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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.