visit this post https://math.stackexchange.com/a/2279318/446173
due to this answer can we infer this equation? if yes how can we prove this???
$$I((X_1,\dots X_n);(Y_1,\dots Y_n))\ge\sum_{i=1}^n H(Y_i|Y_1\dots Y_{i-1})-H(Y_i|Y_1\dots Y_{i-2},X_1\dots X_n)$$
This inequality holds because $$\begin{split}I(\textbf{X};\textbf{Y})&=H(\textbf{Y})-H(\textbf{Y}|\textbf{X})=\sum_{i=1}^nH(Y_i|Y^{i-1})-\sum_{i=1}^nH(Y_i|Y^{i-1},\textbf{X})\\ \end{split}$$ where $\textbf X = X_1,X_2,...,X_n$, and $\textbf Y = Y_1,Y_2,...,Y_n$. Then notice that because conditioning decreases entropy, $$\begin{split}H(Y_i|Y^{i-1},\textbf{X})\leq H(Y_i|Y^{i-2},\textbf{X})\\ \end{split}$$ so the original inequality is true.