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?
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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