Statistics |

**In General** > s.a. game theory;
probability [distributions]; statistics
and error analysis in physics.

* __Idea__:

* __Superstatistics__: Superpositions
of different statistics on different time or spatial scales; > s.a. markov
processes.

@ __Problems__: Bassett et al 00.

@ __Differential geometry methods__: Amari 85; Murray & Rice 93.

> __Online resources__:
see Wikipedia page.

**Concepts and Techniques** > s.a. correlations [correlation
functions]; Median.

* __Frequency distribution__:

*P*(*f*;* p*) = exp{–(*N*/2)
(*f*_{j} – *p*_{j})^{2}
/ *f*_{j}} .

* __Exchangeable random variables__:
A more general concept than that of independent variables, for which one can
obtain distributions analogous to the binomial or Poisson distributions; They
account, e.g., for the fact that animals in the same litter behave more similarly
than across litters (the corresponding variables are exchangeable but not independent)
[@ George & Bowman Biometrics(95)].

$ __Mean__: The (arithmetic) mean of a collection of *n* numbers *a*_{j} is AM({*a*}):= (*a*_{1} + ... + *a*_{n})/*n*; If the numbers have weights (probabilities) *p*_{j}, satisfying (*p*_{j} + ... + *p*_{j}) = 1, then the mean becomes AM({*a*};{*p*} ):= (*p*_{1}*a*_{1} + ... + *p*_{n}*a*_{n})/*n*; In the case of a continuous variable *x* with probability density *p*(*x*), the mean value of *x* is ∫d*x* *p*(*x*) *x*, and the mean value of a function *f*(*x*) is ∫d*x* *p*(*x*) *f*(*x*).

@ __References__: Mukhopadhyay 08 [multivariate analysis]; Bertin & Györgyi JSM(10)-a1006 [extreme-value
statistics, renormalization flow].

> __Online resources__:
see Wikipedia page on average [arithmetic, geometric, harmonic mean].

**Parameter Estimation or Statistical Inference**

* __Idea__: Using experimental data/outcomes to estimate the probability
distribution
that generated them.

@ __General references__: Earman 92; Vapnik 98, 99; Edwards 06; Gupta & Kabe 11 [sample surveys].

@ __Quantum__: Brody & Hughston PRL(96)
[geometrical]; Bogdanov phy/02,
qp/03-conf; Kumagai & Hayashi CMP(13)-a1110 [quantum analogues of chi-square, *t* and *F* tests].

@ __Statistical complexity__: Rissanen 98.

@ __Continuous distributions, field theory method__: Bialek et al PRL(96);
Holy PRL(97);
Periwal
PRL(97)ht,
NPB(99);
van Hameren et al NPB(99)
[discrepancies and Fermions];
Bialek et al NC(01)phy/00;
Nemenman & Bialek PRE(02)cm/00.

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send feedback and suggestions to bombelli at olemiss.edu – modified 8
nov 2015