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 $\propto\exp(\kappa\mu^Tx)$ I get the second derivative to be w.r.t. $\kappa$:
$$\mu^T\left(E(xx^T)-E(x)E(x)^T\right)\mu$$
but, if I take the fact that the log partition function is
$$y=-\log I_{\nu}(\kappa)+\nu\log\kappa+(\nu+1)\log(2\pi)\\
y'=-\frac{I_{\nu+1}(\kappa)}{I_{\nu}(\kappa)}+\nu\frac{1}{\kappa}\\
y''=\frac{I_{\nu+1}^2(\kappa)-I_{\nu}^2(\kappa)}{I_{\nu}^2(\kappa)}+\frac{2\nu+1}{\kappa}\frac{I_{\nu+1}(\kappa)}{I_{\nu}(\kappa)}
$$
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?
Best Answer
Your expression for y' is wrong. Use $2I_\nu'=I_{\nu+1}+I_{\nu-1}.$