the only constraint on the observed distribution is for NDE to be zero. Let $p(Y | M, A, X), p(M | A, X), p(A | X)$ be parameterized by $\alpha_y, \alpha_m, \alpha_a$, respectively. The direct effect can then be identified by ${\mathbb{E}}_x[ m(X; \alpha)]$, where [ $$\begin{aligned}
        m(X; \alpha)& = \sum_{M} \Big\{{\mathbb{E}}[Y | A = 1, M, X; \alpha_y] -  \label{eq:est_NDE} \\
        &\qquad  {\mathbb{E}}[Y | A = 0, M, X; \alpha_y]\Big\} p(M | A = 0, X; \alpha_m). 
        \nonumber 
    \end{aligned}$$ ]{}The *profile empirical likelihood ratio* parameters ($\{p_i, \widehat{ \alpha} \}^{opt}$) are then given by [$$\begin{aligned}
     &\operatorname*{arg\,max}_{p_i, \alpha} \! \prod_{i = 1}^{n}  p_i  p(Y_i | M_i,\! A_i,\! X_i;  \!\alpha_y\!) p(\!M_i | A_i,\! X_i;  \!\alpha_m\!) p(\!A | X_i;  \!\alpha_a\!) \nonumber \\
     &\qquad \text{such that  } \quad \sum_{i=1}^n p_i = 1, \quad \sum_{i=1}^n p_i \ m(X_i; \alpha) = 0
    \label{eq:hybrid_fair}\end{aligned}$$ ]{}The above optimization problem involves a semi-parametric *hybrid* likelihood [@OwenEL], that contains both nonparametric and parametric terms. In order to solve the above optimization problem (formulated on both ${ \alpha}$ and $p_i$ parameters), we can apply the Lagrange multiplier method and solve its dual form (formulated on both ${ \alpha}$ and the Lagrange multipliers); see the appendix for more details. Empirical likelihood methods 