## 3 The Epsom-Hyb Estimator  
In the discourse that follows, we propose some key modifications to the hybrid algorithm to develop a novel estimator better suited for tackling higher-dimensional problems. For the initial development, we do not restrict our analysis to posterior distributions and illustrate the modified algorithm in the general case of log-concave target densities. Our additional assumption about the shape of \(\Psi\) is employed only to ensure that the Hessian is positive definite everywhere and can, in fact, be relaxed to unimodal densities that are approximately log-concave in a suitable neighborhood around the mode. This assumption is widely satisfied by Bayesian posteriors in regular parametric models, and due to the Bernstein-von Mises phenomenon, most regular posterior distributions are approximately log-concave with sufficiently large sample size [28].  
As before, we work with a compactification of the parameter space, for which we have a dyadic partition \(\mathcal{A}\) obtained through the same tree-fitting procedure as before. However,instead of the constant approximation to \(\Psi\) within each partition set \(A_{k}\) in \(\mathcal{A}\), we first retain the general estimator given in Equation (2),  
\[\mathcal{Z}\approx\int_{A}e^{-\Psi(u)}\,du\approx\int_{A}e^{-\widehat{\Psi}(u)} \,du=\sum_{k=1}^{K}\int_{A_{k}}e^{-\widehat{\Psi}_{k}(u)}\,du. \tag{4}\]  
Below, we highlight three fundamental features of the EPSOM-Hyb algorithm that distinguish it from the hybrid algorithm, followed by detailed discussions of each step. See Algorithm 1 for a formal statement of the EPSOM-Hyb estimation procedure.  
* In Equation (2), we take \(\widehat{\Psi}_{k}\) to be a local approximation to \(\Psi\) constructed using a second-order [PERSON] expansion of \(\Psi\) around a representative point \(u_{k}\in A_{k}\).
* The second-order Taylor approximation complicates the calculation of the integrals in Equation (4). We address the intractability of the resulting integral using an expectation propagation (EP) algorithm that targets high-dimensional Gaussian integrals.
* We exploit the unimodality of \(\gamma\) to identify suitable points within each partition set around which we perform the Taylor expansion required for \(\widehat{\Psi}_{k}\).  
```
Input : Sampler for \(\gamma\), functions for evaluating \(\Psi\), \(\
abla\Psi\), \(\
abla^{2}\Psi\). Output : Estimate of the logarithm of the normalizing constant of \(\gamma\).  Sample \(u_{1},\ldots,u_{\gamma}\sim\gamma\).  Fit a CART model, \(\mathcal{T}\), to \((u_{1},\Psi(u_{1})),\ldots,(u_{\gamma},\Psi(u_{\gamma}))\).  Extract the partition \(\mathcal{A}=\{A_{1},\ldots,A_{K}\}\) from \(\mathcal{T}\) of the bounding box \(A\) of \(\mathcal{U}\).  Calculate the global mode, \(u_{0}\), of \(\gamma\). for\(k\in\{1,\ldots,K\}\)do \(u_{k}\leftarrow\operatorname*{argmin}_{u\in A_{k}}||u-u_{0}||_{1}\). \(\lambda_{k}\leftarrow\
abla\Psi(u_{k})\). \(H_{k}\leftarrow\
abla^{2}\Psi(u_{k})\). \(C_{k}\leftarrow(2\pi)^{d/2}|H_{k}|^{-1/2}\exp\Bigl{(}\frac{1}{2}\Bigl{(}u_{k} ^{\prime}H_{k}^{-1}u_{k}-2\lambda_{k}^{\prime}u_{k}+\lambda_{k}^{\prime}H_{k} ^{-1}\lambda_{k}\Bigr{)}\Bigr{)}\). \(b_{k}\gets H_{k}u_{k}-\lambda_{k}\). \(G_{k}\leftarrow\int_{A_{k}}\mathcal{N}\Bigl{(}u\mid H_{k}^{-1}b_{k},H_{k}^{-1 }\Bigr{)}\,du\). \(\widehat{\mathcal{Z}}_{k}\leftarrow\mathbb{C}_{k}\cdot G_{k}\). end return\(\log\widehat{\mathcal{Z}}=\texttt{log-sum-exp}\bigl{(}\log\widehat{\mathcal{Z}}_{1}, \ldots,\log\widehat{\mathcal{Z}}_{K}\bigr{)}\).
```  
**Algorithm 1**EPSOM-Hyb