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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.
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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.,
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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,...
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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.
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The law of total probability B Consider a subset B of the sample space S, S divided into disjoint subsets Ai A such that ∪ A = S, i i i → B ∩ Ai → → law of total probability Bayes’ theorem becomes
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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?
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Bayes’ theorem example (cont.) The probability to have AIDS given a + result is ← posterior probability i.e. you’re probably OK! Your viewpoint: my degree of belief that I have AIDS is 3.2% Your doctor’s viewpoint: 3.2% of people like this will have AIDS
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Frequentist Statistics − general philosophy In frequentist statistics, probabilities are associated only with the data, i.e., outcomes of repeatable observations (shorthand: ). Probability = limiting frequency Probabilities such as P (AIDS exists), P (0.117 < a < 0.121), s etc. are either 0 or 1, but we don’t know which. The tools of frequentist statistics tell us what to expect, under the assumption of certain probabilities, about hypothetical repeated observations. The preferred theories (models, hypotheses, ...) are those for which our observations would be considered ‘usual’.
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Bayesian Statistics − general philosophy In Bayesian statistics, use subjective probability for hypotheses: probability of the data assuming hypothesis H (the likelihood) prior probability, i.e., before seeing the data posterior probability, i.e., normalization involves sum after seeing the data over all possible hypotheses Bayes’ theorem has an “if-then” character: If your prior probabilities were p (H), then it says how these probabilities should change in the light of the data. No general prescription for priors (subjective!)
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Random variables and probability density functions A random variable is a numerical characteristic assigned to an element of the sample space; can be discrete or continuous. Suppose outcome of experiment is continuous value x → f(x) = probability density function (pdf) x must be somewhere Or for discrete outcome xi with e.g. i = 1, 2, ... we have probability mass function x must take on one of its possible values
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Cumulative distribution function Probability to have outcome less than or equal to x is cumulative distribution function Alternatively define pdf with
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Multivariate distributions Outcome of experiment charac- terized by several values, e.g. an n-component vector, (x1, ... xn) joint pdf Normalization:
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Marginal pdf Sometimes we want only pdf of some (or one) of the components: i → marginal pdf x1, x2 independent if
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Conditional pdf Sometimes we want to consider some components of joint pdf as constant. Recall conditional probability: → conditional pdfs: Bayes’ theorem becomes: Recall A, B independent if → x, y independent if
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Conditional pdfs (2) E.g. joint pdf f(x,y) used to find conditional pdfs h(y|x1), h(y|x2): Basically treat some of the r.v.s as constant, then divide the joint pdf by the marginal pdf of those variables being held constant so that what is left has correct normalization, e.g.,
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Expectation values Consider continuous r.v. x with pdf f (x). Define expectation (mean) value as Notation (often): ~ “centre of gravity” of pdf. For a function y(x) with pdf g(y), (equivalent) Variance: Notation: Standard deviation: s ~ width of pdf, same units as x.
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Covariance and correlation Define covariance cov[x,y] (also use matrix notation Vxy) as Correlation coefficient (dimensionless) defined as If x, y, independent, i.e., , then → x and y, ‘uncorrelated’ N.B. converse not always true.
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