Computer Vision Group Prof. Daniel Cremers. 11. Sampling Methods: Markov Chain Monte Carlo

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1 Group Prof. Daniel Cremers 11. Sampling Methods: Markov Chain Monte Carlo

2 Markov Chain Monte Carlo In high-dimensional spaces, rejection sampling and importance sampling are very inefficient An alternative is Markov Chain Monte Carlo (MCMC) It keeps a record of the current state and the proposal depends on that state Most common algorithms are the Metropolis- Hastings algorithm and Gibbs Sampling 2 Group

3 Markov Chains Revisited A Markov Chain is a distribution over discretestate random variables x 1,...,x M p(x 1,...,x T )=p(x 1 )p(x 2 x 1 ) = p(x 1 ) so that The graphical model of a Markov chain is this: TY t=2 p(x t x t 1 ) T We will denote as a row vector p(x t x t 1 ) t A Markov chain can also be visualized as a state transition diagram. 3 Group

4 The State Transition Diagram time k=1 A 11 A 11 states k=2 k=3 A 33 A 33 t-2 t-1 t 4 Group

5 Some Notions The Markov chain is said to be homogeneous if the transitions probabilities are all the same at every time step t (here we only consider homogeneous Markov chains) The transition matrix is row-stochastic, i.e. all entries are between 0 and 1 and all rows sum up to 1 Observation: the probabilities of reaching the states can be computed using a vector-matrix multiplication 5 Group

6 The Stationary Distribution KX The probability to reach state k is Or, in matrix notation: t = t 1 A k,t = i=1 i,t 1 A ik We say that t is stationary if t = t 1 Questions: How can we know that a stationary distributions exists? And if it exists, how do we know that it is unique? 6 Group

7 The Stationary Distribution (Existence) To find a stationary distribution we need to solve the eigenvector problem A T v = v The stationary distribution is then = v T where v is the eigenvector for which the eigenvalue is 1. This eigenvector needs to be normalized so that it is a valid distribution. Theorem (Perron-Frobenius): Every rowstochastic matrix has such an eigen vector, but this vector may not be unique. 7 Group

8 Stationary Distribution (Uniqueness) A Markov chain can have many stationary distributions Sufficient for a unique stationary distribution: we can reach every state from any other state in finite steps at non-zero probability (i.e. the chain is ergodic) This is equivalent to the property that the transition matrix is irreducible: i, j 9m (A m ) ij > 0 8 Group

9 Main Idea of MCMC So far, we specified the transition probabilities and analysed the resulting distribution This was used, e.g. in HMMs Now: We want to sample from an arbitrary distribution To do that, we design the transition probabilities so that the resulting stationary distribution is our desired (target) distribution! 9 Group

10 Detailed Balance Definition: A transition distribution t satisfies the property of detailed balance if i A ij = j A ji The chain is then said to be reversible. 1 3 A 31 + A 13 A A 13 + t-1 t 10 Group

11 Making a Distribution Stationary Theorem: If a Markov chain with transition matrix A is irreducible and satisfies detailed balance wrt. the distribution, then is a stationary distribution of the chain. Proof: KX KX K X i A ij = j A ji = j A ji = j 8j i=1 i=1 i=1 it follows. = A This is a sufficient, but not necessary condition. 11 Group

12 Sampling with a Markov Chain The idea of MCMC is to sample state transitions based on a proposal distribution q. The most widely used algorithm is the Metropolis-Hastings (MH) algorithm. In MH, the decision whether to stay in a given state is based on a given probability. If the proposal distribution is move to state x 0 with probability min Unnormalized target distribution q(x 0 x) 1, p(x0 )q(x x 0 ) p(x)q(x 0 x), then we 12 Group

13 The Metropolis-Hastings Algorithm Initialize x 0 for define s =0, 1, 2,... sample compute acceptance probability compute x = x s x 0 q(x 0 x) = p(x0 )q(x x 0 ) p(x)q(x 0 x) r =min(1, ) sample u U(0, 1) set new sample to x s+1 = ( x 0 if u<r x s if u r 13 Group

