Effective Sample Size (ESS) is central to diagnosing Markov chain Monte Carlo (MCMC) efficiency, yet standard multivariate ESS definitions either collapse in high dimensions when some coordinates are (near-)deterministic or become numerically unstable under hardware approximations (e.g., fixed-point quantization and probability truncation). We introduce mvESS-Z, a multivariate ESS that (i) is invariant to zero-variance coordinates, (ii) admits finite-sample concentration guarantees, and (iii) is stable under quantization/truncation commonly used in probabilistic accelerators. mvESS-Z is defined on the signal subspace determined by the posterior covariance and uses pseudo-determinant or spectral aggregation to avoid degeneracy. We prove basic properties, give a practical estimator with automatic rank selection, and show on synthetic and vision-style workloads that mvESS-Z correlates with downstream task error more reliably than per-coordinate ESS and split-\(\hat{R}\), enabling quality-constrained tuning of precision and random number generators.
Effective Sample Size (ESS) is central to diagnosing Markov chain Monte Carlo (MCMC) efficiency, yet standard multivariate ESS definitions either collapse in high dimensions when some coordinates are (near-)deterministic or become numerically unstable under hardware approximations (e.g., fixed-point quantization and probability truncation). We introduce mvESS-Z, a multivariate ESS that (i) is invariant to zero-variance coordinates, (ii) admits finite-sample concentration guarantees, and (iii) is stable under quantization/truncation commonly used in probabilistic accelerators. mvESS-Z is defined on the signal subspace determined by the posterior covariance and uses pseudo-determinant or spectral aggregation to avoid degeneracy. We prove basic properties, give a practical estimator with automatic rank selection, and show on synthetic and vision-style workloads that mvESS-Z correlates with downstream task error more reliably than per-coordinate ESS and split-\(\hat{R}\), enabling quality-constrained tuning of precision and random number generators.
