Student Probability Seminar
Stein’s Paradox and Random Walks
Speaker: Tim Kunisky, CIMS
Location: Warren Weaver Hall 201
Date: Thursday, September 26, 2019, 12:15 p.m.
I have a secret number in mind. I add some Gaussian noise to it and tell you the result. Now, guess my number. Only one strategy makes sense, right? — just guess the noisy number that I told you. Stein’s paradox is that, if instead you have to guess a secret vector, then you can always do better. I’ll explain this simple but surprising fact, and some of the impact it had in the statistics community.
But this isn’t the student statistics seminar, so we’ll get back to our favorite probability theory topics in short order. Namely, what Stein really found was that when the secret vector’s dimension is two, the naive strategy is still the best; once the dimension is at least three, there is a better strategy. It’s not a coincidence that this transition between dimension two and three matches one of our other beloved transitions, between recurrence and transience of random walks. I’ll conclude with a beautiful result of Brown, who explained how, in fact, the optimality of the naive guessing strategy and the recurrence of certain random walks are equivalent questions.