SLIDE 1
Markov Chains, Mixing Time, and Gerrymandering
SLIDE 2 What is a Markov chain?
Definition (Event Space)
An event space Ω is a collection of events Σ.
Definition (Random Variable)
A random variable X is some mapping X : Ω → R, where X(ω ∈ Ω) represents the value of some outcome in Ω.
Example
Say we have a fair coin c. Then, we can define X to be X(ω) =
1 ω = tails
SLIDE 3
What is a Markov chain?
Example (Weather)
SLIDE 4
What is a Markov chain?
Definition (Markov property)
P(Xn+1 = σ | X1 = σ1, . . . , Xn = σn) = P(Xn+1 = σ | Xn = σn)
Definition (Markov chain)
Suppose we have some random process R = (X0, X1, . . . , Xn). Then, a Markov chain is R equipped with the Markov property.
SLIDE 5
What is a Markov chain?
Definition (Transition Matrix)
Given some Markov chain M = (X0, X1, . . . , Xn), its transition matrix P can be defined as Pi,j = P(Xn+1 = j | Xn = i)
Definition (Reversibility)
A Markov chain M = (X0, X1, . . . , Xn) is considered reversible if, given some probability distribution π, the following holds: πi · P(Xn+1 = j | Xn = i) = πj · P(Xn+1 = i | Xn = j)
SLIDE 6
What is a Markov chain?
Definition (Stationary Distribution)
A Markov chain M has reached a stationary distribution π if, for transition matrix P, π = π · P
SLIDE 7
What is a Markov chain?
Example (Weather)
P = 0.9 0.1 0.5 0.5
SLIDE 8 Pegden et al.
Theorem (1.1)
Let M = (X0, X1, . . . ) be a reversible Markov chain with stationary distribution π, and suppose the states of M have real-valued labels. If X0 ∼ π, then for any fixed k, the probability that the label of X0 is an ǫ-outlier from among the list of labels
- bserved in the trajectory X0, X1, X2, . . . , Xk is, at most,
√ 2ǫ.
SLIDE 9
Pegden et al.
A bit of clarity...
Assume that M has some stationary distribution π, and that, if we start at some distribution X0, we’ll eventually get to the stationary distribution, denoted as X0 ∼ π Then, pick some ǫ. If we have some labeling function ω : Ω → R, such that each state in M has some real-valued label, the probability that X0’s real-valued label, ω(X0), is ǫ(k + 1)-far away from ω(X1, X2, . . . , Xn) is √ 2ǫ.
SLIDE 10
Pegden et al.
A few conclusions arise: (1) If ω(X0) is a most extreme outlier, then the chain M will not approach its stationary distribution with p ≤ √ 2ǫ. (2) Given (1), consider Ω to be a set of districting plans. Then, define some scoring metric for these plans, called R : Ω → R, which acts as our labeling function. Next, consider some districting plan D = X0. If R(D) is an outlier, then the chain M cannot approach its stationary distribution. As such, D may be considered a ”gerrymander.”
SLIDE 11
Conclusion
References: (1) Assessing significance in a Markov chain without mixing, Pegden et al. (2) VRDI Intro. (3) Finite Markov Chains and Algorithmic Applications, Olle Haggstrom (London Mathematical Society).