14 Why Does This Work? We have to prove that the transition probability of the MH algorithm satisfies detailed balance wrt the target distribution. Theorem: If p MH (x 0 x) is the transition probability of the MH algorithm, then p(x)p MH (x 0 x) =p(x 0 )p MH (x x 0 ) Proof: 14 Group

15 Why Does This Work? We have to prove that the transition probability of the MH algorithm satisfies detailed balance wrt the target distribution. Theorem: If p MH (x 0 x) is the transition probability of the MH algorithm, then p(x)p MH (x 0 x) =p(x 0 )p MH (x x 0 ) Note: All formulations are valid for discrete and for continuous variables! 15 Group

16 Choosing the Proposal A proposal distribution is valid if it gives a nonzero probability of moving to the states that have a non-zero probability in the target. A good proposal is the Gaussian, because it has a non-zero probability for all states. However: the variance of the Gaussian is important! with low variance, the sampler does not explore sufficiently, e.g. it is fixed to a particular mode with too high variance, the proposal is rejected too often, the samples are a bad approximation 16 Group

17 Example MH with N(0, ) proposal Target is a mixture of 2 1D Gaussians. Proposal is a Gaussian with different variances Iterations Samples MH with N(0, ) proposal MH with N(0, ) proposal Iterations Samples Iterations Samples Group

18 Gibbs Sampling Initialize {z i : i =1,...,M} For Sample Sample... =1,...,T Sample z ( +1) 1 p(z 1 z ( ) 2,...,z( ) M ) z ( +1) 2 p(z 2 z ( +1) 1,...,z ( ) M ) z ( +1) M p(z M z ( +1) 1,...,z ( +1) M 1 ) Idea: sample from the full conditional This can be obtained, e.g. from the Markov blanket in graphical models. 18 Group

19 Gibbs Sampling: Example Use an MRF on a binary image with edge potentials (x s,x t )=exp(jx s x t ) ( Ising model ) and node potentials (x t )=N (y t x t, 2 ) x t 2 { 1, 1} y t x s x t 19 Group

20 Gibbs Sampling: Example Use an MRF on a binary image with edge potentials (x s,x t )=exp(jx s x t ) ( Ising model ) and node potentials (x t )=N (y t x t, 2 ) Sample each pixel in turn sample 1, Gibbs 1 sample 5, Gibbs 1 mean after 15 sweeps of Gibbs After 1 sample After 5 samples Average after 15 samples 20 Group

21 Gibbs Sampling for GMMs Again, we start with the full joint distribution: p(x, Z, µ,, ) =p(x Z, µ, )p(z )p( ) KY k=1 p(µ k )p( k ) It can be shown that the full conditionals are: p(z i = k x i, µ,, ) / k N (x i µ k, k ) NX p( z) =Dir({ k + z ik } K k=1) i=1 p(µ k k,z,x)=n(µ k m k,v k ) p( k µ k,z,x)=iw( k S k, k ) (linear-gaussian) 21 Group

22 Gibbs Sampling for GMMs First, we initialize all variables Then we iterate over sampling from each conditional in turn In the end, we look at and µ k k 22 Group

23 How Often Do We Have To Sample? Here: after 50 sample rounds the values don t change any more In general, the mixing time is related to the eigen gap of the transition matrix: = 1 2 apple O( 1 log n ) 23 Group

24 Gibbs Sampling is a Special Case of MH The proposal distribution in Gibbs sampling is q(x 0 x) =p(x 0 i x i )I(x 0 i = x i ) This leads to an acceptance rate of: = p(x0 )q(x x 0 ) p(x)q(x 0 x) = p(x0 i x0 i )p(x0 i )p(x i x 0 i ) p(x i x i )p(x i )p(x 0 i x i) =1 Although the acceptance is 100%, Gibbs sampling does not converge faster, as it only updates one variable at a time. 24 Group

25 Summary Markov Chain Monte Carlo is a family of sampling algorithms that can sample from arbitrary distributions by moving in state space Most used methods are the Metropolis-Hastings (MH) and the Gibbs sampling method MH uses a proposal distribution and accepts a proposed state randomly Gibbs sampling does not use a proposal distribution, but samples from the full conditionals 25 Group