A unified theory of the high-dimensional Laplace approximation with application to Bayesian inverse problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916943141797888 |
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| author | Katsevich, Anya Spokoiny, Vladimir |
| author_facet | Katsevich, Anya Spokoiny, Vladimir |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07952 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A unified theory of the high-dimensional Laplace approximation with application to Bayesian inverse problems Katsevich, Anya Spokoiny, Vladimir Statistics Theory 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. |
| title | A unified theory of the high-dimensional Laplace approximation with application to Bayesian inverse problems |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2509.07952 |