Is marginal constraints equivalent to linear constraints?

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Marginal constraints on probability distributions can be expressed as linear constraints through the use of expected values. This involves formulating the problem such that the expected values of certain functions equal specified constants. The discussion seeks clarity on the equivalence of marginal constraints and linear constraints in probability distributions. Understanding this relationship is crucial for effectively modeling and analyzing probability spaces. The inquiry highlights the need for a mathematical framework to transition between these two types of constraints.
Edwinkumar
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If I have a set of Probability distributions on a product space with marginal constraints, is there any way to (how to) express the same as a linear family of PD's ( i.e. all P s.t. E_P[ f_i] =a_i for some f_i, a_i )
 
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Could someone answer this?
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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