I'm reading An Introduction to Statistical Learning book and I've been struggling to understand this product notation. The usual form was $\prod_{i=1}^nx_i$.
What I don't understand is the subscript $i:y_{i}=1$. Is $i$ a variable or $y_{i}$? or both. What are they ranges? $$ l(B_0, B_1) = \prod_{i:y_{i}=1} p(x_i) \prod_{j:y_{j}=0} (1 - p(x_j)) $$
I presume the intention was to make the product over the $i$-s (resp. $j$-s) such that $y_i=1$ (resp. $y_{j}=0$). Therefore, those might be mistypings of LaTeX for $$\prod_{\{i\,:\, y_i=1\}}p(x_i)\quad\text{and}\quad\prod_{\{j\,:\, y_j=0\}}(1-p(x_j))$$ where the parenthesis did not appear because the person who wrote it has typed
{and}instead of\{and\}. However, even leaving just "$i:y_i=1$" and "$j:y_j=0$" might be a conscious choice of notation to the same end.