= Solution
An <Edgeworth expansion> refines a central <normal approximation> using higher <cumulants>. Let $Y_i$ be independent and identically distributed, with mean $\mu$, positive <variance> $\sigma^2$, and <cumulants> $\kappa_r$. Define
$$
Z_n=\frac{\sqrt n(\bar Y-\mu)}\sigma,\qquad
\gamma_3=\frac{\kappa_3}{\sigma^3},\qquad
\gamma_4=\frac{\kappa_4}{\sigma^4}.
$$
The logarithm of its <characteristic function> has the expansion
$$
\log\mathbb E e^{itZ_n}
=-\frac{t^2}{2}+\frac{\gamma_3(it)^3}{6\sqrt n}
+\frac{\gamma_4(it)^4}{24n}+\cdots.
$$
Expanding the exponential adds the squared cubic term $\gamma_3^2(it)^6/(72n)$. Fourier inversion turns $(it)^r e^{-t^2/2}$ into $\phi(z)H_r(z)$, where the probabilists' <Hermite polynomials> satisfy
$$
H_r(z)=(-1)^r\phi(z)^{-1}\frac{d^r}{dz^r}\phi(z).
$$
Thus the density <Edgeworth expansion> through order $n^{-1}$ is
$$
\boxed{f_{Z_n}(z)=\phi(z)\left[
1+\frac{\gamma_3}{6\sqrt n}H_3(z)
+\frac1n\left\{\frac{\gamma_4}{24}H_4(z)
+\frac{\gamma_3^2}{72}H_6(z)\right\}\right]+o(n^{-1}).}
$$
Here $H_3=z^3-3z$, $H_4=z^4-6z^2+3$, and $H_6=z^6-15z^4+45z^2-15$. Since $(\phi H_{r-1})'=-\phi H_r$, the corresponding <cumulative distribution function> expansion is
$$
\Pr(Z_n\le z)=\Phi(z)-\phi(z)\left[
\frac{\gamma_3}{6\sqrt n}H_2(z)
+\frac1n\left\{\frac{\gamma_4}{24}H_3(z)
+\frac{\gamma_3^2}{72}H_5(z)\right\}\right]+o(n^{-1}).
$$
These remainder statements require adequate moments and smooth nonlattice conditions allowing transform inversion and error control, not merely a formal cumulant expansion. The density version needs the corresponding local smoothness. A symmetric distribution has $\gamma_3=0$, removing the first correction. Truncation can produce negative densities or nonmonotone distribution functions far from the center; it is not automatically a uniformly reliable relative tail approximation. A lattice distribution needs its lattice correction.
A <Laplace approximation> instead approximates an integral dominated by a sharp maximum. Suppose $h$ has a unique interior maximizer $\widehat\theta\in\mathbb R^d$, with $A=-h''(\widehat\theta)$ positive definite, and the integral is localized near that point. For smooth amplitude $g$,
$$
I_n=\int g(\theta)e^{nh(\theta)}\,d\theta.
$$
Set $\theta=\widehat\theta+z/\sqrt n$. A <Taylor expansion> gives
$$
n\{h(\theta)-h(\widehat\theta)\}
=-\tfrac12z^{\mathsf T}Az+O(n^{-1/2}\|z\|^3).
$$
Expanding $g$ and integrating the resulting <Gaussian integral> gives
$$
\boxed{I_n\sim e^{nh(\widehat\theta)}g(\widehat\theta)
\left(\frac{2\pi}{n}\right)^{d/2}|A|^{-1/2}.}
$$
Under sufficient smoothness and localization the relative correction is $O(n^{-1})$: odd terms of order $n^{-1/2}$ integrate to zero over the limiting symmetric Gaussian region. Higher terms use derivatives of $h,g$ and Gaussian moments.
For a <likelihood function> with $\ell_n=nh$ and a smooth positive <prior density>, this gives the integrated likelihood approximation
$$
\int e^{\ell_n(\theta)}\pi(\theta)\,d\theta
\simeq e^{\ell_n(\widehat\theta)}\pi(\widehat\theta)
(2\pi)^{d/2}|j(\widehat\theta)|^{-1/2},
$$
where $j=-\ell_n''$ is the <observed information>. Ratios of such integrals give <posterior expectation by Laplace approximation>; the same method is useful when integrating nuisance variables. Boundary maxima, singular <Hessian matrices>, nonlocalized tails and several dominant modes require altered formulas, with contributions summed when appropriate.
Both methods start with a quadratic Gaussian approximation and then retain higher-order terms. The <Edgeworth series> corrects a sampling distribution through its <cumulants> and Fourier inversion; <Laplace's method> corrects an integral through local derivatives at its dominant point. Neither should be confused with simply replacing a likelihood by a normalized posterior density.
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