# Balance equations

## Outline

### Topics

- Global balance (synonym: invariance)
- Detailed balance (synonym: time-reversibility)

### Rationale

We need formulas to help us prove that MH is \(\pi\)-invariant (and hence combined with irreducibility, admits a LLN).

## Example

We continue our eccentric tourists example…

**Question:** which travelling scheme(s) are \(\pi\)-invariant?

**Intuition:** \(\pi\)-invariance in this context just means the fraction of tourists in each city will stay constant over time.

- Pairs of cities, where each pair of cities is assigned a fixed number of planes going back and forth between the two cities (see figure, planes always full).
- A
**tour**where a fixed number of planes, equal to the number of cities, go around the world on a common route (see figure, again, planes always full). - A central “hub”, where all flights land at the hub (see figure, again, planes always full).

a: b: c:

- a only.
- b only.
- c only.
- a, b only.
- b, c only.

The correct answer is 4.

**Intuition:** in a and b there is always the same number of planes leaving and arriving each airport on a given weekend. On the other hand, with the “hub” model, it is easy to see that the city with the hub airport will have a number of people in it increasing over time, hence not invariant.

## Global balance

**Question:** combine

- the definition of \(\pi\)-invariance, with

- the formula we derived for the marginals of a Markov chain, i.e., \[\mu_2(x') = \sum_x \mu_1(x) K(x' | x),\]

to get an equation characterizing \(\pi\)-invariance in terms of \(K\).

- \(\pi(x) K(x' | x) = \pi(x') K(x | x')\)
- \(\pi(x) K(x' | x) = \sum_x \pi(x) K(x' | x)\)
- \(\pi(x) = \sum_{x'} \pi(x) K(x | x')\)
- \(\pi(x') = \sum_x \pi(x) K(x' | x)\)
- None of the above.

The correct answer is 4.

This equation, \(\pi(x') = \sum_x \pi(x) K(x' | x)\) is called **global balance.**

## Detailed balance

**Question:** write mathematically “pairs of cities, where each pair of cities is assigned a fixed number of planes going back and forth between the two cities (planes always full).”

- \(\pi(x) K(x' | x) = \pi(x') K(x | x')\)
- \(\pi(x) K(x' | x) = \sum_x \pi(x) K(x' | x)\)
- \(\pi(x) = \sum_{x'} \pi(x) K(x | x')\)
- \(\pi(x') = \sum_x \pi(x) K(x' | x)\)
- None of the above.

The correct answer is 1.

- We want the number of tourist moving from city 1 to 2 to be the same as from city 2 to 1.
- Flow from city 1 to 2:
- The fraction of tourist in city 1: \(\pi(x)\)
- The fraction of those going to city 2: \(K(x' | x)\)
- Therefore the flow in one direction: \(\pi(x) K(x' | x)\).

- By a similar argument, flow from 2 to 1: \(\pi(x') K(x | x')\)

This equation, \(\pi(x) K(x' | x) = \pi(x') K(x | x')\) is called **detailed balance**.

## Relationship between detailed and global balance

**Question:** what is the relationship between local and global balance? Does one imply the other?

- Global balance implies local balance (converse is false).
- Local balance implies global balance (converse is false).
- Local balance holds iff global balance holds.
- Additional conditions are needed.
- None of the above ever hold.

Local balance implies global balance (converse is false).

**Proposition:** if local balance holds \[\pi(x) K(x' | x) = \pi(x') K(x | x'),\] then global balance holds \[\pi(x') = \sum_x \pi(x) K(x' | x).\]

**Proof:** Suppose local balance holds, then

\[\begin{align*} \sum_x \pi(x) K(x' | x) &= \sum_x \pi(x') K(x | x') \;\;\text{(by local balance)} \\ &= \pi(x') \sum_x K(x | x') \\ &= \pi(x'). \;\;\text{(since $K(\cdot|x')$ is a probability)} \end{align*}\]

**Reverse implication does not hold:** The “**tour**” travelling scheme shows that there are situations where global balance holds but not detailed balance.