# n-dimentional integral over Multivariate Gaussian

given the prior distribution of $$\mathbf{a}=[a_1,\ldots,a_K]^T$$ as $$$$p_{\mathbf{a}}(\mathbf{a})= \frac{1}{\pi^K \det{\mathbf{R_{\mathbf{a}}}}} e^{\mathbf{a}^H \mathbf{R}_{\mathbf{a}}^{-1} \mathbf{a}}$$$$ I am looking into the expectation of a exponential function of a nonlinear function of $$\mathbf{a}$$ as

$$$$E_{\mathbf{a}}\{e^{\mathbf{a}^H\mathbf{z}+\mathbf{z}^H\mathbf{a}-\mathbf{a}^H\mathbf{R}_{n}^{-1}\mathbf{a}}\}=c_1\int_{\mathbf{a}} e^{\mathbf{a}^H\mathbf{z}+\mathbf{z}^H\mathbf{a}-\mathbf{a}^H\mathbf{R}_{n}^{-1}\mathbf{a}} p_{\mathbf{a}}(\mathbf{a}) d\mathbf{a}$$$$

where $$\mathbf{R}_a$$ and $$\mathbf{R}_n$$ are covariance matrices and therefore positive definite. Is there any way to compute this integral?

• How is this related to signal processing? It may be a better fit in one of the math-oriented SE sites.
– MBaz
Apr 28, 2022 at 21:44
• What is $\mathbf{z}$? What is $\mathbf{R}_n$ and is it different from $\mathbf{R}_a$? You may want to check Inverse transform sampling. Apr 29, 2022 at 7:56
• $\mathbf{R}_a$ and $\mathbf{R}_n$ are covariance matrices and therefore positive definite. Apr 29, 2022 at 15:03