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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?

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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 $\phi$.

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Is the sign always constant? –  mtiano Oct 23 '13 at 23:21
    
@mtiano see edit. –  Alex R. Oct 23 '13 at 23:24

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