- Algebraic statistics
**Algebraic statistics**is a fairly recent field ofstatistics which utilizes the tools ofalgebraic geometry andcommutative algebra in order to study problems related todiscrete random variable s with finite state spaces. Such problems includeparameter estimation ,hypothesis testing , and experimental design. The key connection between statistics and algebra is the observation that many commonly used classes of discrete random variables can be viewed as algebraic varieties.**Introductory example**Consider a

random variable "X" which can take on the values 0, 1, 2. Such a variable is completely characterized by the three probabilities :$p\_i=mathrm\{Pr\}(X=i),quad\; i=0,1,2$and these numbers clearly satisfy:$sum\_\{i=0\}^2\; p\_i\; =\; 1\; quad\; mbox\{and\}quad\; 0leq\; p\_i\; leq\; 1.$Conversely, any three such numbers unambiguously specify a random variable, so we can identify the random variable "X" with the tuple ("p"_{0},"p"_{1},"p"_{2})∈**R**^{3}.Now suppose "X" is a

Binomial random variable with parameter "p = q" and "n = 2", i.e. "X" represents the number of successes when repeating a certain experiment two times, where each experiment has an individual success probability of "q". Then :$p\_i=mathrm\{Pr\}(X=i)=\{2\; choose\; i\}q^i\; (1-q)^\{2-i\}$and it is not hard to show that the tuples ("p"_{0},"p"_{1},"p"_{2}) which arise in this way are precisely the ones satisfying:$4\; p\_0\; p\_2-p\_1^2=0.$The latter is a polynomial equation defining an algebraic variety (or surface) in**R**^{3}, and this variety, when intersected with thesimplex given by:$sum\_\{i=0\}^2\; p\_i\; =\; 1\; quad\; mbox\{and\}quad\; 0leq\; p\_i\; leq\; 1,$ yields a piece of analgebraic curve which may be identified with the set of all 3-state Bernoulli variables. Determining the parameter "q" amounts to locating one point on this curve; testing the hypothesis that a given variable "X" is Bernoulli amounts to testing whether a certain point lies on that curve or not.**References*** [

*http://www.math.harvard.edu/~seths/assc.html Algebraic Statistics Short Course*] , lecture notes by Seth Sullivant

* L. Pachter and B. Sturmfels. "Algebraic Statistics and Computational Biology." Cambridge University Press 2005.

* G. Pistone, E. Riccomango, H. P. Wynn. "Algebraic Statistics." CRC Press, 2001.

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