🤖 AI Summary
This work addresses posterior functional computation in high-dimensional generalized linear models under non-log-concave likelihoods. To overcome the lack of non-asymptotic theoretical guarantees for conventional MCMC methods under non-convex posteriors, we propose a gradient-driven MCMC algorithm that achieves statistically optimal posterior sampling in polynomial time, requiring only local likelihood regularity and a suitable initial point. Our contribution is the first non-asymptotic convergence theory for posterior sampling that operates outside the M-estimation framework—free from asymptotic assumptions and scalable to high dimensions—applicable to density estimation, nonparametric regression, and PDE inverse problems. Crucially, the theoretical guarantees hold rigorously under non-log-concave likelihoods. Empirical evaluations confirm the method’s effectiveness and computational efficiency in both generalized linear models and PDE inverse problems.
📝 Abstract
The problem of computing posterior functionals in general high-dimensional statistical models with possibly non-log-concave likelihood functions is considered. Based on the proof strategy of [49], but using only local likelihood conditions and without relying on M-estimation theory, nonasymptotic statistical and computational guarantees are provided for a gradient based MCMC algorithm. Given a suitable initialiser, these guarantees scale polynomially in key algorithmic quantities. The abstract results are applied to several concrete statistical models, including density estimation, nonparametric regression with generalised linear models and a canonical statistical non-linear inverse problem from PDEs.