Polynomial time guarantees for sampling based posterior inference in high-dimensional generalised linear models

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 [56], but using only local likelihood conditions and without relying on M-estimation theory, non-asymptotic statistical and computational guarantees are provided for gradient based MCMC algorithms. 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 cononical statistical non-linear inverse problem from PDEs.