Algebraic Statistics: the new interface

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Algebraic Statistics the new interface


Stochastics afternoon, 23 September 2003

• Henry Wynn

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Contents

• 1. Gröbner bases, varieties, ideals
• 2. Designs and interpolation
• 3. Probability models
• 5. Toric varieties and saturation
• 6. Graphical models
• 7. Moments and cumulants
• 8. Sufficient statistics and Maximum likelihood
• 9. Markov bases and simulation
• 10. Live research areas

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Use Qa,b,c,r,s,t,x,y,z Points 1,0,0,1,0,0
,1,0,0,1,0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0,1,
0,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,0,0,0,1,0,0,0,
1,0,1,0, 0,0,1,1,0,0,0,1,0,0,0,1,0,1,0,0,0,1,
0,0,1,0,0,1,1,0,0  
Ideal(x y z - 1, r s t - 1, a b c -
1, z2 - z, yz, cz - sz, bz sz tz - z, y2 -
y, ty - sz - 1/3b 1/3c 1/3s - 1/3t - 1/3y
1/3z, sy sz tz 2/3b 1/3c - 2/3s - 1/3t -
1/3y - 2/3z, cy - tz - 1/3b - 2/3c 1/3s 2/3t
- 1/3y 1/3z, by - sz - 1/3b 1/3c 1/3s -
1/3t - 1/3y 1/3z, t2 - t, st, ct sz tz
1/3b - 1/3c - 1/3s - 2/3t 1/3y - 1/3z, bt - sz
- 1/3b 1/3c 1/3s - 1/3t - 1/3y 1/3z, s2 -
s, cs - sz, bs - tz - 2/3b - 1/3c - 1/3s 1/3t
1/3y 2/3z, c2 - c, bc, b2 - b)
Leading terms x, r, a, z2, yz, cz, bz, y2, ty,
sy, cy, by, t2, st, ct, bt, s2, cs, bs, c2,
bc, b2   Basis 1, b,
c, s, t, y, z, sz, tz
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Ideal(x2y 1/3y3 - 4/3y, x3 3xy2 - 4x,
xy3 - xy, y5 - 5y3 4y) Leading terms
x2y, x3, xy3, y5 Basis 1,
x, y, x2, xy, y2, xy2, y3, y4
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-x418x319x59v2, -x312x55vx411x2,
-x47x2x37x54v, x44x22-x35x5,
-x43x23x32x53v, -x33x25vx52x4,
-x3x42x28x55v2, -x22x3x52x1x44,
-x27x54vx32x1x43, -x24x5x1x34,
x1x2x3x4x5v-1
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Research areas

• Design corner cut, inverse problems
• Complex probability models
• Boundary exponential models, MLEs
• Kernel v Markov v Gröbner bases
• Moments and cumulants asymptotics
• Secant varieties and hidden Markov
• Design/Probability link structural zeros etc

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References

• Basic book on G-bases Cox, Little, OShea
• Monograph Algebraic Statistics Pistone,
  Riccomagno and W
• Book by Sturmfels Algebraic Statistics
• PhD Thesis Fabio Rapallo, Torino (supervisor P)
• Assorted notes PRW and Rapallo
• Lecture notes by Serken Hosken
• Key papers Sturmfels and co-authors,
  particularly on graphical models edition of J
  Symb. Comp
• Role of EURANDOM