Statistics 101 & Exploratory Data Analysis (EDA) A definition of probability Consider a set S with subsets A, B, ... Kolmogorov axioms (1933) From these axioms we can derive further properties, e.g. Conditional probability, independence Also define conditional probability of A given B (with P(B) ≠ 0): E.g. rolling dice: Subsets A, B independent if: If A, B independent, N.B. do not confuse with disjoint subsets, i.e., Interpretation of probability I. Relative frequency A, B, ... are outcomes of a repeatable experiment cf. quantum mechanics, particle scattering, radioactive decay... II. Subjective probability A, B, ... are hypotheses (statements that are true or false) Both interpretations consistent with Kolmogorov axioms. In particle physics frequency interpretation often most useful, but subjective probability can provide more natural treatment of non-repeatable phenomena: systematic uncertainties, probability that Higgs boson exists,... Bayes’ theorem From the definition of conditional probability we have, and but , so Bayes’ theorem First published (posthumously) by the Reverend Thomas Bayes (1702−1761) An essay towards solving a problem in the doctrine of chances, Philos. Trans. R. Soc. 53 (1763) 370; reprinted in Biometrika, 45 (1958) 293. The law of total probability Consider a subset B of the sample space S, B S divided into disjoint subsets Ai such that ∪i Ai = S, → Ai B ∩ Ai → → Bayes’ theorem becomes law of total probability An example using Bayes’ theorem Suppose the probability (for anyone) to have AIDS is: ← prior probabilities, i.e., before any test carried out Consider an AIDS test: result is + or ← probabilities to (in)correctly identify an infected person ← probabilities to (in)correctly identify an uninfected person Suppose your result is +. How worried should you be? Bayes’ theorem example (cont.) The probability to have AIDS given a + result is ← posterior probability i.e. you’re probably OK! Your viewpoint: my de