 then given by [$$\begin{aligned}      &\operatorname*{arg\,max}_{p_i, \alpha} \! \prod_{i = 0}^{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\!) 