For the “realistic” Bayes factor (10^2–10^5), what value of n_gal was used? Was it the average cosmic density of Milky Way-like galaxies (~0.01 Mpc^-3)? If so, this is inappropriate for a targeted observation of M81. M81 resides in a galaxy group; the local density of galaxies in its vicinity is orders of magnitude higher than the cosmic average. Using the cosmic average unfairly dilutes the prior and inflates the Bayes factor against the coincidence hypothesis.
Conditional Probability vs. Volumetric Prior: The entire analysis is triggered by the known presence of rFRB 20200120E in M81. Therefore, the correct question is not “What is the probability of any galaxy having a BNS?” but “Given that we have a galaxy (M81) hosting an rFRB, what is the probability that it also hosts a BNS in the pre-merger phase?” This conditional probability likely depends on factors not captured by a simple volumetric rate: e.g., the stellar population, formation history, and dynamical environment of M81’s globular cluster specifically. Using a universal volumetric rate to calculate this specific probability may be a significant oversimplification.
In essence, if the prior volume (or the effective number of “trials”) is substantially larger than justified, first by assuming a uniform sky, then by possibly using an inappropriate galaxy density, the evidence against the coincidence hypothesis (i.e., the Bayes factor) could be severely overestimated. This would directly impact the central claim of establishing a link at 2.81–4.17σ significance.
Could the authors please clarify their calculation of the coincidental probability and detail how the conditional nature of the observation (knowing the FRB’s host galaxy) is explicitly incorporated into their Bayesian model comparison? Without this, the strength of the statistical evidence presented is difficult to assess.