CS 70 Discrete Mathematics and Probability Theory Spring 2022 Course Notes Note 20 Prediction !

Prediction is one of the most compelling applications of probability, particularly in computer science. As an example, suppose you know the height of all students at UC Berkeley, and a Berkeley student is chosen uniformly at random. What would be the best prediction for that student’s height that minimizes the error of your prediction? It would make sense to guess the average height, as it lies in the "middle" of all the heights. More formally, prediction is the problem of making an estimate for a random variable from available information; for this note, that information is a probability distribution or joint distribution. The goal of prediction is to minimize the error (sometimes called loss) in the prediction. More formally, we consider the mean squared error. That is, for a random variable X with a known distribution, we define prediction as giving an estimate x̂1 which minimizes the following expression: E[(X− x̂)2].