We simplify further by rewriting it as
$$ p(\boldsymbol{X};\boldsymbol{\Theta})=\frac{1}{Z(\boldsymbol{\Theta})}\prod_{x_i\in \boldsymbol{X}}f(x_i;\boldsymbol{\Theta}), $$where we used \( p(x_i;\boldsymbol{\Theta}) = \sum_{h_j\in \boldsymbol{H}}f(x_i,h_j;\boldsymbol{\Theta}) \). The optimization problem is then
$$ {\displaystyle \mathrm{arg} \hspace{0.1cm}\max_{\boldsymbol{\boldsymbol{\Theta}}\in {\mathbb{R}}^{p}}} \hspace{0.1cm}p(\boldsymbol{X};\boldsymbol{\Theta}). $$