Recently, I have encountered the following integral solution problem in my research. Because it involves special functions, I cannot successfully solve it in calculation.
$$\mathbb{E}_{Z_{1},Z_{2}} \log(I_{n_{d}-1}(2||y|||\hat{h}|\sqrt{n_{d}}\rho_{d}))$$
where $I_{n_{d}-1}$ the modified Bessel function of the first kind with $n_{d}-1$ degrees of freedom, $Z_{1}$ obeys the $\Gamma(1,1)$ distribution and $Z_{2}$ obeys the $\Gamma(n_{d}-1,1)$ distribution, $||y||^{2}$ equals to the $\left(1+n_{\mathrm{d}}\alpha_{k}\right) Z_{1}+Z_{2}$ in the distribution sense, $\hat{h},n_{d},\rho_{d}$ are all known.
At present, my preliminary idea is to deflate the above expectations according to Markov inequality and put the expectations into logarithm and there is no idea about how to deal with the integral of Bessel function.