Slides David Kahle
Transcript of Slides David Kahle
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8/12/2019 Slides David Kahle
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Introducing Algebraic Statistics
David Kahle
Assistant Professor
Department of Statistical Science
June 3, 2013
Introducing Algebraic StatisticsDavid Kahle
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Minimum Distance Estimation in Categorical CI Models2
Outline
u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
It is true that few unscientific people have this particular type of religious
experience. Our poets do not write about it; our artists do not try to
portray this remarkable thing. I don't know why. Is no one inspired by our
present picture of the universe? This value of science remains unsung by
singers: you are reduced to hearing not a song or poem, but an evening
lecture about it. This is not yet a scientific age.
Richard Feynman, The Value of Science
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Introducing Algebraic StatisticsDavid Kahle
Whats algebraic statistics?
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Introducing Algebraic StatisticsDavid Kahle
Whats algebraic statistics?
Algebraic statistics is concerned with the development of techniques inalgebraic geometry, commutative algebra, and combinatorics, to addressproblems in statistics and its applications.
Drton, Sturmfels, and Sullivant
Lectures on Algebraic Statistics, 2009
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u n i v e r s i t y
BAYLOR
Introducing Algebraic StatisticsDavid Kahle
Whats algebraic statistics?
Algebraic statistics is concerned with the development of techniques inalgebraic geometry, commutative algebra, and combinatorics, to addressproblems in statistics and its applications.
Drton, Sturmfels, and Sullivant
Lectures on Algebraic Statistics, 2009
Disclaimer :Im not an algebraist!
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u n i v e r s i t y
BAYLOR
Introducing Algebraic StatisticsDavid Kahle
Whats algebraic statistics?
Algebraic statistics is concerned with the development of techniques inalgebraic geometry, commutative algebra, and combinatorics, to addressproblems in statistics and its applications.
Drton, Sturmfels, and Sullivant
Lectures on Algebraic Statistics, 2009
Disclaimer :Im not an algebraist!
Great text :Cox, Little, and OSheas
Ideals, Varieties, and Algorithms
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u n i v e r s i t y
BAYLOR
Introducing Algebraic StatisticsDavid Kahle
Whats algebraic statistics?
Algebraic statistics is concerned with the development of techniques inalgebraic geometry, commutative algebra, and combinatorics, to addressproblems in statistics and its applications.
Drton, Sturmfels, and Sullivant
Lectures on Algebraic Statistics, 2009
Areas of application
Multiway contingency tables
Graphical models Factor analysis
Structural equations models
Statistical disclosure limitation
Evolutionary biology
Causal models Mixture models
Bayesian integrals
Disclaimer :Im not an algebraist!
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u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
My multivariatecalculus nightmare :
5. Sketch x2 y2 z2= 1/2.
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u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
My multivariatecalculus nightmare :
5. Sketch x2 y2 z2= 1/2.
Lets ask !!
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u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
My multivariatecalculus nightmare :
5. Sketch x2 y2 z2= 1/2.
Lets ask !!
ContourPlot3D[
x^2 - y^2 - z^2 == 1/2,
{x, -3, 3}, {y, -3, 3}, {z, -3, 3}
]
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u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
My multivariatecalculus nightmare :
5. Sketch x2 y2 z2= 1/2.
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u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
My multivariatecalculus nightmare :
5. Sketch x2 y2 z2= 1/2.
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u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
My multivariatecalculus nightmare :
5. Sketch x2 y2 z2= 1/2.
Over 600,000evaluations!
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u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
My multivariatecalculus nightmare :
5. Sketch x2 y2 z2= 1/2.x2 y2 z2= 1/2
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u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
x2 y2 z2= 1/2
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Introducing Algebraic StatisticsDavid Kahle
x2 y2 z2= 1/2
Polynomial equation
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u n i v e r s i t y
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Introducing Algebraic StatisticsDavid Kahle
x2 y2 z2= 1/2
Polynomial equation
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Introducing Algebraic StatisticsDavid Kahle
x2 y2 z2= 1/2
Polynomial equation
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Introducing Algebraic StatisticsDavid Kahle
x2 y2 z2= 1/2
Polynomial equation
This is called a realalgebraic set or anaffine variety.
Th i i f f
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Introducing Algebraic StatisticsDavid Kahle
The intersection of surfaces
Where do these surfaces intersect?
ATh i i f f
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Introducing Algebraic StatisticsDavid Kahle
The intersection of surfaces
Where do these surfaces intersect?
