4. In the acute triangle $\triangle A B C$, if
$$
\sin A=2 \sin B \cdot \sin C,
$$

then the minimum value of $\tan A+2 \tan B \cdot \tan C+\tan A \cdot \tan B \cdot \tan C$ is $\qquad$.
4. 16 .
$$
\begin{array}{l}
\text { Given } \sin A=2 \sin B \cdot \sin C \\
\Rightarrow \tan B+\tan C=2 \tan B \cdot \tan C \\
\Rightarrow \tan A \cdot \tan B \cdot \tan C=\tan A+\tan B+\tan C \\
\quad=\tan A+2 \tan B \cdot \tan C \\
\geqslant 2 \sqrt{2 \tan A \cdot \tan B \cdot \tan C} .
\end{array}
$$

Therefore, $\tan A+2 \tan B \cdot \tan C+\tan A \cdot \tan B \cdot \tan C$
$$
\geqslant 16 \text {, }
$$

When $\tan A=4, \tan B=2+\sqrt{2}, \tan C=2-\sqrt{2}$, the above expression achieves its minimum value.