Tight skew adjustment to the Laplace approximation in high dimensions
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In Bayesian inference, a simple and popular approach to reduce the burden of computing high dimensional integrals against the posterior is to make the Laplace approximation. This Gaussian approximation to the posterior is accurate when sample size is large, but for smaller sample sizes it falls short by not capturing the skew of the true posterior. Building on Katsevich, 2023 (arXiv:2305.17604) we derive the leading order contribution to the Laplace approximation error for an arbitrary observable expectation, and thereby obtain a canonical, computable skew correction. Making the correction reduces the observable expectation error by at least one order of magnitude. Moreover, the leading order term allows us to derive tight dimension-dependent upper and lower bounds on the unadjusted Laplace error, which is of independent interest.
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