BAYLORTh i i f f
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Introducing Algebraic StatisticsDavid Kahle
The intersection of surfaces
Where do these surfaces intersect?
BAYLORTh i i f f
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Introducing Algebraic StatisticsDavid Kahle
The intersection of surfaces
Where do these surfaces intersect?
Intersection = Solutions =
BAYLORTh i t ti f f
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The intersection of surfaces
Geometrically....
BAYLORTh i t ti f f
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Introducing Algebraic StatisticsDavid Kahle
The intersection of surfaces
BAYLORTh i t ti f f
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Introducing Algebraic StatisticsDavid Kahle
The intersection of surfaces
Secret : found with the cylindrical
algebraic decomposition (CAD) algorithm
BAYLORTh i t ti f f
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The intersection of surfaces
BAYLORThe intersection of surfaces
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The intersection of surfaces
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Notation : the collection of polynomials
with real coefficients is written
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Notation : the collection of polynomials
with real coefficients is written
Ring of polynomials
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Notation : the collection of polynomials
with real coefficients is written
Ring of polynomials
Defn : The idealgenerated by the npolynomials
is the set of polynomial combinations of thefks:
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Notation : the collection of polynomials
with real coefficients is written
Ring of polynomials
Defn : The idealgenerated by the npolynomials
is the set of polynomial combinations of thefks:
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Defn : The idealgenerated by the npolynomials
is the set of polynomial combinations of thefks:
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Defn : The idealgenerated by the npolynomials
is the set of polynomial combinations of thefks:
Giant set of polynomials.
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Defn : The idealgenerated by the npolynomials
is the set of polynomial combinations of thefks:
Giant set of polynomials.
Think : vector space.
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Defn : The idealgenerated by the npolynomials
is the set of polynomial combinations of thefks:
Giant set of polynomials.
Think : vector space.
Think : basis vectors.
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Defn : The idealgenerated by the npolynomials
is the set of polynomial combinations of thefks:
Giant set of polynomials.
Think : vector space.
Think : basis vectors.
Fact : If are mpolynomials which generate
the same ideal, then
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Defn : The idealgenerated by the npolynomials
is the set of polynomial combinations of thefks:
Giant set of polynomials.
Think : vector space.
Think : basis vectors.
Fact : If are mpolynomials which generate
the same ideal, then
Think : a different basis.
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Defn : The idealgenerated by the npolynomials
is the set of polynomial combinations of thefks:
Giant set of polynomials.
Think : vector space.
Think : basis vectors.
Fact : If are mpolynomials which generate
the same ideal, then
Think : a different basis.
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Fact : If are mpolynomials which generate
the same ideal, then
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Fact : If are mpolynomials which generate
the same ideal, then
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Fact : If are mpolynomials which generate
the same ideal, then
A Grbner basis is a good selection of thegks computed with Buchbergers algorithm
or Faugeres F4 algorithm
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
Fact : If are mpolynomials which generate
the same ideal, then
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
BAYLORPolynomial ideals
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Introducing Algebraic StatisticsDavid Kahle
Polynomial ideals
... help find exactglobal solutions byelimination(but even in moderately sized problems can become intractable)
Grbner bases can...
... identify if a polynomialfis in the ideal.
... allow for implicitization of a rationallyparameterized variety.
... compute related algebraic and geometric structures (e.g. the radical of the ideal)
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Introducing Algebraic StatisticsDavid Kahle
Differentialgeometrystudies smooth manifoldsvia
smooth topological structureAlgebraicgeometrystudies varietiesvia algebraic
structure
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Introducing Algebraic StatisticsDavid Kahle
Differentialgeometrystudies smooth manifoldsvia
smooth topological structureAlgebraicgeometrystudies varietiesvia algebraic
structure
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BAYLOR
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Introducing Algebraic StatisticsDavid Kahle
Differentialgeometrystudies smooth manifoldsvia
smooth topological structureAlgebraicgeometrystudies varietiesvia algebraic
structure
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BAYLOR
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Introducing Algebraic StatisticsDavid Kahle
Differentialgeometrystudies smooth manifoldsvia
smooth topological structureAlgebraicgeometrystudies varietiesvia algebraic
structure
Real algebraic geometryis the study of real (semi)algebraic sets
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Introducing Algebraic StatisticsDavid Kahle
Differentialgeometrystudies smooth manifoldsvia
smooth topological structure
Algebraicgeometrystudies varietiesvia algebraic
structure
A semialgebraic setis a finite union of sets of the form
Real algebraic geometryis the study of real (semi)algebraic sets
u n i v e r s i t y
BAYLOR(Early?) Mantra of algebraic statistics
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Introducing Algebraic StatisticsDavid Kahle
Fact 1 : Statistical inference depends on thegeometryof the parameter space(the model)
( y ) g
E.g. regularity conditions, smoothness, etc.
