# 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\$:

\$\$mu^Tleft(E(xx^T)-E(x)E(x)^Tright)mu\$\$

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

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?

Contents

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