We study the classical Laplace asymptotic expansion of $\int_{\mathbb R^d} f(x)e^{-nv(x)}dx$ in high dimensions $d$. We derive an error bound to the expansion when truncated to arbitrary order. The error bound is fully explicit except for absolute constants, and it depends on $d$, $n$, and operator norms of the derivatives of $v$ and $f$ in a neighborhood of the minimizer of $v$.

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