u n i v e r s i t y
BAYLOR(Early?) Mantra of algebraic statistics
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Fact 1 : Statistical inference depends on thegeometryof the parameter space(the model)
Fact 2 : Ifthe parameter spaceis a semialgebraic set,
then
statistical inference can be analyzed with
algebraic methods
( y ) g
E.g. regularity conditions, smoothness, etc.
u n i v e r s i t y
BAYLOR(Early?) Mantra of algebraic statistics
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Fact 1 : Statistical inference depends on thegeometryof the parameter space(the model)
Fact 2 : Ifthe parameter spaceis a semialgebraic set,
then
statistical inference can be analyzed with
algebraic methods
( y ) g
E.g. regularity conditions, smoothness, etc.
Statistical models are algebraic varieties
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BAYLOREx 1 : A parametric ASM
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A simple experiment
p
u n i v e r s i t y
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A simple experiment
Requirements on the s
p
u n i v e r s i t y
BAYLOREx 1 : A parametric ASM
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A simple experiment
1.
Requirements on the s
p
u n i v e r s i t y
BAYLOREx 1 : A parametric ASM
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A simple experiment
1.
2.
Requirements on the s
u n i v e r s i t y
BAYLOREx 1 : A parametric ASM
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A simple experiment
Requirements on the s
1.
2.
u n i v e r s i t y
BAYLOREx 1 : A parametric ASM
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Requirements on the s
Introducing Algebraic StatisticsDavid Kahle
A simple experiment
1.
2.
This is a geometric structure!
u n i v e r s i t y
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Requirements on the s
Introducing Algebraic StatisticsDavid Kahle
A simple experiment
1.
2.
This is a geometric structure!
u n i v e r s i t y
BAYLOR
Ex 1 : A parametric ASM
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This is a geometric structure!
Statistical models = subsets of the simplex
u n i v e r s i t y
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Ex 1 : A parametric ASM
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This is a geometric structure!
Statistical models = subsets of the simplex
Ex : The binomial distribution
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Ex 1 : A parametric ASM
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This is a geometric structure!
Statistical models = subsets of the simplex
Ex : The binomial distribution
u n i v e r s i t y
BAYLOR
Ex 1 : A parametric ASM
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This is a geometric structure!
Statistical models = subsets of the simplex
Ex : The binomial distribution
u n i v e r s i t y
BAYLOR
Ex 1 : A parametric ASM
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This is a geometric structure!
Statistical models = subsets of the simplex
Observation :
u n i v e r s i t y
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Ex 1 : A parametric ASM
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This is a geometric structure!
Statistical models = subsets of the simplex
Observation :
Implic
itization
u n i v e r s i t y
BAYLOR
Ex 1 : A parametric ASM
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This is a geometric structure!
Statistical models = subsets of the simplex
Observation :
Implic
itization
u n i v e r s i t y
BAYLOR
Ex 1 : A parametric ASM
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This is a geometric structure!
Statistical models = subsets of the simplex
Observation :
Implicitizat
ion
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Ex 1 : A parametric ASM
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This is a geometric structure!
Statistical models = subsets of the simplex
Observation :
Implicitizat
ion
u n i v e r s i t y
BAYLOREx 1 : A parametric ASM
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The binomial model is the intersection of anaffine variety and the probability simplex!
u n i v e r s i t y
BAYLOREx 1 : A parametric ASM
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Two descriptions of the binomial model
Parametric description Implicit description
Were used to this... ...but sometimes we get this.
u n i v e r s i t y
BAYLOREx 1 : A parametric ASM
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Two descriptions of the binomial model
Parametric description Implicit description
Were used to this... ...but sometimes we get this.
Parametric description Implicit description
Possible (may be hard)
Not always possible
u n i v e r s i t y
BAYLOREx 1 : A parametric ASM
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A better visualization is available via
barycentric coordinates
3d to 2d
A last note : Visualization
... but since were insidethe simplex, wecant see the varieties outside of it
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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The 2 x 2 contingency table
0 1
0
1
X2
X1
where
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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The 2 x 2 contingency table
0 1
0
1
X2
X1
where
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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The 2 x 2 contingency table
0 1
0
1
X2
X1
where
Requirements on the s
1.
