I'm working with the paper of Wu - On the Convergence Properties of the EM Algorithm (1983). In a theorem he says that the EM-algorithm converges to a stationary point if the conditional expectation $Q(\phi^{'} | \phi)$ is both continuous in $\phi^{'}$ and $\phi$. Furthermore he writes that this condition is always fulfilled for curved exponential families. So, given is the density of a curved exponential family $f(x| \phi) = b(x) \exp(\phi^{T} t(x))/a(\phi)$, where the parameters $\phi$ lie in a compact submanifold $\Omega_0$ of the r-dimensional convex region $\Omega = \{\phi |\int b(x) \exp(\phi^{T} t(x)) \, dx < \infty \}.$ \
Then $Q(\phi^{'} | \phi) := E[\log(f(x|\phi^{'})) | y, \phi] = - \log(a(\phi^{'})) + E[\log(b(x)) | y, \phi] + \phi'^{\,T} E[t(x) | y, \phi].$ Wu claims that the continuity follows from the compactness of $\Omega_0$ and properties of the exponential family. I think I can say that $- \log(a(\phi^{'}))$ is continuous as $- \log$ is continuous and $\Omega_0$ is compact, so that $- \log(a(\phi^{'}))$ takes its maximum on this compact set which means that this part must be continuous. For the two other terms I should need the exponential family properties. But I don't know any properties, which give me useful information about $b(x)$ and $t(x)$. Can I somehow show that they have to be continuous? The definition of exponential families say that they have to be only measureable... Thanks!
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(explanation of the EM-algorithm: E-Step: Determine $Q(\phi | \phi_p)$, M-Step: Choose $\phi_{p+1}$ to be any value of the parameter space which maximizes $Q(\phi | \phi_p)$. (Repeat these steps until $Q(\phi_{p+1} | \phi_p)$ convergence against a $Q(\phi^* | \phi^*)$))