Let x be a discrete real random variable, and suppose that p and q are possibly correlated discrete r.v.'s that may shed light on x.
Prove or (unlikely) disprove: If p and q are independent, conditional on x, then E[x|p=p_0] >= E[x] ===> E[x|p=p_0] >= E[E[x|q]|p=p_0] >= E[x].
Note that this is not the law of iterated expectations, but close.
Thanks for any help,
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