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Advances in Algebraic Statistics

Advances in Algebraic Statistics. General Applications and Statistical Computing Sections 8 June 2005. Henry Wynn. Contents. 1. Gr öbner bases, varieties, ideals 2. Designs and supports

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Advances in Algebraic Statistics

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  1. Advances in Algebraic Statistics General Applications and Statistical Computing Sections 8 June 2005 Henry Wynn

  2. Contents 1. Gröbner bases, varieties, ideals 2. Designs and supports 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

  3. Use Q[a,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, z^2 - z, yz, cz - sz, bz + sz + tz - z, y^2 - 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, t^2 - 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, s^2 - s, cs - sz, bs - tz - 2/3b - 1/3c - 1/3s + 1/3t + 1/3y + 2/3z, c^2 - c, bc, b^2 - b) [x, r, a, z^2, yz, cz, bz, y^2, ty, sy, cy, by, t^2, st, ct, bt, s^2, cs, bs, c^2, bc, b^2] [1, b, c, s, t, y, z, sz, tz]

  4. Ideal(x^2y + 1/3y^3 - 4/3y, x^3 + 3xy^2 - 4x, xy^3 - xy, y^5 - 5y^3 + 4y) [1, x, y, x^2, xy, y^2, xy^2, y^3, y^4, x^2y, x^3, xy^3, y^5]

  5. gg := [-p3^8*p5+p2^3*p4^6, -p2^3*p5+p1*p4^3, p1*p3^8-p2^6*p4^3, v*p5^4*p3^17-p2^2*p4^14,-p4^8+p2*v*p5^3*p3^9, -p4^2+p2^4*v*p5^2*p3, v*p2^7*p4^4*p5-p3^7, -p1*p3^7+v*p2^10*p5^2*p4, p2^13*v*p5^3-p1^2*p4^2*p3^7, v*p1*p2*p3*p4*p5-1]

  6. Live research areas • More on design: corner cut, inverse problems • Complex probability models: large area • Boundary exponential models, MLE etc • Kernel v Markov v Gröbner bases • Secant varieties and hidden Markov • Design/Probability link: structural zeros etc

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