Appendix A Proof of Theorem 1
Under conditions A1 and A2, we can expand the mean of each subposterior at the corresponding MLE, \(\hat{\varvec{\theta }}^{(j)}\), as follows:
$$\begin{aligned} \varvec{\mu }^{(j)}= & {} E_{\tilde{\pi }_j} (\varvec{\theta }) =\hat{\varvec{\theta }}^{(j)}+\frac{\hat{I}^{(j)-1}}{n} \\&\times \left[ \frac{\partial \log g({\varvec{\theta }})}{\partial {\varvec{\theta }}}|_{\hat{{\varvec{\theta }}}^{(j)}} -\frac{1}{2}\hat{H}^{(j)}\hat{I}^{(j)-1} \right] +O(n^{-2}), \end{aligned}$$
where \(\tilde{\pi }_j=\tilde{\pi }({\varvec{\theta }}|{\varvec{X}}_{[j]})\), \(\hat{I}^{(j)}=-\frac{1}{m} \frac{\partial ^{2} \log f({\varvec{X}}_{[j]}|\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^{T}}|_{\varvec{\theta }=\hat{\varvec{\theta }}^{(j)}}\), \(\hat{H}^{(j)}=-\frac{1}{m}\frac{\partial ^{3} logf({\varvec{X}}_{[j]}|\varvec{\theta })}{\partial \varvec{\theta } \partial \varvec{\theta }^{T}\partial \varvec{\theta }}|_{\varvec{\theta }=\hat{\varvec{\theta }}^{(j)}}\), and \(\hat{H}^{(j)}\hat{I}^{(j)-1}\) is a vector whose rth element equals \(\sum _{st}\hat{H}_{rst}^{(j)}\hat{I}_{st}^{(j)-1}\). To simplify the notation, we denote \(\hat{I}^{(j)-1}[\partial \log g({\varvec{\theta }})/\partial {\varvec{\theta }}|_{\hat{{\varvec{\theta }}}^{(j)}} -\frac{1}{2}\hat{H}^{(j)}\hat{I}^{(j)-1}]\) by \(\varvec{\nu }^{(j)}\). Here we would like to mention that although A1 and A2 are directly associated with the Laplace approximation of the full posterior, they can also lead to the conditions that guarantee the Laplace approximation of each subposterior, as long as m goes to \(\infty \) with n.
Moreover, for each \(\hat{\varvec{\theta }}^{(j)}\), we have
$$\begin{aligned} \hat{\varvec{\theta }}^{(j)}=\varvec{\theta }^{*}+\frac{\varvec{\xi }^{(j)}}{\sqrt{m}}+O_{p}(m^{-1}), \end{aligned}$$
where
$$\begin{aligned} \varvec{\xi }^{(j)}= & {} \frac{1}{\sqrt{m}}I^{-1}\sum _{i=1}^{m}\frac{\partial \log f(X_{ji}|\varvec{\theta }^{*})}{\partial \varvec{\theta }},\\ I= & {} -E_{{\varvec{X}}|\varvec{\theta }^{*}}\frac{\partial ^2 \log f({\varvec{X}}|\varvec{\theta }^{*}) }{\partial \varvec{\theta } \partial \varvec{\theta }^{T} }. \end{aligned}$$
Therefore, the mean of the mixture distribution \(\tilde{\pi }(\varvec{\theta }|{\varvec{X}})\) is given by
$$\begin{aligned} E_{\tilde{\pi }} (\varvec{\theta })= & {} \frac{1}{k}\sum _{j=1}^{k}\varvec{\mu }^{(j)}\\= & {} \varvec{\theta }^{*}+\frac{1}{k}\sum _{j=1}^{k}\frac{\varvec{\nu }^{(j)}}{n}+\frac{1}{k}\sum _{j=1}^{k} \frac{\varvec{\xi }^{(j)}}{\sqrt{m}}\\&+\,O_{p}(m^{-1})+O(n^{-2}). \end{aligned}$$
We can also implement a similar procedure on the full data posterior \(\pi (\varvec{\theta }|{\varvec{X}})\) and obtain
$$\begin{aligned} E_{\pi }(\varvec{\theta })= & {} \varvec{\theta }^{*}+\frac{\varvec{\nu }}{n} +\frac{\varvec{\xi }}{\sqrt{n}}+O_{p}(n^{-1})+O(n^{-2}), \end{aligned}$$
where \(\varvec{\nu }=\hat{I}^{-1}[\partial \log g({\varvec{\theta }})/\partial {\varvec{\theta }}|_{\hat{{\varvec{\theta }}} } -\frac{1}{2}\hat{H}\hat{I}^{-1}]\), \(\varvec{\xi }=\frac{1}{\sqrt{n}}I^{-1}\sum _{i=1}^{n}\frac{\partial \log f(X_{i}|\varvec{\theta }^{*})}{\partial \varvec{\theta }}\), \(\hat{I}=-\frac{1}{n} \frac{\partial ^{2} \log f({\varvec{X}}|\varvec{\theta })}{\partial \varvec{\theta }\partial \varvec{\theta }^{T}}|_{\varvec{\theta }=\hat{\varvec{\theta }}}\), \(\hat{H}=-\frac{1}{m}\frac{\partial ^{3} logf({\varvec{X}}|\varvec{\theta })}{\partial \varvec{\theta } \partial \varvec{\theta }^{T}\partial \varvec{\theta }}|_{\varvec{\theta }=\hat{\varvec{\theta }}}\), and \(\hat{\varvec{\theta }}\) denotes the MLE of \(\theta \) calculated with the full dataset. By noting that \(\frac{\varvec{\xi }}{\sqrt{n}}=\frac{1}{k}\sum _{j=1}^{k}\frac{\varvec{\xi }^{(j)}}{\sqrt{m}}\), \(\varvec{\nu }^{(j)}\) and \(\varvec{\nu }\) are O(1), \(m<n\), Eq. (4) in Theorem 1 is thereby verified:
