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Problem
In this exercise, we will provide a characterization of when two distributions P1 and P2 will have the same M-projection.
a. Let P1 and P2 be two distribution over X, and let Q be an exponential family defined by the functions τ (ξ) and t(θ). If EP1[τ(X)] = EP2[τ (X )], then the M-projection of P1 and P2 onto Q is identical.
b. Now, show that if the function ess(θ) is invertible, then we can prove the converse, showing that the M-projection of P1 and P2 is identical only if EP1 [τ (X)] = EP2[τ (X )]. Conclude that this is the case for linear exponential families.
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