How narrow is the M87* ring? I. The choice of closure likelihood function

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 fractional widths of $10$-$20\%$. 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.