Gibbs sampler - simple properties

1. Gibbs sampler - simple properties • It’s not hard to show that this MC chain is aperiodic. • Often is reversible distribution. • If in addition the chain is irreducible (which depends on which elements of have non-zero probability), then this MC is a correct MCMC algorithm for simulating random variables with distribution .

2. Systematic Gibbs sampler • A commonly used variant of the Gibbs sampler is the MC obtained by systematic cycle trough the vertex set. • For instance, for we may decide to update vertex:

3. Back to the Boolean cube • In this case: , . • In each step we pick and flip the appropriate coordinate of with probability .

4. Stationary distribution for the Boolean Cube • We will show that the uniform distribution is reversible (and therefore stationary). • Denote by the transition probability from state to . • We need to show that: for any two states. • Denote by the number of indexes in which and differ: • Case no.1: • Case no.2: • Case no.3:

5. Reminder - Coupling • For a given MC with state space and transitions matrix , a coupling for the MC is a MC on the state space with transition probabilities defined by:

6. Reminder (cont.) - The coupling lemma • For a coupling , based on a ground MC on . Suppose is a function satisfying the condition: for all     and all : then the mixing time for is bounded by .

7. Fast convergence of the Gibbs sampler for the Boolean cube • Coupling: we run two Markov chains and simultaneously. • We go threw the coordinates systematically. • If at some integer time it holds that: • We flip both of them with probability • Otherwise we flip each one separately with same probability.

8. Fast convergence of the Gibbs sampler for the Boolean cube • For any coordinate : after one cycle threw the coordinates: • After cycles threw the chain: • The event implies that for at least one coordinate : meaning that:

9. Fast convergence of the Gibbs sampler for the Boolean cube • Setting and solving for gives: • Now, using the coupling lemma, it means that the mixing time is bounded by:

10. Gibbs sampler for q-colorings • For a vertex and an assignment of colors to the vertices other than , the conditional distribution of the color at is uniform over the set of all colors that are not attained in at some neighbor of . • A Gibbs Sampler for random q-colorings is therefore an -valued MC where at each time , transitions take place as follows: • Pick a vertex uniformly at random. • Pick according to the uniform distribution over the set of colors that are not attained at any neighbor of . • Leave the color unchanged at all other vertices, i.e. let for all vertices except .

11. Gibbs sampler for q-colorings • It has as a stationary distribution, the proof is similar to the hard-core model case, based on the fact that it is reversible. • Whether or not the chain is irreducible depends on and , and it is a nontrivial question to generally determine this. But for big enough ( is always enough) it is.

12. Fast convergence of Gibbs sampler for q-coloring • Theorem [HAG 8.1]: Let be a graph. Let be the number of vertices in , and suppose any vertex has at most neighbors. Suppose furthermore that Then, for any fixed , the number of iterations needed for the systematic sweep Gibbs sampler described above (starting from any fixed -coloring ) to come within total variation distance of the target distribution is at most:

13. Fast convergence of Gibbs sampler for q-coloring • Comments: • Can be proved for (rather than ). • The important bound is: for some .

14. Fast convergence of Gibbs sampler for q-coloring • Coupling: we run two -valued Markov chains and simultaneously. • When a vertex is chosen to be updated, we pick a new color for it uniformly at random among the set of colors that are not attained by any neighbor of . Concretely: • Let be an i.i.d sequence of random permutations of , each of them uniformly distributed on the set the of such permutations.

15. Fast convergence of Gibbs sampler for q-coloring (cont.) • At each time , the updates of the 2 chains use the permutation and the vertex to be updated is assigned the new value where: in the first chain. • In the second chain, we set where:

16. Successful and not-successful updates • If at some (random, hopefully not too large) time , we will also have for all • First consider the probability that: for a given vertex . • We call an update of the 2 chains at time successful if it results in having: • Otherwise – we say that the update failed.

17. Fast convergence of Gibbs sampler for q-coloring • Notations: • The event of having successful update has probability: • Respectively: • Using that and that we get: (we only used and some algebra).

18. Fast convergence of Gibbs sampler for q-coloring • Hence, we have, after steps of the MC-s, that for each vertex : • Now, consider updates during the 2nd sweep of the Gibbs sampler, i.e. between times and . • For an update at time during the 2nd sweep to fail the configurations and need to differ in at least one neighbor of . • Each neighbor has with probability at most • By summing over at most neighbors we get that:

19. Fast convergence of Gibbs sampler for q-coloring • “discrepancy” means that there exists a neighbor of with . • By Repeating the arguments above we get: • Hence, after steps of the Markov chains, each vertex has probability at most: • By arguing the same for the third sweep:

20. Fast convergence of Gibbs sampler for q-coloring • and for any : • The event implies that for at least one vertex , meaning that:

21. Fast convergence of Gibbs sampler for q-coloring • Summary: • By setting and solving for , we get by using the coupling lemma that for: It holds that the mixing time after scans is bounded by .

22. Fast convergence of Gibbs sampler for q-coloring • To go from the number of scans to the number of steps of the MC, we have to multiply by , giving that: should be enough. • In order to make sure that gets an integer value we should take to be: