4.7 Model Invariants
Each algebraic statistical model gives rise to model invariants that describe the relationships between the probabilities. Consider an algebraic statistical modelf :Cd→Cmgiven by
f : (θ1, . . . , θd)7→(f1(θ), . . . , fm(θ)). (4.42) Here the ambient spaces are taken over the complex numbers, but the coordinates f1, . . . , fm are polynomials inQ[θ1, . . . , θd]. We study the imagef(Cd) by the polynomial parametrization
pi=fi(θ1, . . . , θd), 1≤i≤m. (4.43) The Implicitization theorem yields the following result.
Proposition 4.15.Consider the idealI=hp1−f1, . . . , pm−fmiin C[θ1, . . . , θd, p1, . . . , pm]. For the d-th elimination idealId=I∩C[p1, . . . , pm], the affine varietyV(Id)is the Zariski closure of the image f(Cd).
The polynomials in the elimination ideal Id are calledinvariants of the model f. By the Elimination theorem, these invariants can be established by computing the reduced Groebner basis of the elimination idealId with respect to an elimination ordering forθ1> . . . > θd > p1> . . . > pm.
Example 4.16 (Singular).Consider the mappingf :C2→C3: (θ1, θ2)7→(θ31, θ1θ2, θ1θ2). The image off is a (dense) subset of a plane in three-space,
f(C2) ={(x, y, z)∈C3|y=z∧(x= 0⇒y= 0)}
= [V(Y −Z)\ V(X, Y −Z)]∪ V(X, Y, Z).
This is a Boolean combination of affine varieties, but not an affine variety. In view of the ideal I = hp1−θ13, p2−θ1θ2, p3−θ1θ2iin C[θ1, θ2, p1, p2, p3], the reduced Groebner basis with respect to thelp ordering withθ1> . . . > θd> p1> . . . > pmcan be calculated as follows,
> ring r = 0, (t(1..2),p(1..3)), lp;
> ideal i = p(1)-t(1), p(2)-t(1)*t(2), p(3)-t(1)*t(2);
> std(i);
_[1]=p(2)-p(3) _[2]=t(2)*p(1)-p(3) _[3]=t(1)-p(1)
Thus the reduced Groebner basis of the second elimination idealI2=I∩Q[p1, p2, p3] is{p2−p3}Hence the Zariski closure of the imagef(Cd) isV(p2−p3) andp2−p3 is a model invariant forf. ♦ Example 4.17 (Singular). Reconsider the toric modelf :C2→C3: (θ1, θ2)7→(θ12, θ1θ2, θ22) studied in Ex. 4.5. Take the idealI= (p1−θ12, p2−θ1θ2, p3−θ22) inC[θ1, θ2, p1, p2, p3] and calculate a Groebner basis ofIwith respect to the lpordering withθ1> θ2> p1> p2> p3.
> ring r = 0, (t(1..2),p(1..3)), lp;
> ideal i = p(1)-t(1)^2, p(2)-t(1)*t(2), p(3)-t(2)^2;
> std(i);
_[1]=p(1)*p(3)-p(2)^2 _[2]=t(2)^2-p(3) _[3]=t(1)*p(3)-p(2)^2 _[4]=t(1)*p(2)-t(2)*p(1) _[5]=t(1)*t(2)-p(2) _[6]=t(1)^2-p(1)
The first element provides the Groebner basis of the second elimination idealI2 and yields the model
invariantp1p3−p22. ♦
Example 4.18 (Singular). Reconsider the DiaNA model. The polynomial parametrization of the DiaNA model is given in (4.2). Take the idealIin C[θ1, θ2, p1, p2, p3, p4] generated by
p1−(−0.10·θ1+ 0.02·θ2+ 0.25), p2−(0.08·θ1−0.01·θ2+ 0.25), p3−(0.11·θ1−0.02·θ2+ 0.25), p4−(−0.09·θ1+ 0.01·θ2+ 0.25).
A Groebner basis ofI with respect to thelporderingθ1> θ2> p1> p2> p3> p4 is
−0.5500000002 + 1.399999999·p2−0.1999999995·p3+ 1.0000·p4,
−0.05059523811 + 0.2380952383·p4+ 0.9523809525·p3+ 0.8333333334·p1,
−0.4285714291 + 0.9428571435·p4+ 0.7714285717·p3+ 0.6000000000e−2·θ2,
−1.071428573 + 2.857142859·p4+ 1.428571429·p3 + 0.100·θ1.
The first two polynomials generate the second elimination ideal I2 and thus provide invariants of the
DiaNA model. ♦
Example 4.19 (Singular). Consider the dishonest casino in Ex. 6.1. Assume that the dealer always starts with the fair coin and then switches eventually to the loaded one. If a game consists of n= 4 coin tosses, the probability for an outcomeτ∈Σ′4 is
pτ =pF F F F,τ +pF F F L,τ+pF F LL,τ+pF LLL,τ. The invariants for this model can be computed as follows.
