Foundations of Data Science

5
8
7
12
1
3
1
6
1
8
1
3
3,1 3,2 3,3
1
6
1
12
2,1 2,2 2,3
1
6
1
12
1,1 1,2 1,3
1
4
1
6
5
12
3
8
3
4
p(1, 1) = 1 3
p(1, 2) = 1 4
p(1, 3) = 1 6
p(2, 1) = 1 8
p(2, 2) = 1 6
p(2, 3) = 1
12
p(3, 1) = 1 6
p(3, 2) = 1 6
p(3, 3) = 1
12
p (11)(12) = 1p d 12/(p 11 + p 12 + p 13 = 1 1
2
Calculation of edge probability p (11)(12)
/( 1 1 1
= 1 1
/ 9 = 1 1 4
= 1
4 3 4 6 2 4 12 2 4 3 6
p (11)(12) = 1 2
p (11)(13) = 1 2
p (11)(21) = 1 2
p (11)(31) = 1 2
Edge probabilities.
1 4
= 1
4 3 6
1 4
= 1
6 3 9
1 8
= 1
8 5 10
1 8
= 2
6 5 15
p (12)(11) = 1 2
p (12)(13) = 1 2
p (12)(22) = 1 2
p (12)(32) = 1 2
1 4
= 2
3 3 9
1 4
= 1
6 3 9
1 12
= 1
6 7 7
1 12
= 1
6 7 7
p (13)(11) = 1 2
p (13)(12) = 1 2
p (13)(23) = 1 2
p (13)(33) = 1 2
1 4
= 2
3 3 9
1 4
= 1
4 3 6
1 3
= 1
12 1 8
1 3
= 1
12 1 8
p (21)(22) = 1 2
p (21)(23) = 1 2
p (21)(11) = 1 2
p (21)(31) = 1 2
1 8
= 2
6 3 9
1 8
= 1
12 3 9
1 8
= 4
3 5 15
1 8
= 2
6 5 15
p 11 p (11)(12) = 1 1
= 1 2
= p 3 6 4 9 12p (12)(11)
p 11 p (11)(13) = 1 1
= 1 2
= p 3 9 6 9 13p (13)(11)
p 11 p (11)(21) = 1 1
= 1 4
= p 3 10 8 15 21p (21)(11)
Verification of a few edges.
Note that the edge probabilities out of a state such as (1,1) do not add up to one.
That is, with some probability the walk stays at the state that it is in. For example,
p (11)(11) = 1 − (p (11)(12) + p (11)(13) + p (11)(21) + p (11)(31) ) = 1 − 1 − 1 − 1 − 1 = 9 .
6 24 32 24 32
Figure 5.3: Using the Gibbs algorithm to set probabilities for a random walk so that the
stationary probability will be a desired probability.
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