Beyond Best-Fits and Model Selection -- Introducing "Reliability" of cusp-core inference of dark matter halos

We introduce the notion of a Bayesian analysis motivated `reliability' that gives a truer distinction of cusp-core and other halo-parameters (like mass-concentration) in an ensemble of observed galaxies. Our approach goes beyond the standard statistical techniques of parameter estimation and model fitting. We create hundreds of thousands of realistic mock SPARC RCs, with both cuspy and cored DM density profiles as model inputs. These RCs carefully incorporate the details of SPARC data such as the nature of observed uncertainties and different sources of scatters arising from observation, presence of baryons, DM mass-concentration, etc. Bayesian analysis of these mock RCs enables us to reconstruct and identify the parameter space in galaxy observable and theory where one can venture beyond best-fits to a preferred DM halo model or model selections between different density models. We find that it is imperative to choose low stellar surface density ($\Sigma_{\star}$) galaxies for reliable cusp-vs-core distinction; for example, RC data for galaxies with $\Sigma_{\star} \leq 2.5$ is needed for a 75\% confidence in distinguishing cusps from cores. Similarly, we also find that for correct estimations of the halo masses and concentrations, the RCs need to be measured to at least a radial distance $\geq 0.8r_s$ where $r_s$ is the scale radii of the corresponding DM halo density profiles. Out of the total $\sim$ 135 SPARC galaxies, using our reliability criteria, we find that only 21 RCs clear the bar to be used for any unbiased cusp-core distinction as well as DM halo mass-concentration estimates at $\geq$75\% reliability confidence level. With $\geq$66\% ( $\geq$50\%) reliability settings, the sample size increases to 44 (59). Interestingly, in the $\geq 75$\% reliable subsample, there are 5 times more galaxies that are reliably cored than cuspy. [Abridged]