Divide and Conquer: A Mixture-Based Approach to RAPTOR Regional - PowerPoint PPT Presentation

Divide and Conquer: A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai The Problem RAPT Divide and Conquer: A Mixture-Based Approach to RAPTOR Regional Adaptation for MCMC Theoretical Results A Fish-Bone-Shaped Distribution and A Square-Shaped Distribution Yan Bai Summary References Department of Statistics, University of Toronto Collaborators: V. Radu Craiu (University of Toronto) and Antonio F. Di Narzo (University of Bologna) Graduate Research Day on April 29, 2010

Divide and Conquer: The Problem A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai • We assume that the population of interest is heterogenous, or it The Problem can be represented as a non-standard density. The posterior RAPT distribution can be multimodal. RAPTOR • MCMC sampling from multimodal distributions can be extremely Theoretical Results difficult as the chain can get trapped in one mode due to low A Fish-Bone-Shaped Distribution and A probability regions between the modes. Square-Shaped Distribution Some approaches: Summary • Gelman and Rubin (1992); Geyer and Thompson (1995); Neal References (1996); Richardson and Green (1997); Kou et al. (2006). • One possible approach is to approximate the multimodal posterior distribution with a mixture of Gaussians in West (1993) who shows that such an approximation may be useful for computation even if the posterior is skewed and not necessarily multimodal. • Adaptive MCMC algorithms based on the same natural approach have been developed by Giordani and Kohn (2006), Andrieu and Thoms (2008) and Craiu et al. (2009).

Divide and Conquer: Regional AdaPTive Sampler (RAPT) in Craiu et al. (2009) A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai The Problem • Assume that one has reasonable knowledge about regions where RAPT different sampling regions are needed. RAPTOR Theoretical Results • One could use sophisticated methods to detect the modes of a A Fish-Bone-Shaped multimodal distribution (see Sminchisescu and Triggs (2001), Neal Distribution and A Square-Shaped (2001)), but it is not clear how to use such techniques for defining Distribution the desired partition of the sample sample. Summary • Simply, assume the sample space S = S 1 ∪ S 2 . RAPT’s proposal References 2 λ ( j ) Q ( j ) ( X n , · ) = ˜ � i Q i ( X n , · ) for j = 1 , 2 , i =1 where Q i and the mixture weights λ ( j ) are adapted. i • The regions remain unchanged.

Divide and Conquer: RAPT A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai The Problem RAPT RAPTOR Theoretical Results A Fish-Bone-Shaped Distribution and A Square-Shaped Distribution Summary References

Divide and Conquer: Regional Adaptive with Online Recursion (RAPTOR) A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai The Problem RAPT RAPTOR • We consider a different framework allowing the regions to evolve as Theoretical Results the simulation proceeds. A Fish-Bone-Shaped • The regional adaptive random walk Metropolis algorithm proposed Distribution and A Square-Shaped here relies on the approximation of the target distribution π with a Distribution mixture of Gaussians. Summary References • The partition of the sample space used for RAPTOR is defined based on the mixture parameters which are updated using the simulated samples. • The algorithm 7 in Andrieu and Thoms (2008) differs from RAPTOR in a few important aspects.

Divide and Conquer: RAPTOR - Recursive Adaptation A Mixture-Based Approach to Regional Adaptation for MCMC • Assume that π has K modes in the sample space S ⊂ R d . Yan Bai Consider its approximation by the mixture model The Problem K RAPT � β ( j ) N d ( x , µ ( j ) , Σ ( j ) ) , ˜ q η ( x ) = (1) RAPTOR j =1 Theoretical Results j =1 β ( j ) = 1 and N d ( x , µ, Σ) is the probability density of a A Fish-Bone-Shaped where � K Distribution and A Square-Shaped d -variate Gaussian distribution with mean µ and covariance matrix Distribution Σ. Summary • We are facing an online setting in which the parameters need to be References updated each time new data are added to the sample. • Suppose that at time n − 1 the current parameter estimates are � � β ( j ) n − 1 , µ ( j ) n − 1 , Σ ( j ) η n − 1 = 1 ≤ j ≤ K and the available samples are n − 1 { x 0 , x 1 , · · · , x n − 1 } . • We define the mixture indicator Z n such that given x n , P ( Z n = j | x n , η n − 1 ) = ν ( j ) n β ( j ) n − 1 N d ( x n , µ ( j ) n − 1 , Σ ( j ) n − 1 ) ν ( j ) = , ∀ 1 ≤ i ≤ n , 1 ≤ j ≤ K . (2) n i =1 β ( i ) n − 1 N d ( x n , µ ( i ) n − 1 , Σ ( i ) � K n − 1 )

