Abstract: Algebraic statistics identifies probability distributions with points and studies equations vanishing on these points. The first half of the talk will be on algebraic approaches to discrete and Gaussian models, with focus on when they have a toric structure. We will also briefly discuss the maximum likelihood degree of such models. The second part will concern analyzing quantitative and qualitative data of algebraic objects in the infinite setting, with a view toward applications. To do so, we will ask for an unusual involvement of formal languages and finite automata.