{"ID":2873140,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.07952","arxiv_id":"2509.07952","title":"A unified theory of the high-dimensional Laplace approximation with application to Bayesian inverse problems","abstract":"The Laplace approximation (LA) to posteriors is a ubiquitous tool to simplify Bayesian computation, particularly in the high-dimensional settings arising in Bayesian inverse problems. Precisely quantifying the LA accuracy is a challenging problem in the high-dimensional regime. We develop a theory of the LA accuracy to high-dimensional posteriors which both subsumes and unifies a number of results in the literature. The primary advantage of our theory is that we introduce a new degree of flexibility, which can be used to obtain problem-specific upper bounds which are much tighter than previous \"rigid\" bounds. We demonstrate the theory in a prototypical example of a Bayesian inverse problem, in which this flexibility enables us to improve on prior bounds by an order of magnitude. Our optimized bounds in this setting are dimension-free, and therefore valid in arbitrarily high dimensions.","short_abstract":"The Laplace approximation (LA) to posteriors is a ubiquitous tool to simplify Bayesian computation, particularly in the high-dimensional settings arising in Bayesian inverse problems. Precisely quantifying the LA accuracy is a challenging problem in the high-dimensional regime. We develop a theory of the LA accuracy to...","url_abs":"https://arxiv.org/abs/2509.07952","url_pdf":"https://arxiv.org/pdf/2509.07952v1","authors":"[\"Anya Katsevich\",\"Vladimir Spokoiny\"]","published":"2025-09-09T17:40:38Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}