In this exercise we will provide a characterization of when

Question

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 E_P1[τ(X)] = E_P2[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 E_P1 [τ (X)] = E_P2[τ (X )]. Conclude that this is the case for linear exponential families.