# Conditional Expectation

Lucas Schiffer
January 20, 2016

Data Analysis for the Life Sciences

### Today's Topics

• Introduction
• Conditional Probability / Conditional Expectation
• Galton Height Data
• Regression as Machine Learning
• Conditional Expectation / Optimal Prediction
• Exercises

### Introduction

• Conditional probability and conditional expectation are slightly different
• Both are often useful as machine learning techniques
• Helpful in solving classification and prediction problems
• Generally useful for categorical and continuous outcomes

### Conditional Probability / Conditional Expectation

• Conditional probability
• What is the probability that the next marble is green given that the first was yellow?
• $f_{k}(x)=Pr(Y=k\ |\ X = x)$
• Conditional Expectation
• Given that the first marble was yellow, what color is expected next?
• $E[(\hat{Y}-Y)^2|X=x]$

### Galton Height Data

• Francis Galton created a height dataset (1885)
• Useful for conditional expectation practice

### Regression as Machine Learning

• Suppose we want to predict a random son's height
• Suppose we want to predict a son's height who's father is 71 inches tall
• $E[(\hat{Y}-Y)^2|X=71]$