Markov Random Fields and Sampling Techniques: Monte Carlo Markov Chains and Gibbs Sampling, Assignments of Computer Science

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Joe Kniss

CS 530

Probabilistic and Geometric Methods Markov Random Fields

Sampling from a distribution

• Problem 8 on homework:
  • Generate N samples from a distribution.
  • 1. Generate N samples from Uniform Dist.
  • 2. Map through inv CDF of desired dist.

Other ways to Sample

• Discrete PMFs:
  • Make sure # of samples for each bin matches PMF, then shuffle
• Continuous PDFs?

Rejection Method

• 1. determine max prob/density d
• 2. determine min-max domain m,M
• 3. Draw sample from Uniform: x
• 4. Scale & Bias x to [ m,M ]
• 5. Draw sample from Uniform: t
• 6. Scale t by d: t=t*d
• 7. Keep x if t>p(x)

MCMC Example

• Generate N samples from:

P ( x " [ 0 : 255]) = e # x^^2 /1000^ / $ i^255 = 0 e # i^2 /

MCMC

• Choose an initial xt
  • According to some “candidate” dist
• Choose “next” xt+
  • xt jumps to xt+1 according to:

xt + 1 =

xt + 1 P ( xt + 1 ) > P ( xt ) xt + 1 P ( xt + 1 ) / P ( xt ) > rand() Draw new xt + 1 & start over

#^ $

%^ $

MCMC

• Samples form a markov chain
• We are sampling this conditional
  • But we never need the inv CDF!
  • Does P(x) need to be “normalized”?
  • Works for complicated PMFs and PDFs

P (xt + 1 | xt ) " min( P ( xt + 1 ) / P ( xt ), 1 )

MCMC

• What about the multidimensional case?
  • Not so easy to come up with inv CDF!

Gibbs Sampling

• Conditional distribution:
• Update each xi independently
• This is a 1D PMF
• Maybe you can go back to inv CDF! …

P ( x i^ t^ | x^ ( jt "# i 1))

Convergence, Stopping Criteria

• The first many samples may not be a good sampling of distribution.
• For complicated PMFs/PDFs may take some time to “settle down”

Markov Random Fields

• Remember joint to conditional:

s !

t 1

!

t 2

!

t 3

t 4

!

t 5 t 6 t 7 t 8

P ( xs | xt , t " Ns )

= P P ( x ( sx ,^ tx , tt^ , t "^ " N^ Ns ) s )

Markov Random Fields

• It can be difficult to manage high- dimensional joint/conditional distributions

s !

t 1

!

t 2

!

t 3

t 4

!

t 5 ! t 6 t 7 t 8

P ( xs | xt 1 , xt 2 , xt 3 , xt 4 , xt 5 , xt 6 , xt 7 , xt 8 )

Gibbs Sampler for MRFs

• Made easy if we apply the Gibbs method: - Update one site at a time - Pick one site (s) at random - Draw new xs from conditional - … and repeat, repeat, repeat !

P ( xs | xt , t " Ns )

Gibbs Potential

• Still difficult to deal with conditional:
• Gibbs potential
  • Compose conditional using “easier” functions ! – Operate on subsets of neighborhood

P ( xs | xt 1 , xt 2 , xt 3 , xt 4 , xt 5 , xt 6 , xt 7 , xt 8 )