From Stanford's admissions page:
My mathematics background isn't that strong. Should I still apply?
A strong mathematics background, especially in probability, statistics and linear algebra, is important in the admission process. However, it is not the only factor that determines which applicants are admitted. You may consider strengthening your math background and applying later, or just hope that factors such as breadth or research experience will compensate for a relatively weak math background.
My math background isn't "weak" per se, but I assume they are talking about a much stronger level of mathematics than I might have on the gradesheets of my MS and bachelor's programmes. I do take an interest in mathematics, and I try to study several concepts related to stats and math (eg., Machine Learning).
However, how can I prove this? I have done well in courses related to Machine Learning and Probability, and have taken courses in Real Analysis and Linear Algebra (and there is of course overlap in all four of those areas), but how can I say "I also read stuff because I like it"? I guess one way would be to ask a professor in the stats dept. to guide me through some research and then make them assess me, but how else can I show that I have "strengthened my background"?
Also, is a background in machine learning, signal processing and computer science in general considered for these admissions?