Restricted Collapsed Draw: Accurate Sampling for Hierarchical Chinese Restaurant Process Hidden Markov Models

📝 Abstract

We propose a restricted collapsed draw (RCD) sampler, a general Markov chain Monte Carlo sampler of simultaneous draws from a hierarchical Chinese restaurant process (HCRP) with restriction. Models that require simultaneous draws from a hierarchical Dirichlet process with restriction, such as infinite Hidden markov models (iHMM), were difficult to enjoy benefits of \markerg{the} HCRP due to combinatorial explosion in calculating distributions of coupled draws. By constructing a proposal of seating arrangements (partitioning) and stochastically accepts the proposal by the Metropolis-Hastings algorithm, the RCD sampler makes accurate sampling for complex combination of draws while retaining efficiency of HCRP representation. Based on the RCD sampler, we developed a series of sophisticated sampling algorithms for iHMMs, including blocked Gibbs sampling, beam sampling, and split-merge sampling, that outperformed conventional iHMM samplers in experiments

💡 Analysis

We propose a restricted collapsed draw (RCD) sampler, a general Markov chain Monte Carlo sampler of simultaneous draws from a hierarchical Chinese restaurant process (HCRP) with restriction. Models that require simultaneous draws from a hierarchical Dirichlet process with restriction, such as infinite Hidden markov models (iHMM), were difficult to enjoy benefits of \markerg{the} HCRP due to combinatorial explosion in calculating distributions of coupled draws. By constructing a proposal of seating arrangements (partitioning) and stochastically accepts the proposal by the Metropolis-Hastings algorithm, the RCD sampler makes accurate sampling for complex combination of draws while retaining efficiency of HCRP representation. Based on the RCD sampler, we developed a series of sophisticated sampling algorithms for iHMMs, including blocked Gibbs sampling, beam sampling, and split-merge sampling, that outperformed conventional iHMM samplers in experiments

📄 Content

arXiv:1106.0474v1 [stat.ML] 2 Jun 2011 Restricted Collapsed Draw: Accurate Sampling for Hierarchical Chinese Restaurant Process Hidden Markov Models Abstract We propose a restricted collapsed draw (RCD) sampler, a general Markov chain Monte Carlo sampler of simultaneous draws from a hierar- chical Chinese restaurant process (HCRP) with restriction. Models that require simultaneous draws from a hierarchical Dirichlet process with restriction, such as infinite Hidden markov mod- els (iHMM), were difficult to enjoy benefits of the HCRP due to combinatorial explosion in calculating distributions of coupled draws. By constructing a proposal of seating arrangements (partitioning) and stochastically accepts the pro- posal by the Metropolis-Hastings algorithm, the RCD sampler makes accurate sampling for com- plex combination of draws while retaining effi- ciency of HCRP representation. Based on the RCD sampler, we developed a series of sophis- ticated sampling algorithms for iHMMs, includ- ing blocked Gibbs sampling, beam sampling, and split-merge sampling, that outperformed conven- tional iHMM samplers in experiments. 1 Introduction Existing sampling algorithms for infinite hidden Markov models (iHMMs, also known as the hierarchical Dirichlet process HMMs) [??] do not use a hierarchical Chinese restaurant process (HCRP) [?], which is a way of repre- senting the predictive distribution of a hierarchical Dirich- let process (HDP) by collapsing, i.e. integrating out, the un- derlying distributions of the Dirichlet process (DP). While an HCRP representation provides efficient sampling for many other models based on an HDP [??] through reduc- ing the dimension of sampling space, it has been consid- ered rather “awkward” [?] to use an HCRP for iHMMs, due to the difficulty in handling coupling between random variables. In the simplest case, consider step-wise Gibbs sampling from an iHMM defined as πk ∼DP(β, α0) and xi-1 xi xi+1 yi-1 yi yi+1 xi+2 yi+2 xi-1 x’i xi+1 x’i+1 yi-1 yi yi+1 xi+2 yi+2 Figure 1: Step-wise Gibbs sampling in iHMM. Since the Dirichlet process prior is posed on transitions in iHMM, resampling xi involves taking two transitions, xi−1 →xi and xi →xi+1, simultaneously. In this case, we consider distribution of two draws (x′ i, x′ i+1) with restriction that the draws are consistent with remaining sequence, i.e., x′ i+1 = xi+1. β ∼GEM(γ). Given x1, . . . , xi−1, xi+1, . . . , xT , resam- pling hidden state xi at time step i actually consists of two draws (Figure 1), x′ i ∼πxi−1 and x′ i+1 ∼πx′ i, under the restriction (x′ i, x′ i+1) ∈C that these draws are consistent with the following sequence, i.e., C = {(x′ i, x′ i+1)|x′ i+1 = xi+1}. Under the HCRP, the two draws are coupled even if xi−1 ̸= x′ i, because distributions πxi−1, πx′ i as well as the base measure β are integrated out in an HCRP, and cou- pling complicates sampling from the restricted distribution. To generalize, the main part of the difficulty is to obtain a sample from a restricted joint distribution of simulta- neous draws from collapsed distributions, which we call restricted collapsed draw (RCD). Consider resampling L draws simultaneously, x = (xj1i1, . . . , xjLiL), from the respective restaurants j = (j1, . . . , jL), when we have a restriction C such that x ∈C. Step-wise Gibbs sampling from iHMM can be fitted into RCD with L = 2 by allowing restaurant index j2 to be dependent on the preceding draw xj1i1. In this paper, we point out that it is not enough to consider the distribution of draws. Since the HCRP introduces an additional set of latent variables s that accounts for the seat- ing arrangements of the restaurants, we have to compute an exact distribution of s as well, under the restriction. We want to perform sampling from the following conditional distribution, p(x, s|C) = 1 ZC I[ x ∈C ] p(x, s) , (1) where ZC is a normalization constant and I is the indicator function, whose value is 1 if the condition is true and 0 oth- erwise. Although non-restricted probability p(x, s) can be easily calculated for a given x and s, calculating the nor- malization constant ZC leads to a combinatorial explosion in terms of L. To solve this issue, we propose the restricted collapsed draw (RCD) sampler, which provides accurate distribu- tions of simultaneous draws and seating arrangements from HCRP. The RCD sampler constructs a proposal of seating arrangements using a given proposal of draws, and the pair of proposals are stochastically accepted by the Metropolis- Hastings algorithm [?]. Since the RCD sampler can han- dle any combination of restricted collapsed draws simul- taneously, we were able to develop a series of sampling method for HCRP-HMM, including a blocked collapsed Gibbs sampler, a collapsed beam sampler, and a split- merge sampler for HCRP-HMM. Through experiments we found that our collapsed samplers outperformed their non- collapsed counterparts. 2 HCRP representation for iHMM 2.1 Infinite HMM An infinite hidden Markov model (iHMM) [??] is

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