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# Consistency of MLE proof watch

1. During the proof there is this step:

How do we get from one part to the next?
I suspect it's part of how we can define the expectation but why do we multiply by the PDF with the 'true' theta rather than the one with any theta?
2. (Original post by DQd)
During the proof there is this step:

How do we get from one part to the next?
I suspect it's part of how we can define the expectation but why do we multiply by the PDF with the 'true' theta rather than the one with any theta?
You've pretty much got the answer already - on the left hand side of the equation, that expectation operator is relative to the true parameter value - it's really . The result follows immediately from the definition of expectation.

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Updated: February 21, 2018
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