Sorry for not responding for a long time. I didn't mean to be snarky above, sorry about that too. I wrote a bit more here, but wasn't satisfied with how it wen

Sorry for not responding for a long time. I didn't mean to be snarky above, sorry about that too. I wrote a bit more here, but wasn't satisfied with how it went so its no more. It was anyways mainly about Sturmfels' programme of reasoning about graphical models with algebraic geometry and how I see it and less TO the paper, so it was probably unsubstantial not only badly written. By this time I'm sure you already know what program of algebraic statistics is anyways and surely can judge it for yourself. This paper fits it and is not directly consumable by machine learning just yet. Other parts of AS are. To actually use either parts the book I mentioned is minimum anyways and will clarify alot. Even if summary of the paper or AS itself is to be presented to a prof, there is no way around this book. The shortest intro to AG is by M. Reid, it freely builds on commutative algebra (short intro on J.S. Milne page at jmilne.org) that builds on abstract algebra. I haven't read the new book by Sturmfels and Sullivant but I'm half-sure it just assumes AG nor doesn't explain statistics.