Divide and Conquer: RAPTOR - Recursive Adaptation A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai � � β ( j ) n , µ ( j ) n , Σ ( j ) • The recursive estimator η n = n The Problem 1 ≤ j ≤ K RAPT n 1 RAPTOR β ( j ) � ν ( j ) = n i n + 1 Theoretical Results i =0 A Fish-Bone-Shaped n = µ ( j ) n ( x n − µ ( j ) µ ( j ) n − 1 + ρ n γ ( j ) Distribution and A n − 1 ) , Square-Shaped Distribution n − 1 ) ⊤ − Σ ( j ) � � Σ ( j ) n = Σ ( j ) n − 1 + ρ n γ ( j ) (1 − γ ( j ) n )( x n − µ ( j ) n − 1 )( x n − µ ( j ) n n − 1 Summary (3) References ν ( j ) where γ ( j ) = and ρ n is a non-increasing positive sequence. n n � n i =0 ν ( j ) n • Sample Mean and Sample Covariance: given { x 0 , x 1 , · · · , x n } 1 µ < w > = µ < w > n + 1( x n − µ < w > n − 1 + n − 1 ) , n � � 1 1 n − 1 ) ⊤ − Σ < w > Σ < w > = Σ < w > n + 1)( x n − µ < w > n − 1 )( x n − µ < w > (1 − n − 1 + . n n − 1 n + 1 (4)

Divide and Conquer: RAPTOR - Definition of Regions A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai The Problem • Suppose that the K -partition of the sample space Π = {S (1) , S (2) , · · · , S ( K ) } satisfying S = S (1) ∪ S (2) ∪ · · · ∪ S ( K ) RAPT and S ( i ) ∩ S ( j ) = ∅ for i � = j . RAPTOR Theoretical Results • Denote the projection of π on the set A by A Fish-Bone-Shaped � π A ( x ) = π ( x ) I A ( x ) / A π ( y ) dy . We try to find an “optimal” Distribution and A Square-Shaped estimator of K -partition minimizing Distribution Summary � � � KL ( π S ( i ) , N ( i ) max d ) � � References � 1 ≤ i ≤ K where N ( i ) d ( x ) = N d ( x , µ ( i ) , Σ ( i ) ) (defined in Eq. (1)) and the � Kullback-Leibler divergence KL ( f , g ) = log( f ( x ) / g ( x )) f ( x ) dx . • With this aim, we define � � S ( j ) N d ( x , µ ( i ) n , Σ ( i ) = x ∈ S : arg max n ) = j . (5) n i

Divide and Conquer: RAPTOR - Definition of Regions A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai The Problem RAPT RAPTOR Theoretical Results A Fish-Bone-Shaped Distribution and A Square-Shaped Distribution Summary References

Divide and Conquer: RAPTOR - Definition of the Proposal Distribution A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai • At each time n , the sample { x 0 , x 1 , · · · , x n } is obtained, the The Problem corresponding parameter estimators { µ ( j ) n , Σ ( j ) n : j = 1 , 2 , · · · , K } , RAPT µ < w > , Σ < w > are computed. The recursive estimates can RAPTOR n n determine the recursive region partition {S (1) , S (2) , · · · , S ( K ) } . Theoretical Results A Fish-Bone-Shaped • Propose the value y n +1 from the Proposal distribution Distribution and A Square-Shaped Distribution K � I S ( j ) ( x n ) N d ( y ; x n , s d ˜ Σ ( j ) Summary Q n ( x n , dy ) =(1 − α ) n ) dy + References j =1 α N d ( y ; x n , s d ˜ Σ < w > ) dy , n where s d = 2 . 38 2 / d , ˜ Σ ( j ) n = Σ ( j ) n + ǫ I d , ˜ Σ < w > = Σ < w > + ǫ I d , and n n α = 1 / 3. • Accept or reject y n +1 for x n +1 according to Metropolis acceptance rate min(1 , π ( y ) q ( y , x ) π ( x ) q ( x , y ) ). • Compute the recursive parameter estimators indexed by n + 1.

Divide and Conquer: Theoretical Results A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai The Problem (A1): There is a compact set S ⊂ R d such that the target density π is RAPT RAPTOR continuous on S , positive on the interior of S , and zero outside of S . Theoretical Results (A2): The sequence { ρ j : j ≥ 1 } is positive and non-increasing. A Fish-Bone-Shaped (A3): For all k = 1 , 2 , · · · , K , Distribution and A Square-Shaped Distribution l � ρ j γ k Summary P (lim sup sup j = 0) = 1 . i →∞ l ≥ i References j = i Theorem a) Assuming (A1-2), the RAPTOR algorithm is ergodic to π . b) Assuming (A2-3), the adaptive parameters µ ( j ) n , Σ ( j ) converge in n probability for any j ∈ { 1 , 2 , · · · , K } .

Divide and Conquer: A Fish-Bone-Shaped Distribution A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai The Problem RAPT RAPTOR Theoretical Results A Fish-Bone-Shaped Distribution and A Square-Shaped Distribution Summary References

Divide and Conquer: A Square-Shaped Distribution A Mixture-Based Approach to Regional Adaptation for MCMC Yan Bai The Problem RAPT RAPTOR Theoretical Results A Fish-Bone-Shaped Distribution and A Square-Shaped Distribution Summary References