1
$\begingroup$

Eq 2 of this paper states this integral: \begin{align*} r^{-\beta} = \frac{1}{\Gamma(\beta)}\int_{-\infty}^{\infty} e^{-re^t + \beta t} dt \end{align*} Is there is name for this identity, or the class of identities of this form? And are there generalizations? I am dealing with a problem that requires computation of \begin{align*} \int_{\mathbb{R}^p}\exp\left(-\left\{\sum_{i=1}^{n}\lambda_i e^{\boldsymbol{\theta}_i^\intercal \mathbf{t}}\right\} + \boldsymbol{\alpha}^\intercal \mathbf{t} - \frac{1}{2}\mathbf{t}^\intercal \Gamma \mathbf{t}\right) d\mathbf{t} \end{align*} where $\lambda_i \ge 0$ and $\boldsymbol{\theta}_i, \boldsymbol{\alpha}, \mathbf{t} \in \mathbb{R}^p$ and $\Gamma \in \mathbf{R}^{p\times p}$. I'm willing to also settle for a very good approximation, if such exists.

$\endgroup$
2
  • 3
    $\begingroup$ The first formula can be obtained by a change of variable from the integral definition of $\Gamma(\beta)$ (let $re^t = x$). But putting a $t^2$ term in the exponent probably makes things much harder even before you generalize to $n$ dimensions. $\endgroup$ Commented Dec 18, 2017 at 3:53
  • 1
    $\begingroup$ You can use Markov Chain Monte Carlo to estimate this integral — at the price of Monte Carlo statistical error which might be very large, but is independent of p. $\endgroup$ Commented Dec 19, 2017 at 12:42

0

You must log in to answer this question.