Probability Theory |

**In General** > s.a. integration; measure
theory; random processes; statistics.

* __Idea__: The branch of mathematics
that deals with computing the odds of an event's occurrence.

* __History__: The idea that
rigorous math can be used to calculate odds was introduced by Pascal, who conferred
with Fermat; De Moivre found the Bell curve for distributions of random variables;
The theory was founded by Laplace.

* __Applications__: Telecommunications
(e.g., Dublin IAS applied probability group).

$ __Definition (measure theory)__:
A probability is a normalized measure on a set of possible events Γ.

$ __Definition (logic)__:
A probability is a map *P* : *X* × *X* → [0,1],
where *X* is a set of propositions, closed under ∧, \(\lor\),
and ¬, such that (1) *P*(*b* ∧ *c*|*a*)
= *P*(*b*|*a*) *P*(*c*|*a* ∧ *b*);
(2) *P*(*b*|*a*) + *P*(¬*b*|*a*)
= 1; (3) *P*(¬*a*|*a*) = 0.

> __Online resources__:
see Wikipedia page;
H. Pishro-Nik's text.

**Interpretations and Applications**
> s.a. probability in physics [including axiomatic approach].

* __Bayesian approach__:
An approach to the problem of inferring something about a parameter or state
of nature *s* after observing a random variable *x* whose
distribution *p* depends on *s*; Probabilities are "degrees
of belief," and refer to our confidence in certain statements based on
previous experience; Useful for measurements and updating our predictions,
allows us to assign probabilities that numbers be "true values" and
to use induction; & Bayes, Bernoulli, Gauss, Laplace (used it to conclude
that the boy-girl ratio [< 1] is universal to humankind and determined by
biology); XX-century statistics was overwhelmingly behavioristic and frequentist,
especially in applications, but the XXI century is seeing a resurgence of Bayesianism;
> s.a. foundations of quantum mechanics; statistics.

* __Frequentist approach__:
Probabilities are frequencies of occurrence of values in the ensembles of all
observations; It is considered the conventional one in physics, but if strictly
followed will lead nowhere.

* __Lindley paradox__:
A counterintuitive statistical effect where the Bayesian and frequentist approaches
to hypothesis testing give radically different answers, depending on the choice
of the prior distribution.

* __Combinatorial approach__:
Probabilities are ratios of favorable cases to some statement over all cases,
in a series of performed tests.

@ __Frequentist__: Brody in(89);
Wall SHPSA(06) [Venn's opposition to degree of belief].

@ __Bayesian__: Hartigan 83;
D'Agostini hp/95 [primer];
Howson BJPS(97) [and logic];
Weintraub BJPS(01) [paradox];
D'Agostini 03 [primer];
Jeffrey 04 [readable];
Appleby O&S(05)qp/04-in [vs frequentist];
Howson & Urbach 05;
D'Agostini ISBA-phy/05 ["Fermi's Bayes theorem"];
Mielczarek et al a0901 [introduction, conceptual];
Fuchs & Schack AIP(09)-a0906 [priors in quantum Bayesian inference];
Henderson et al PhSc(10)apr [Bayesian inference in a hierarchical hypothesis space];
Leitgeb & Pettigrew PhSc(10)apr,
PhSc(10)apr [justification];
Efron BAMS(13) [250th anniversary];
Benétreau-Dupin Syn(15)-a1412 [adequately handling ignorance];
Lucy A&A(16)-a1511 [frequentist tests for Bayesian models].

@ __And physics__: Alamino a1008 [as a framework for physical theories];
Mauri & Paris PLA(15)-a1510 [Lindley paradox in optical interferometry];
Beck a1804 [and quantum mechanics];
Thrane & Talbot a1809
[Bayesian inference in gravitational-wave astronomy, intro].

**Related Concepts**

* __Marginal distribution__:
For a subset of a collection of random variables, it is the probability distribution
of the variables in the subset, obtained adding the values of the overall probability
distribution for all values of the remaining variables; > s.a. Wikipedia
page.

* __Moments of a distribution__:
The *n*-th moment of a probability distribution for *x* is the
mean value of *x*^{n}.

@ __ Moments of a distribution__: Yehia JPA(02)-a1308 [relation between the first and second moments].

@ __Combining probability distributions__:
Eliazar & Klafter PhyA(08) [limit laws and non-linear scaling schemes];
> s.a. Central-Limit Theorem.

@ __Space of probability distributions__: Calvo et al JSP(10) [renormalization group transformation];
Pistone Ent(15)-a1308 [application to Statistical Physics].

@ __Marginal distribution__: Cheon JKPS(15)-a1507 [in quantum mechanics, graphical representation].

> __Related topics__:
see analysis [fractional moments]; Law
of Large Numbers; measure theory.

**Examples of Probability Distributions** > see Binomial,
Gamma, gaussian, Poisson
Distribution; Exponential Family.

**References** > s.a. probabilistic
combinatorics; probability in quantum theory [quasi-probability].

@ __General__: Bernoulli 1713;
Howson BJPS(95) [survey];
Williams SHPSA(05) [history, Cardano];
Novak & LaCroix a1205-proc [free probability, lectures].

@ __Books, I__: Ekeland 93;
Nahin 00;
Holland 02;
Prakasa Rao 08;
Mukhopadhyay 11;
Woolfson 12.

@ __Books, II__:
Grimmett & Welsh 86;
Rotar 98;
Stirzaker 99,
99 [II];
Dokuchaev 15 [1-semester course].

@ __Books__: Marle 74;
de Groot 75;
Bitsakis & Nicolaides ed-89;
Fristedt & Grey 97;
Grimmett & Stirzaker 01 [and random processes],
01 [exercises];
Rosenthal 06 [rigorous];
Heyer 09 [structural, algebraic-topological aspects].

@ __Books, problems__:
Chaumont & Yor 12 [r CP(13)];
Mills 13.

@ __Books, special emphasis__: Billingsley 68 [convergence];
Mackey 78 [and group representations];
Stroock 93 [analytic];
Vershynin 18 [data science].

@ __Results__: Tribelsky PRL(02)m.PR/01 [sums of variables].

@ __Fuzzy__: in Gudder IJTP(00),
Gudder FP(00) [rev];
Habil & Nasr IJTP(02).

@ __Complex probabilities__: Salcedo JMP(97);
Weingarten PRL(02) [and path integrals];
Bender et al AP(10)-a0912 [and quantum mechanics].

@ __Related topics__: Nelson 87 [non-standard];
Friedman AAM(99) [frequency interpretation];
Streater JMP(00)mp [historical survey].

**Generalizations** > s.a. homotopy [homotopy probability theory];
probability in physics [negative probabilities, quantum probabilities];
Plausibility Measures.

@ __General references__: Liu a1705 [infinite-dimensional probability, intro].

@ __Extended probabilities__: Niestegge JPA(01)-a1001 [non-Boolean extension, and quantum measurement];
Burgin a0912 [mathematical foundations];
Noldus a1509 [two physical interpretations].

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