(\[eq:est\_NDE\]), and ${\mathbb{E}}[Y | A, M, X;  \alpha_y]$ is given in (\[eq:reparam\_NDE\]). Assuming $p(X_i = x_i) = p_i$ as in (\[eq:reparam\_NDE\]), $m(X;  \alpha)$ will be a function of $p_i$s as well. The *profile empirical likelihood ratio* ($\{p_i, \widehat{  \alpha} \}^{opt} $) in this setting is then given by [$$\begin{aligned}
     &\operatorname*{arg\,max}_{p_i, \alpha} \! \prod_{i = 1}^{n}  p_i  p(Y_i | M_i,\! A_i,\! X_i;  \!\alpha_y\!) p(\!M_i | A_i,\! X_i;  \!\alpha_m\!) p(\!A | X_i;  \!\alpha_a\!)
     \nonumber \\ 
    & \quad \text{such that  } \quad \sum_{i=1}^n p_i = 1, \quad \sum_{i=1}^n p_i \ m(X_i; p_i,   \alpha) = 0
    \label{eq:hybrid_reparam}\end{aligned}$$ ]{}Unlike the constrained optimization problem in (\[eq:hybrid\_fair\]), it is not straightforward to find the dual form of the optimization problem in (\[eq:hybrid\_reparam\]), which is a standard approach for solving such problems in the empirical likelihood literature. The reason is that $p_i$ appears in multiple places in the constraint corresponding to setting PSE to zero; that is $m(X; \alpha)$ is now a function of both $\alpha$ and $p_i$’s. As an alternative, we provide a heuristic approach for optimizing (\[eq:hybrid\_reparam\]) via an iterative procedure that starts with initialization of $\alpha$ and $p_i$s, and at the $k$th iteration updates the values for $\alpha^{k}$ and $p^{k}_i$s by treating $m(X_i; p_i, \  