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Math Help - Markov Chain Conditional Expectation

  1. #1
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    Markov Chain Conditional Expectation

    Hello, in relation to Markov chains, could you please clarify the following equations:



    In particular, could you please expand on why the first line is equal. Surely from , along with the first equation, this implies that:



    I just don't see why they are all equal. Please could you provide some intuition on this. Thanks
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  2. #2
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    Quote Originally Posted by tunaaa View Post
    Hello, in relation to Markov chains, could you please clarify the following equations:



    In particular, could you please expand on why the first line is equal. Surely from , along with the first equation, this implies that:



    I just don't see why they are all equal. Please could you provide some intuition on this. Thanks
    The first two are equal by definition. The actual meaning of conditioning is introducing a new measure i.e. \mathbb{P}^1(\cdot)=\mathbb{P}(\cdot | A). When you take expectation with respect to that, you indicate it by \mathbb{E}[\cdot|A].

    In your calculation your indicator function is meaningless. What is the set operation A | B?
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