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Can anybody help me in this question? How to derive the expectation of log(1/X) when X follows inverse Gaussian distribution?

Found out a type of approximation for $\mathbb{E}[\log(X)]$ while $X \sim \mathcal{N}(\mu,\sigma^2)$ in this link (Expected value and variance of log(a)). Can this be extended to the present case to get an approximate solution?

Any help will be appreciated.

Thanks in advance.

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Sandipan Karmakar
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  • How does such a question arise? Is this for a class, for your own study, or something else? – Glen_b Jul 21 '15 at 00:16
  • This is coming in finding the lower bound for Variational Inference for a LASSO model. Can you help??? – Sandipan Karmakar Jul 21 '15 at 00:23
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    Okay, then how does the question of the lower bound for variational inference arise? Is this for a class, for your own study, or something else? – Glen_b Jul 21 '15 at 00:25
  • This is for my research... – Sandipan Karmakar Jul 21 '15 at 00:27
  • Thanks. Do you need an exact analytical answer, or will approximations suffice? – Glen_b Jul 21 '15 at 00:30
  • I have got an approximate general solution for expectation of $\log(x)$ using Taylor Series approximation, in this forum only. I am not sure whether it works as approximation while applying it to this case. Can you provide me both of them so that I can verify my results. – Sandipan Karmakar Jul 21 '15 at 00:33
  • Perhaps not of direct help but the distribution of 1/X is given in Tweedie (1957) "Statistical Properties of Inverse Gaussian Distributions. I", *Ann. Math. Stat* **28**:2, 362-377 -- see [here](http://projecteuclid.org/download/pdf_1/euclid.aoms/1177706964). – Glen_b Jul 21 '15 at 00:34
  • Thank you for the reference you provided. The approximation for $\mathbb{E}[\log X]$ is given here in this page: http://stats.stackexchange.com/questions/57715/expected-value-and-variance-of-loga/57766#57766 – Sandipan Karmakar Jul 21 '15 at 00:38
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    It might be good to include that link (and your concerns about its applicability) in your question. – Glen_b Jul 21 '15 at 00:41
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    $\log(1/X)=-\log(X)$, did you consider that? – mpiktas Jul 21 '15 at 07:33

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