$$\begin{aligned} E_{\tilde{\pi }} (\varvec{\theta })-E_{\pi } (\varvec{\theta })= & {} \frac{1}{k}\sum _{j=1}^{k}\frac{\varvec{\nu }^{(j)}}{n}+O_{p}(m^{-1})\\&-\frac{\varvec{\nu }}{n} +O_{p}(n^{-1})+O(n^{-2}) \\= & {} O(n^{-1})+O_{p}(m^{-1})+O(n^{-1})\\&+\,O_{p}(n^{-1})+O(n^{-2}) \\= & {} O_{p}(m^{-1}). \end{aligned}$$
The variance of each subposterior can be approximated as follows:
$$\begin{aligned} \mathrm{Var}_{\tilde{\pi }_{j}}(\varvec{\theta })=\frac{\hat{I}^{(j)-1}}{n}+O(n^{-2}). \end{aligned}$$
Therefore, the variance of the mixture distribution \(\tilde{\pi }(\varvec{\theta }|{\varvec{X}})\) is given by
$$\begin{aligned} \mathrm{Var}_{\tilde{\pi }}(\varvec{\theta })=\frac{1}{k}\sum _{j=1}^{k}\mathrm{Var}_{\tilde{\pi }_{j}}(\varvec{\theta }) =\frac{1}{k}\sum _{j=1}^{k}\frac{\hat{I}^{(j)-1}}{n}+O(n^{-2}). \end{aligned}$$
In addition, we have
$$\begin{aligned} \mathrm{Var}_{\pi }(\varvec{\theta })=\frac{\hat{I}^{-1}}{n}+O(n^{-2}). \end{aligned}$$
Since \(\hat{I}^{(j)}=I+o_{p}(1)\) and \(\hat{I}=I+o_{p}(1)\), we further have \(\hat{I}^{(j)-1}=I^{-1}+o_{p}(1)\) and \(\hat{I}^{-1}=I^{-1}+o_{p}(1)\). Equation (5) in Theorem 1 is thereby verified:
$$\begin{aligned} \mathrm{Var}_{\tilde{\pi }}(\varvec{\theta })-\mathrm{Var}_{\pi }(\varvec{\theta })= & {} \frac{1}{k}\sum _{j=1}^{k}\frac{\hat{I}^{(j)-1}}{n}-\frac{\hat{I}^{-1}}{n}+O(n^{-2})\\= & {} \frac{I^{-1}}{n}+o_{p}(n^{-1})-\frac{I^{-1}}{n}\\&+\,o_{p}(n^{-1})+O(n^{-2})\\= & {} o_{p}(n^{-1}). \end{aligned}$$
Finally, for the square of the Wasserstein distance of order 2, we have
$$\begin{aligned}&\bigg |d^{2}(\tilde{\pi }(\varvec{\theta }|{\varvec{X}}),\delta _{{\varvec{\theta }}^{*}})-d^{2}(\pi (\varvec{\theta }|{\varvec{X}}),\delta _{{\varvec{\theta }}^{*}})\bigg |\\&\quad =\bigg |\int _{\Theta }\Vert \varvec{\theta }- \varvec{\theta }^{*}\Vert _{2}^{2}\tilde{\pi }(\varvec{\theta }|{\varvec{X}})\mathrm{d}\varvec{\theta }\\&\quad \quad -\int _{\Theta }\Vert \varvec{\theta }- \varvec{\theta }^{*}\Vert _{2}^{2}\pi (\varvec{\theta }|{\varvec{X}})\mathrm{d}\varvec{\theta }\bigg |\\&\quad =\left| \left\| E_{\tilde{\pi }}(\varvec{\theta })-\varvec{\theta }^{*}\right\| _2^2+tr\left( \mathrm{Var}_{\tilde{\pi }}(\varvec{\theta }) \right) \right. \\&\quad \quad \left. -\left\| E_{\pi }(\varvec{\theta })-\varvec{\theta }^{*}\right\| _2^2 -tr\left( \mathrm{Var}_{\pi }(\varvec{\theta }) \right) \right| \\&\quad \le \left| \left\| E_{\tilde{\pi }}(\varvec{\theta })-E_{\pi }(\varvec{\theta })+E_{\pi }(\varvec{\theta })-\varvec{\theta }^{*}\right\| _2^2\right. \\&\left. \quad \quad -\left\| E_{\pi }(\varvec{\theta })-\varvec{\theta }^{*}\right\| _2^2\right| +|tr\left( \mathrm{Var}_{\tilde{\pi }}(\varvec{\theta }) -\mathrm{Var}_{\pi }(\varvec{\theta }) \right) |\\&\quad \le \Vert E_{\tilde{\pi }}(\varvec{\theta })-E_{\pi }(\varvec{\theta })\Vert _2^2+2\Vert E_{\tilde{\pi }}(\varvec{\theta })\\&\quad \quad -\,E_{\pi }(\varvec{\theta })\Vert _{2}\Vert E_{\pi }(\varvec{\theta })-\varvec{\theta }^{*}\Vert _{2}+o_{p}(n^{-1})\\&\quad =O_{p}(m^{-2})+2O_{p}(m^{-1})O_{p}(n^{-1/2})+o_{p}(n^{-1})\\&\quad =o_{p}(n^{-1}), \end{aligned}$$
where the last equality follows from the fact \(n=o(m^{2})\). This verifies Eq. (5) in Theorem 1.