Foundations of Data Science

Proof: This is the main Lemma. The proof of the Lemma uses a crucial idea of probability
flows. We will use two ways of calculating the probability flow from heavy states
to ligh states when we execute one step of the Markov chain starting at probabilities a.
The probability vector after that step is aP . Now, a − aP is the net loss of probability
for each state due to the step.
Consider a particular group G t = {k, k + 1, . . . , l}, say. First consider the case when
k < i 0 . Let A = {1, 2, . . . , k}. The net loss of probability for each state from the set A in
one step is ∑ k
i=1 (a i − (aP ) i ) which is at most 2 by the proof of Theorem 5.2.
t
Another way to reckon the net loss of probability from A is to take the difference of
the probability flow from A to Ā and the flow from Ā to A. For any i < j,
net-flow(i, j) = flow(i, j) − flow(j, i) = π i p ij v i − π j p ji v j = π j p ji (v i − v j ) ≥ 0,
Thus, for any two states i, j, with i heavier than j, i.e., i < j, there is a non-negative net
flow from i to j. [This is intuitvely reasonable, since, it says that probability is flowing
from heavy to light states.] Since, l ≥ k, the flow from A to {k + 1, k + 2, . . . , l} minus
the flow from {k + 1, k + 2, . . . , l} to A is nonnegative. Since for i ≤ k and j > l, we have
v i ≥ v k and v j ≤ v l+1 , the net loss from A is at least
∑
Thus,
Since
i≤k
j>l
k∑
π j p ji (v i − v j ) ≥ (v k − v l+1 ) ∑ i≤k
j>l
i=1 j=k+1
(v k − v l+1 ) ∑ i≤k
j>l
l∑
π j p ji ≤
l∑
j=k+1
π j p ji .
π j p ji ≤ 2 t . (5.9)
π j ≤ εΦπ(A)/4
and by the definition of Φ, using (5.8)
∑
π j p ji ≥ ΦMin(π(A), π(Ā)) ≥ εΦγ k/2,
i≤k