[Math] log partition function of exponential family

Question

In an exponential family $$p_{\theta}(x)=\exp \left(h(x)+\sum\limits_{i=1}^s \theta_iT_i(x) – \phi(\theta) \right) $$ is the log partition function $$ \phi(\theta)=\log \int \exp \left(h(x)+\sum\limits_{i=1}^s \theta_iT_i(x) \right)$$
always positive?

Answer 1

No. Take $f(x)=\lambda e^{-\lambda x}$, the exponential distribution. Then $\phi=-\log(\lambda)$ and $\lambda>1$ gives you a negative $\phi$ and $0<\lambda<1$ gives you a positive $\phi$.