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.

@ __Related topics__: news sn(19)apr [the debate on 'statistical significance'].

> __Online resources__:
see Wikipedia page;
Good Calculators site [statistics calculators].

**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.

* __Duhem-Quine problem__:
Hypothesis testing is always conditional on a bundle of real auxiliary assumptions.

@ __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.

@ __Hypothesis testing__:
Liu & Adams a1812 [and Bertrand's question].

@ __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.

main page
– abbreviations
– journals – comments
– other sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 30 apr 2020