new likelihood
approximation prefers somewhat thicker rings, more in line with theoretical
expectations. Further analysis is needed, however, to determine which
approximation is better for the EHT data.
