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I need your advice in integrating $\int ln(f(x)) dx = \int x \frac{f'(x)}{f(x)} dx$, where $f(x)$ is a probability density function.

How can I solve it?

user1721713
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1 Answers1

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Hint: $$\frac{f'(x)}{f(x)} = \frac{d}{dx} \log f(x)$$

So use that as "dv" in integration by parts.

John Hughes
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  • $\int ln(f(x)) dx$ was my starting point and I am interested in that how can I leave the integral. – user1721713 May 29 '16 at 00:58
  • Suppose that $g$ is any function you know to not be integrable in elementary terms. Let $f(x) = e^{g(x)}$. Now you have $\int \log f(x) ~dx = \int g(x) dx$, which is hopeless. So there's not going to be any "general" answer for integrating the log of a density. – John Hughes May 29 '16 at 10:05
  • And maybe could I estimate it? As I see it is similar to the score function. https://en.wikipedia.org/wiki/Score_%28statistics%29 – user1721713 May 29 '16 at 10:29
  • Sure, you can always estimate an integral. Whether the estimate will be tight enough to be worthwhile is a matter of luck and skill. Shakespeare put it well. GLENDOWER: I can call spirits from the vasty deep. HOTSPUR: Why, so can I, or so can any man; But will they come when you do call for them? – John Hughes May 29 '16 at 12:20