## observed vs. complete in EM algorithm

While answering a question related with the EM  algorithm on X validated, I realised a global (or generic) feature of the (objective) E function, namely that

$E(\theta'|\theta)=\mathbb E_{\theta}[\log\,f_{X,Z}(x^\text{obs},Z|\theta')|X=x^\text{obs}]$

can always be written as

$\log\,f_X(x^\text{obs};\theta')+\mathbb E_{\theta}[\log\,f_{Z|X}(Z|x^\text{obs},\theta')|X=x^\text{obs}]$

therefore always includes the (log-) observed likelihood, at least in this formal representation. While the proof that EM is monotonous in the values of the observed likelihood uses this decomposition as well, in that

$\log\,f_X(x^\text{obs};\theta')=\log\,\mathbb E_{\theta}\left[\frac{f_{X,Z}(x^\text{obs},Z;\theta')}{f_{Z|X}(Z|x^\text{obs},\theta)}\big|X=x^\text{obs}\right]$

I wonder if the appearance of the actual target in the temporary target E(θ’|θ) can be exploited any further.

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