Solved – Is log partition function guaranteed to be convex

In a paper by Wainwright and Jordan on page 62 it mentions that a log partition function is always convex. This is done by showing that the second derivative of the log partition function is the covariance matrix of the sufficient statistic vector $phi(x)$.

Question is, is the covariance function guaranteed to be positive semi definite? With respect to the von-mises fisher distribution $proptoexp(kappamu^Tx)$ I get the second derivative to be w.r.t. $kappa$:


but, if I take the fact that the log partition function is
$$y=-log I_{nu}(kappa)+nulogkappa+(nu+1)log(2pi)\
where $nu=d/2-1$ and plot the second derivative (for d=2), I get,

enter image description here
which is negative implying its concave.

Two possible explanations: 1. Covariance matrix is not necessarily positive definite. 2. numerical errors in plotting log bessel function

so which assumption is wrong?

Your expression for y' is wrong. Use $2I_nu'=I_{nu+1}+I_{nu-1}.$

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