Simulation-calibrated Bayesian inference for progenitor properties of the microquasar SS 433

SS$\,$433 is one of the most extreme Galactic X-ray binaries, exhibiting semi-relativistic jets and super-critical accretion, and harboring a compact object, likely a black hole. Despite decades of observation and modeling, the precise nature of its progenitor binary remains uncertain. To estimate the zero-age main sequence (ZAMS) properties of binaries that evolve into SS$\,$433-like systems, we apply simulation-based calibration to Bayesian inference and convolve a multivariate Gaussian likelihood constructed from six measured binary parameters of SS$\,$433 with the isolated binary evolution model COSMIC. Employing the dynamic nested sampler of $\texttt{dynesty}$, we perform posterior inference over a ten-dimensional progenitor parameter space defined by the masses, orbital parameters, mass transfer possibilities, and natal kick velocity. We find that SS$\,$433-like systems arise from specific regions of binary evolution parameter space depending on key assumptions, such as the mass transfer rate and uncertainty taken from observations. Our simulation-based calibration framework, implemented with a suite of machine learning algorithms and scored by a heuristic reliability metric, allows us to iteratively build posterior distributions of the progenitors of SS$\,$433-like systems. This analysis reveals 90% confidence intervals for the ZAMS primary mass (8, 11) M$_\odot$, secondary mass (32, 40) M$_\odot$, orbital period (136, 2259) days, eccentricity (0.26, 0.6), common envelope evolution efficiency (0.44, 0.76), accreted fraction in stable mass transfer (0.22, 0.6), and black hole natal kick velocity magnitude (5, 68) km/s. These results demonstrate the feasibility of direct probabilistic inference of X-ray binary progenitors to offer new insights into the evolution of high-accretion-rate systems such as SS$\,$433.