Exponential-Family Membership Inference: From LiRA and RMIA to BaVarIA

Membership inference attacks (MIAs) are becoming standard tools for auditing the privacy of machine learning models. The leading attacks -- LiRA (Carlini et al., 2022) and RMIA (Zarifzadeh et al., 2024) -- appear to use distinct scoring strategies, while the recently proposed BASE (Lassila et al., 2025) was shown to be equivalent to RMIA, making it difficult for practitioners to choose among them. We show that all three are instances of a single exponential-family log-likelihood ratio framework, differing only in their distributional assumptions and the number of parameters estimated per data point. This unification reveals a hierarchy (BASE1-4) that connects RMIA and LiRA as endpoints of a spectrum of increasing model complexity, and yields a practical rule -- match the attack's complexity to the available shadow-model budget. Within this framework, we identify variance estimation as a primary bottleneck at small shadow-model budgets and propose BaVarIA, a Bayesian variance inference attack that replaces threshold-based parameter switching with conjugate normal-inverse-gamma priors. BaVarIA yields a Student-t predictive (BaVarIA-t) or a Gaussian with stabilized variance (BaVarIA-n), providing stable performance without per-dataset hyperparameter tuning. Across 12 testbeds and 7 shadow-model budgets, BaVarIA is a drop-in replacement for LiRA that matches or, on average, improves upon it. The gains are largest in the practically important low-shadow-model and offline regimes: offline, the Bayesian prior replaces LiRA's heuristic and outperforms it on 10 of 12 testbeds.