Continuous Hidden Variables CEM

For completeness, we also derive the CEM algorithm for the continuous hidden variable case. In Equation 7.12 we present the analog to the above Q function bound on conditional likelihood for continuous hidden variables (${\bf m}$). The derivation mirrors that of the discrete hidden variable case except that summations over the hidden data are replaced with integration. We apply Jensen's inequality and the linear upper bound on the logarithm to obtain a Q function as before.

 

$$\begin{array}{l} \begin{array}{lll} \Delta l^c & = & \log p({\cal... ...{\bf m} ,{\bf x}_i , \vert \Theta^{(t-1)}) {\bf dm}} + 1 \end{array}\end{array}$$ (7.12)