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Statistical inference is the process of making conclusions about populations from noisy data.

Statistical Inference is the process of drawing formal conclusions from data. It is used to: * Infer facts about data * Where the data is noisy or uncertain * Identify limitations of various approaches

Week 1: Probability and inference

Introduction

There is two main ways to think about probability and the adoption of each one leads to two different categories of inference:

  • Frequency probability: is the long run proportion of times an event occurs in independent, identically distributed repetitions.
  • Bayesian probability: is the probability calculus of beliefs, given that beliefs follow certain rules.

We are going to focus on the frequency style of thinking.

Introduction to Probability

Given a random experiment (say rolling a die) a probability measure is a population quantity that summarizes the randomness. Specifically, probability takes a possible outcome from the experiment and:

  • assigns it a number between 0 and 1
  • so that the probability that something occurs is 1 (the die must be rolled) and
  • so that the probability of the union of any two sets of outcomes that have nothing in common (mutually exclusive) is the sum of their respective probabilities.

Probability rules

  • The probability that nothing occurs is 0.
  • The probability that something occurs is 1.
  • The probability of something is 1 minus the probability that the opposite occurs.
  • The probability of two (or more) things that can not simultaneously occur (mutually exclusive) is the sum of their respective probabilities.
  • If an event A implies the occurrence of event B, then the probability of A occurring is less than the probability that B occurs.
  • For any two events the probability that at least one occurs is the sum of their probabilities minus their intersection.