Foundations of Data Science

p(a) = 1 2
p(b) = 1 4
p(c) = 1 8
p(d) = 1 8
1
2
a
d
1
8
1
4
b
c
1
8
1
a → b
3
1
a → c
3
1 2
4
1 2
8
1
8
1 = 1 c → a 1 6 3
1 = 1 1
c → b
12 3
2
1 = 1 1
c → d
12 3
a → d 1 3
a → a 1− 1 6 − 1
12 − 1 12 = 2 c → c 1− 1 3 3 − 1 3 − 1 3 = 0
1
b → a d → a 1 3 3
1 1 4
b → c
3 8 1 = 1 1
d → c
6 3
b → b 1− 1 3 − 1 6 = 1 d → d 1− 1 2 3 − 1 3 = 1 3
p(a) = p(a)p(a → a) + p(b)p(b → a) + p(c)p(c → a) + p(d)p(d → a)
= 1 2
2
3 + 1 4
1
3 + 1 8
1
3 + 1 8
1
3 = 1 2
p(b) = p(a)p(a → b) + p(b)p(b → b) + p(c)p(c → b)
= 1 2
1
6 + 1 4
1
2 + 1 8
1
3 = 1 4
p(c) = p(a)p(a → c) + p(b)p(b → c) + p(c)p(c → c) + p(d)p(d → c)
= 1 1
+ 1 1
+ 1 0 + 1 1
= 1
2 12 4 6 8 8 3 8
p(d) = p(a)p(a → d) + p(c)p(c → d) + p(d)p(d → d)
= 1 2
1
12 + 1 8
1
3 + 1 8
1
3 = 1 8
Figure 5.2: Using the Metropolis-Hasting algorithm to set probabilities for a random walk
so that the stationary probability will be the desired probability.
Example: Consider the graph in Figure 5.2. Using the Metropolis-Hasting algorithm,
assign transition probabilities so that the stationary probability of a random walk is
p(a) = 1, p(b) = 1, p(c) = 1, and p(d) = 1 . The maximum degree of any vertex is three,
2 4 8 8
so at a, the probability of taking the edge (a, b) is 1 1 2
or 1 . The probability of taking the
3 4 1 6
edge (a, c) is 1 1 2
or 1 and of taking the edge (a, d) is 1 1 2
or 1 . Thus, the probability
3 8 1 12 3 8 1 12
of staying at a is 2. The probability of taking the edge from b to a is 1 . The probability
3 3
of taking the edge from c to a is 1 and the probability of taking the edge from d to a is
3
1
. Thus, the stationary probability of a is 1 1
+ 1 1
+ 1 1
+ 1 2
= 1 , which is the desired
3 4 3 8 3 8 3 2 3 2
probability.
5.2.2 Gibbs Sampling
Gibbs sampling is another Markov Chain Monte Carlo method to sample from a
multivariate probability distribution. Let p (x) be the target distribution where x =
(x 1 , . . . , x d ). Gibbs sampling consists of a random walk on an undirectd graph whose
vertices correspond to the values of x = (x 1 , . . . , x d ) and in which there is an edge from
x to y if x and y differ in only one coordinate. Thus, the underlying graph is like a
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