Probability chain rule: Does $p(x_{1},...,x_{n}) = p(x_{1},...,x_{n-1}|x_{n})p(x_{n})$?

Question

Typically the chain rule is written $p(x_{1},...,x_{n}) = p(x_{1}|x_{2},...,x_{n})p(x_{2}|x_{3},...,x_{n})...p(x_{n-1}|x_{n})p(x_{n})$. While it seems logical that I should be able to write it the way noted in the title, I didn't really know how to deduce/induce this formally.

Answer 1

That was derived by induction. You have $p(x_1,\ldots,x_n)=p(x_1\,|\,x_2,\ldots,x_n)p(x_2,\ldots,x_n)$ and you keep on doing that, you will obtain the chain rule. As for the title, it is similar to this, just by definition of conditional probability.