1 node
In the simplest case, assume there is only one binary node, X1. I think the probability of the data given the order can be simplified to:
Now, running on some small datasets (1,2,and 3 entries), the results are:
- X1={0}:
-0.693147180559945 - X1={0,1}:
-2.07944154167984 - X1={0,0}:
-0.980829253011726 - X1={0,0,0}:
-1.16315080980568 - X1={0,0,1}:
-2.77258872223978
2 node
In the case of two nodes, X1 and X2, the equation for X1<X2 becomes
where n_(a,b) is n_{X1=a,X2=b}. A couple test results are
- X1={0}
X2={0}
X1<X2: -0.693147180559945
X2<X1: -0.693147180559945 - X1={0,0}
X2={0,0}
X1<X2: -1.21444410419323
X2<X1: -1.21444410419323 - X1={0,1}
X2={0,0}
X1<X2: -2.54944517092557
X2<X1: -2.54944517092557 - X1={0,1,0}
X2={0,0,1}
X1<X2: -5.03435182071357
X2<X1: -5.03435182071357
3 node
A couple tests:
- X1={0}
X2={0}
X3={0}
X1<X2<X3: 4.44089209850063e-16
X1<X3<X2: 4.44089209850063e-16
X2<X1<X3: 4.44089209850063e-16
X2<X3<X1: 4.44089209850063e-16
X3<X1<X2: 4.44089209850063e-16
X3<X2<X1: 4.44089209850063e-16 - X1={0,0}
X2={0,0}
X3={0,0}
X1<X2<X3: -0.708631022250785
X1<X3<X2: -0.708631022250785
X2<X1<X3: -0.708631022250785
X2<X3<X1: -0.708631022250785
X3<X1<X2: -0.708631022250785
X3<X2<X1: -0.708631022250785 - X1={0,1}
X2={0,0}
X3={0,0}
X1<X2<X3: -2.29351179678837
X1<X3<X2: -2.29351179678837
X2<X1<X3: -2.29351179678837
X3<X1<X2: -2.29351179678837
X2<X3<X1: -2.28836378027097
X3<X2<X1: -2.28836378027097 - X1={0,1}
X2={0,1}
X3={0,0}
X1<X2<X3: -2.94248775903518
X2<X1<X3: -2.94248775903518
X1<X3<X2: -2.8942856572173
X2<X3<X1: -2.8942856572173
X3<X1<X2: -2.8942856572173
X3<X2<X1: -2.8942856572173 - X1={0,0,1}
X2={0,0,0}
X3={0,0,0}
X1<X2<X3: -3.31084145201962
X1<X3<X2: -3.31084145201962
X2<X1<X3: -3.31084145201962
X3<X1<X2: -3.31084145201962
X2<X3<X1: -3.30491111588477
X3<X2<X1: -3.30491111588477 - X1={0,0,1,0,0}
X2={0,1,0,1,1}
X3={0,1,0,0,0}
X1<X3<X2: -10.049176787447
X3<X1<X2: -10.049176787447
X3<X2<X1: -10.0403062276219
X2<X3<X1: -10.0403062276219
X1<X2<X3: -10.0271462031263
X2<X1<X3: -10.0271462031263