2.
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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The 2 x 2 contingency table
0 1
0
1
X2
X1
where
Requirements on the s
1.
2. Points in/on the tetrahedronare distributions on the table!
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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The 2 x 2 contingency table
0 1
0
1
X2
X1
where
Requirements on the s
1.
2.
2d
3d
Points in/on the tetrahedronare distributions on the table!
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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0 1
0
1
X2
X1
Ex : The independence model
3d
Statistical models = subsets of the simplex
The 2 x 2 contingency table
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0 1
0
1
X2
X1
Ex : The independence model
3d
Statistical models = subsets of the simplex
means , where is the marginal probability of X1 = x1:
The 2 x 2 contingency table
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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0 1
0
1
X2
X1
Ex : The independence model
3d
Statistical models = subsets of the simplex
means , where is the marginal probability of X1 = x1:
This holds for everycombination of x1and x2!
The 2 x 2 contingency table
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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0 1
0
1
X2
X1
Ex : The independence model
3d
Statistical models = subsets of the simplex
means , where is the marginal probability of X1 = x1:
This holds for everycombination of x1and x2!
The 2 x 2 contingency table
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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0 1
0
1
X2
X1
Ex : The independence model
3d
Statistical models = subsets of the simplex
means , where is the marginal probability of X1 = x1:
The 2 x 2 contingency table
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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0 1
0
1
X2
X1
Ex : The independence modelStatistical models = subsets of the simplex
means , where is the marginal probability of X1 = x1:
The 2 x 2 contingency table
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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0 1
0
1
X2
X1
Ex : The independence modelStatistical models = subsets of the simplex
means , where is the marginal probability of X1 = x1:
The 2 x 2 contingency table
u n i v e r s i t y
BAYLOREx 2 : An implicitly defined ASM
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0 1
0
1
X2
X1
Ex : The independence modelStatistical models = subsets of the simplex
means , where is the marginal probability of X1 = x1:
The 2 x 2 contingency table
u n i v e r s i t y
BAYLOREx 3 : A Gaussian ASM
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Introducing Algebraic StatisticsDavid Kahle
Do the inferential methods change?
For example, typically the MLE isasymptotically normal...
u n i v e r s i t y
BAYLOREx 3 : A Gaussian ASM
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Simulation :1. Draw n= 50, 1000 samples from
the bivariate normaldistribution with 0 mean.
2. Compute MLE for mean (closestpoint to the parameter space)
u n i v e r s i t y
BAYLOREx 3 : A Gaussian ASM
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Introducing Algebraic StatisticsDavid Kahle
One sample, its mean, and the MLE
u n i v e r s i t y
BAYLOREx 3 : A Gaussian ASM
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Introducing Algebraic StatisticsDavid Kahle
Simulation :1. Draw n= 50, 1000 samples from
the bivariate normaldistribution with 0 mean.
2. Compute MLE for mean (closestpoint to the parameter space)
Do N= 2000times
u n i v e r s i t yBAYLOREx 3 : A Gaussian ASM
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Introducing Algebraic StatisticsDavid Kahle
n= 50
u n i v e r s i t yBAYLOREx 3 : A Gaussian ASM
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Introducing Algebraic StatisticsDavid Kahle
n= 1000
u n i v e r s i t yBAYLOREx 3 : A Gaussian ASM
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Introducing Algebraic StatisticsDavid Kahle
Do the inferential methods change?
For example, typically the MLE isasymptotically normal...
Conclusion :
Problems occur at singularities
which dont ever go away.
Problems happen near singularitieseven for relatively large n
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Algebraic statistical models
Defined in reference to another family of models (e.g. exponential family)
Defined as semialgebraic sets on the parameter space.
u n i v e r s i t yBAYLOR
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Introducing Algebraic StatisticsDavid Kahle
Algebraic statistical models
Defined in reference to another family of models (e.g. exponential family)
Defined as semialgebraic sets on the parameter space.
Drton and Sullivant, Statistica Sinica, 2007
u n i v e r s i t yBAYLOR
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Introducing Algebraic StatisticsDavid Kahle
Summary : Algebraic statistical models are an interestingand general class of models, but require care with many of theclassical aspects of inference...
u n i v e r s i t yBAYLOR
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Summary : Algebraic statistical models are an interestingand general class of models, but require care with many of theclassical aspects of inference...
... with cool pictures.
Thank you!