Laws of probability

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Probabilistic Inference

The most central ideas in statistics involve how to derive statistical judgments from probability theory. Only in 1933 did the Russian mathematician Kolmogorov ...

[edit] derive the entire theory of probability from a few basic rules

• Definition. The probability P(E) of any event E satisfies 0 ≤ P(E) ≤ 1.
• Law 1. Product Law of independent events. This law states that the joint probability of independent events is the product of their independent probabilities.
• Law 2. Summation to One. The total probability law for mutually exclusive and coexhaustive classes of any sample space Ω is that probabilities must sum to one: P(Ω) = 1 = Σ(E|E in PΩ(C_j)) j=1,m. The complement of P(E) is P(not E) = 1- P(E).
• Law 3 Additivity Law of exclusive events. The probability of events in two mutually exclusive classes is the sum of their independent probabilities: PΩ(E|E in C_j=1 or C_j=2: mutually exclusive) = PΩ(C_j=1) + PΩ(C_j=2).
• Empirical Abstraction 1. Law of sample spaces. The probability laws apply if the probability of each class is computed as its fraction in the sample space Ω.
• Bootstrap Abstraction 2. To compare an empirical distribution of N observations to a theoretical distribution in y=f(x), draw N values of x randomly without bias (equiprobable draws for possible values in the range of x), calculate y(x) for each x, and the cumulative distribution Y(x) over the N values of x. Repeat the calculation of Y from N theoretical samples for 100 or more times (the bootstrap) to get a sampling distribution of Y with wich to test model fit.
• And from this last step we derive today's cutting-edge modeling of causality by Judea Perl and Halbert White.

[edit] Nominal Variables and the Laws of Probability

For applications see chapter 5 at http://eclectic.ss.uci.edu/~drwhite/xc/book.htm