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I am studying expectation maximisation and I encountered the following conditional exception term.

$E[l(\theta) | X, \theta_{t-1}]$

Where:

$l(\theta) = \sum_{i=0}^N log p(x_i, z_i | \theta)$

This equation is accompanied by the following text:

“The expectation is taken wrt to $\theta_{t-i}$ and the observed data X

I am trying to understand how to interpret the conditional expectation. specifically this statement above. I.e what does it mean to take an expectation with respect to two variables.

To me an expectation is computed by taking the product of The Probability density of an outcome along with the outcome itself and then summing over all possible outcomes. In this context what does taking the “expectation with respect to...” mean exactly ?

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    $\begingroup$ Where did you encounter this? $\endgroup$ – Iosif Pinelis Nov 28 at 23:29
  • $\begingroup$ Probabilistic Machine Learning, Murphy Et al $\endgroup$ – SFD Nov 29 at 0:51
  • $\begingroup$ "With respect to …" introduces the random vector (or variable) whose probability density weights the sum/integral of interest.. $\endgroup$ – Mickybo Yakari Nov 29 at 18:11