Probability assigns numerical values to possible outcomes and events. For discrete outcomes, each value lies between zero and one, and all values sum to one. For continuous variables, probability density integrates to one. section 2.1.1
A random variable represents an uncertain outcome: X might denote a three-sided die roll, while X = 2 denotes the observed event. Capital letters name variables; lowercase symbols stand for particular values. section 2.1.1
Use probability operationally by connecting a question to a quantity and an estimate:
- Pose the question: identify the quantity of interest, such as the height of Bostonians or the digit represented by an image. section 1.5.1
- Write down the probability quantity that answers it, such as P(D = 4 | I), the probability that image I represents digit 4. section 1.5.1
- Perform statistical inference to estimate that quantity from data, such as averaging a random sample or selecting the most probable digit. section 1.5.1
Repeated rolls and physical probability
Imagine repeatedly rolling a three-sided die. As the number of rolls grows, the proportion producing an outcome converges to a value. Interpreting that limiting proportion as the chance of the outcome gives its physical, or frequentist, probability. section 2.2.1
Probability also represents uncertainty about counterfactual outcomes. When an alternative decision was not taken, its possible result can remain uncertain; modeling that uncertainty supports regret analysis and policy evaluation. section 8.1.4 section 8.0
Figure 1.5 Is this an image of the digit 4 or 9? The canonical task of the MNIST dataset is to classify the digit label given the image.
Inspect the MNIST classification task as a concrete view of conditional probability: infer a digit label from an image and compare candidate probabilities. section 1.5.1
