A cautionary tale in fitting galaxy rotation curves with Bayesian techniques: does Newton's constant vary from galaxy to galaxy?

The application of Bayesian techniques to astronomical data is generally non-trivial because the fitting parameters can be strongly degenerated and the formal uncertainties are themselves uncertain. An example is provided by the contradictory claims over the presence or absence of a universal acceleration scale (g$_\dagger$) in galaxies based on Bayesian fits to rotation curves. To illustrate the situation, we present an analysis in which the Newtonian gravitational constant $G_N$ is allowed to vary from galaxy to galaxy when fitting rotation curves from the SPARC database, in analogy to $g_{\dagger}$ in the recently debated Bayesian analyses. When imposing flat priors on $G_N$, we obtain a wide distribution of $G_N$ which, taken at face value, would rule out $G_N$ as a universal constant with high statistical confidence. However, imposing an empirically motivated log-normal prior returns a virtually constant $G_N$ with no sacrifice in fit quality. This implies that the inference of a variable $G_N$ (or g$_{\dagger}$) is the result of the combined effect of parameter degeneracies and unavoidable uncertainties in the error model. When these effects are taken into account, the SPARC data are consistent with a constant $G_{\rm N}$ (and constant $g_\dagger$).