my question comes from Ziebart's thesis.
The theorem what I want to prove is Thm 5.8 from his thesis as following:
$E_{P(X^T,Y^T)}[F(X,Y)] = c \quad \quad \quad $for some function F and given c
where $H(Y^T||X^T) = E_{X^T,Y^T}[\Sigma_t-log P(Y_t|X_{1:t},Y_{1:t-1})]$ and $P(X^T,Y^T) = \Pi_t P(Y_t|X_{1:t},Y_{1:t-1})P(X_t|X_{1:t-1},Y_{1:t-1})$ satisfies.
Also, $P(X_t|X_{1:t-1},Y_{1:t-1}) $ is given. And $Y^T=[Y_1,Y_2,...Y_T], X^T=[X_1,X_2,...X_T], X_{1:t} = [X_1,..X_t], Y_{1:t} = [Y_1, .. Y_t]$
what I want to show is that that above problem is convex optimization and when slater's condition holds, strong duality holds.
And the proof is given in the Ziebart's thesis. He said that we can see the problem as a function of $P(Y^T||X^T) = \Pi_t P(Y_t|X_{1:t},Y_{1:t-1})$.
Then the formation becomes as following:
$E_{P(X^T,Y^T)}[F(X,Y)] = c$
The second constraint appears to satisfy $P(Y^T||X^T) = \Pi_t P(Y_t|X_{1:t},Y_{1:t-1})$.
Now, Ziebart said that the second problem is a convex problem and strong duality holds with respect to $P(Y^T||X^T)$ because the objective is convex and all constraints are affine. And the proof ends.
But what we want to know is if the first problem with respect to $P(Y_t|X_{1:t},Y_{1:t-1})$ satisfies the theorem.
So, can I conclude the problem is a convex problem and strong duality holds if the equivalent but with respect to the other(may be substituted) variable is a convex problem and strong duality holds??