documentclass{article}
usepackage{showframe}
renewcommand*{ShowFrameLinethickness}{0.3pt}
usepackage[utf8]{inputenc}%
usepackage{mathtools}

renewcommand{P}{mathcal{P}}
date{May 2019}

begin{document}

begin{align*}
    &H_{n + 1}= \
    &-sum_{x_{n + 1}} P(x_{n + 1}mid x_n) smashoperator{sum_{x_1, x_2, dots, x_{n}}} P(x_1, x_2, dots, x_n) log(P(x_1, x_2, dots, x_n)P(x_{n + 1} mid x_n)) =\
    &-begin{multlined}smash[b]{sum_{x_{n + 1}}} P(x_{n + 1}mid x_n) smash[b]{smashoperator{sum_{x_1, x_2, dots, x_{n}}}} P(x_1, x_2, dots, x_n)bigl (log(P(x_1, x_2, dots, x_n)) \+ log(P(x_{n + 1}mid x_n))bigr) =
    end{multlined}\
    &-sum_{x_{n + 1}} P(x_{n + 1}mid x_n) H_n -smashoperator{ sum_{x_1, x_2, dots, x_{n + 1}}} P(x_1, x_2, dots, x_{n + 1}) log(P(x_{n + 1}mid x_n))) =
end{align*}

end{document}