# Convexity of Negative Log Likelihood involving Gamma function

## Problem

The negative log-likelihood of Dirichlet multinominal distribution is given by $$f(\alpha)=-L(\alpha)=\sum_{i=1}^n\sum_{j=1}^d \log\frac{\Gamma(\alpha_j)}{\Gamma(\alpha_j+\mathbf{x}_{ij})} -\sum_{i=1}^n\log \frac{\Gamma(\sum_{j=1}^d\alpha_j)}{\Gamma(m+\sum_{j=1}^d\alpha_j)} +\text{constant}$$ where $$\mathbf{x}_{ij}\in \{0, 1\}$$ and $$\alpha\in \mathbb{R}^d, \alpha_j >0$$, $$m$$ is a constant.

Then determined whether $$f(\alpha)$$ is convex or not.

## What I Have Done

It is natural to compute the Hessian $$\nabla_\alpha^2 f(\alpha)$$. However, when I checked Wikipedia, the derivative of Gamma function looks much too complicated.

Then I tried to find some counter examples for $$d=1$$ case. It seems that this function is indeed convex according to several simulations I carried out.

I am wondering if we could analytically determine the convexity of this function.

$$\Gamma(x+1) = x\Gamma(x)$$ and $$u\mapsto- \log(a+u)$$ is convex.