By Levin D. A., Peres Y., Wilmer E. L.
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Define for λ ≥ 1, πλ (x) = λ f (x) , Z(λ) {Rmk:MetRat} 42 4. 4. A hill climb may become trapped at a local maximum. Fig:HillClimb where Z(λ) := x∈Ω λ f (x) is a normalizing constant making µ a probability measure. Note that π(x) is increasing in f (x), so that π favors x with large values of f (x). If f (y) < f (x), the Metropolis chain accepts a transition x → y with probability λ−[ f (x)− f (y)] . As λ → ∞, the chain more closely resembles the deterministic hill climb. Suppose that Ω = {x ∈ Ω : f (x) = max f (x) := f }.
When P is irreducible, all the states of the chain lie in a single communication class. 12. When a communication class consists of a single state z ∈ Ω, it follows that P(z, z) = 1 and we call z an absorbing state. Once a trajectory arrives at z, it is “absorbed” there and can never leave. 24(c) that every chain trajectory follows a weakly increasing path in the partial order on communication classes. Once the chain arrives in a class that is maximal in this order, it stays there forever. 8, which connects this structure to the concepts of recurrence and transience defined in Chapter 18.
1. Recall that B = {x : µ(x) > ν(x)}. Region I has area µ(B) − ν(B). Region II has area ν(Bc ) − µ(Bc ). Since the total area under each of µ and ν is 1, regions I and II must have the same area—and that area is µ − ν TV . 1) how to compute the total variation distance between two given distributions. We now give three extremely useful alternative characterizations. 2 reduces total variation distance to a simple sum over the state space. 3 describes total variation distance in terms of integrating a single function with respect to both underlying measures.
