given the prior distribution of $\mathbf{a}=[a_1,\ldots,a_K]^T$ as \begin{equation} 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}} \end{equation} I am looking into the expectation of a exponential function of a nonlinear function of $\mathbf{a}$ as

\begin{equation} 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} \end{equation}

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

  • 1
    $\begingroup$ How is this related to signal processing? It may be a better fit in one of the math-oriented SE sites. $\endgroup$
    – MBaz
    Apr 28, 2022 at 21:44
  • 1
    $\begingroup$ 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. $\endgroup$
    – AlexTP
    Apr 29, 2022 at 7:56
  • $\begingroup$ $\mathbf{R}_a$ and $\mathbf{R}_n$ are covariance matrices and therefore positive definite. $\endgroup$
    – zahraesb
    Apr 29, 2022 at 15:03


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