The Laplace asymptotic expansion in high dimensions

We prove that the classical Laplace asymptotic expansion of $\int_{\mathbb R^d} g(x)e^{-nz(x)}dx$, $n\gg1$ extends to the high-dimensional regime in which $d$ may grow large with $n$. We formulate simple conditions on the growth of the derivatives of $g$ and $z$ near the minimizer of $z$ under which the terms of the expansion and the remainder are bounded as powers of $(\tau d/\sqrt n)^2$. The parameter $\tau$ controls the growth rates of the derivatives and can be potentially large, but we obtain a useful expansion whenever $\tau d/\sqrt n\ll 1$. Our result relies on a new and transparent proof of the Laplace expansion valid for any $d$. The proof leads to a new representation of the terms and remainder, which is crucial in obtaining tight control on the growth of these quantities with $d$. To demonstrate that our bounds are tight, we consider the case of a quartic $z$ and $g\equiv1$, showing that the $k$th term is precisely of the order $(d^2/n)^k$ for all $k$, and that our bounds on the terms are of the same order of magnitude. We also apply our results to derive a high-dimensional Laplace expansion for a particular function $z$ arising in the context of Bayesian inference, which is both random and depends on $n$. We show that with high probability, the terms of the expansion and the remainder are bounded in powers of $d^2/n$. In both of these examples, $\tau=1$.