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Kay (Fundamentals of Statistical Signal Processing) defines the curvature of a log-likelihood function to be the "negative of the second derivative of the logarithm of the likelihood function at its peak".

I haven't come across this definition before, and Google hasn't been particularly forthcoming; is this a standard definition, or something Kay has introduced himself?

This is in the context of a derivation of the CRLB.

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The first derivative indicates the slope of the curve at a given point. If you take the derivative again (ie, you take the second derivative) you have the rate of change of that slope.

If the slope changes slowly (or does not change at all) you have something that seems straight, while if the slope changes abruptly, you have a curve that twists a lot.

At least, that's how I get an intuition of the relationship between curvature and second derivaitve. I hope this helped!

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