> ring r = 0, (FF,FL,LL,Fh,Ft,Lh,Lt,p(0..15)), dp
> ideal i =
# hhhh
p(0) - Fh*FF*Fh*FF*Fh*FF*Fh - Fh*FF*Fh*FF*Fh*FL*Lh - Fh*FF*Fh*FL*Fh*LL*Lh - Fh*FL*Lh*LL*Lh*LL*Lh,
# hhht
p(1) - Fh*FF*Fh*FF*Fh*FF*Ft - Fh*FF*Fh*FF*Fh*FL*Lt - Fh*FF*Fh*FL*Fh*LL*Lt - Fh*FL*Lh*LL*Lh*LL*Lt,
# hhth
p(2) - Fh*FF*Fh*FF*Ft*FF*Fh - Fh*FF*Fh*FF*Ft*FL*Lh - Fh*FF*Fh*FL*Ft*LL*Lh - Fh*FL*Lh*LL*Lt*LL*Lh,
# hhtt
4.7 Model Invariants 95
p(3) - Fh*FF*Fh*FF*Ft*FF*Ft - Fh*FF*Fh*FF*Ft*FL*Lt - Fh*FF*Fh*FL*Ft*LL*Lt - Fh*FL*Lh*LL*Lt*LL*Lt,
# hthh
p(4) - Fh*FF*Ft*FF*Fh*FF*Fh - Fh*FF*Ft*FF*Fh*FL*Lh - Fh*FF*Ft*FL*Fh*LL*Lh - Fh*FL*Lt*LL*Lh*LL*Lh,
# htht
p(5) - Fh*FF*Ft*FF*Fh*FF*Ft - Fh*FF*Ft*FF*Fh*FL*Lt - Fh*FF*Ft*FL*Fh*LL*Lt - Fh*FL*Lt*LL*Lh*LL*Lt,
# htth
p(6) - Fh*FF*Ft*FF*Ft*FF*Fh - Fh*FF*Ft*FF*Ft*FL*Lh - Fh*FF*Ft*FL*Ft*LL*Lh - Fh*FL*Lt*LL*Lt*LL*Lh,
# httt
p(7) - Fh*FF*Ft*FF*Ft*FF*Ft - Fh*FF*Ft*FF*Ft*FL*Lt - Fh*FF*Ft*FL*Ft*LL*Lt - Fh*FL*Lt*LL*Lt*LL*Lt,
# thhh
p(8) - Ft*FF*Fh*FF*Fh*FF*Fh - Ft*FF*Fh*FF*Fh*FL*Lh - Ft*FF*Fh*FL*Fh*LL*Lh - Ft*FL*Lh*LL*Lh*LL*Lh,
# thht
p(9) - Ft*FF*Fh*FF*Fh*FF*Ft - Ft*FF*Fh*FF*Fh*FL*Lt - Ft*FF*Fh*FL*Fh*LL*Lt - Ft*FL*Lh*LL*Lh*LL*Lt,
# thth
p(10)- Ft*FF*Fh*FF*Ft*FF*Fh - Ft*FF*Fh*FF*Ft*FL*Lh - Ft*FF*Fh*FL*Ft*LL*Lh - Ft*FL*Lh*LL*Lt*LL*Lh,
# thtt
p(11)- Ft*FF*Fh*FF*Ft*FF*Ft - Ft*FF*Fh*FF*Ft*FL*Lt - Ft*FF*Fh*FL*Ft*LL*Lt - Ft*FL*Lh*LL*Lt*LL*Lt,
# tthh
p(12)- Ft*FF*Ft*FF*Fh*FF*Fh - Ft*FF*Ft*FF*Fh*FL*Lh - Ft*FF*Ft*FL*Fh*LL*Lh - Ft*FL*Lt*LL*Lh*LL*Lh,
# ttht
p(13)- Ft*FF*Ft*FF*Fh*FF*Ft - Ft*FF*Ft*FF*Fh*FL*Lt - Ft*FF*Ft*FL*Fh*LL*Lt - Ft*FL*Lt*LL*Lh*LL*Lt,
# ttth
p(13)- Ft*FF*Ft*FF*Ft*FF*Fh - Ft*FF*Ft*FF*Ft*FL*Lh - Ft*FF*Ft*FL*Ft*LL*Lh - Ft*FL*Lt*LL*Lt*LL*Lh,
# tttt
p(15)- Ft*FF*Ft*FF*Ft*FF*Ft - Ft*FF*Ft*FF*Ft*FL*Lt - Ft*FF*Ft*FL*Ft*LL*Lt - Ft*FL*Lt*LL*Lt*LL*Lt;
> ideal j = std(i);
> eliminate(j, FF*FL*LL*Fh*Ft*Lh*Lt);
The output is a list of 53 generating invariants. ♦