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I know of a proof for this which involves finding matrix of second derivatives (Hessian) for the given expression and proving that it is negative semi definite. But that is quite sophisticated for my use. I need to prove it using the fact that the sum of concave functions is a concave function (or another easier method). How do I go about it?

Formula of logistic regression for reference

Useful link- https://homes.cs.washington.edu/~marcotcr/blog/concavity/

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closed as off-topic by Henrik, Jendrik Stelzner, Delta-u, José Carlos Santos, Gibbs Sep 6 '18 at 19:15

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Everything you have in sight there are linear (affine) functions, their sums and composition with the function $s(x)=\log(1+e^x)$. So all you need to show is that $s$ is convex, which is a simple exercise in one variable.

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