Sequentially-Allocated Merge-Split Sampler for Conjugate and Nonconjugate Dirichlet Process Mixture Models

Transcription

1 Sequentially-Allocated Merge-Split Sampler for Conjugate and Nonconjugate Dirichlet Process Mixture Models David B. Dahl* Texas A&M University November 18, 2005 Abstract This paper proposes a new efficient merge-split sampler for both conjugate and nonconjugate Dirichlet process mixture (DPM) models. These Bayesian nonparametric models are usually fit using Markov chain Monte Carlo (MCMC) or sequential importance sampling (SIS). The latest generation of Gibbs and Gibbs-like samplers for both conjugate and nonconjugate DPM models effectively update the model parameters, but can have difficulty in updating the clustering of the data. To overcome this deficiency, merge-split samplers have been developed, but until now these have been limited to conjugate or conditionally-conjugate DPM models. This paper proposes a new MCMC sampler, called the sequentially-allocated merge-split (SAMS) sampler. The sampler borrows ideas from sequential importance sampling. Splits are proposed by sequentially allocating observations to one of two split components using allocation probabilities that condition on previously allocated data. The SAMS sampler is applicable to general nonconjugate DPM models as well as conjugate models. Further, the proposed sampler is substantially more efficient than existing conjugate and nonconjugate samplers. KEYWORDS: Sequential importance sampling; Partial conditioning; Markov chain Monte Carlo; Metropolis-Hastings Algorithm; Bayesian nonparametrics. *David B. Dahl is Assistant Professor, Department of Statistics, Texas A&M University, College Station, TX ( dahl@stat.tamu.edu). The author thanks Michael Newton for many helpful discussions and valuable suggestions. Sonia Jain graciously provided the simulated data used in Jain and Neal (2004). Finally, Marina Vannucci and Gordon Dahl provided helpful advice in the prepartion of the manuscript. The Monte Carlo study was performed using free, open source software written by the author and available at dahl/sams.

2 1 Introduction The Dirichlet process mixture (DPM) model is arguably the best studied class of Bayesian nonparametric models. Theoretical groundwork for the model was laid in pioneering work of Ferguson (1973), Blackwell and MacQueen (1973), Antoniak (1974), Ferguson (1974), Korwar and Hollander (1973) and others. With the advent of modern computational methods in the 1990 s, much research has been devoted to techniques for fitting DPM models. All methods for fitting DPM models must accommodate the following two features: (1) The allocation ( clustering ) of the data into components ( clusters ) and (2) The model parameters of each component ( cluster locations ). Escobar (1994) was the first to provide a Gibbs sampler for fitting DPM models. The sampler removes a data point from its component and reallocates it to one of the existing components or to a new component by itself. One Gibbs scan performs this procedure for every data point. A major deficiency of the original algorithm is that model parameters only change by completely emptying a component and/or creating a new component. Due to sticky model parameters inherent in this approach, this original algorithm has great difficulty in covering the posterior distribution. Computational strategies have been proposed to address the sticky model parameters. One strategy is to integrate away the model parameters, as suggested by Liu (1994), MacEachern (1994), and Neal (1992). Since the model parameters can easily be sampled given the clustering, this strategy of integrating away the model parameters does not preclude inference on them. Unfortunately, this integration is typically only feasible in conjugate DPM models. In the case of nonconjugate DPM models, sticky model parameters can be addressed by adding an additional element to the sampling algorithm specifically designed to move the model parameters, as suggested by Bush and MacEachern (1996), Damien et al. (1999), MacEachern and Müller (2000), and Neal (2000), among others. These two strategies for sticky model parameters have been used in the context of Markov chain Monte Carlo (MCMC) and sequential importance sampling (Liu 1

3 1996). For reviews and comparisons of various computational methods for fitting DPM models, see MacEachern et al. (1999), MacEachern and Müller (2000), Neal (2000), and Quintana and Newton (2000). These two strategies for sticky model parameters do not directly address the allocation of the data into components. With the exception of Jain and Neal (2004), all the MCMC samplers in the DPM model literature repeatedly reallocate one data point at a time. Researchers have noticed, however, that these Gibbs and Gibbs-like samplers can get stuck in particular clustering configurations due to the one-at-a-time nature of their updates. Celeux et al. (2000) state that the main defect of the Gibbs sampler from our perspective is the ultimate attraction of the local modes; that is, the almost impossible simultaneous reallocation of a group of observations to a different component. Even when these samplers are able to move between modes, they may do so rather slowly and thus require a major expenditure of computational resources. Thus, practitioner may find they are unable to use DPM models in moderately large or complex datasets. Efficient algorithms for DPM models are essential for the practical application of DPM models. Recent work has sought to address the deficiencies of the conjugate Gibbs sampler for DPM models. For conjugate DPM models, Jain and Neal (2004) propose a multivariate update mechanism (instead of a mechanism providing one-at-a-time updates) based on merging and splitting existing clusters. Their sampler can provide a significant reduction in the computational effort required to fit conjugate DPM models. Nevertheless, the computational burden is still formidable. New samplers which further improve computational efficiency would be able to fit existing models more readily and would facilitate the use of DPM models that are currently impractical. Jain and Neal have extended their method to conditionally-conjugate DPM models in a recent technical report (Jain and Neal 2005). A conditionally-conjugate DPM is one in which the centering distribution of a Dirichlet process prior is conjugate to the observational distribution when conditioning on all the other parameters in the model. A conditionally-conjugate setup is appropriate in many situations, but a large class of DPM models are not conditionally-conjugate. For 2

4 example, DPM models in a generalized linear model framework will usually not be conditionallyconjugate. It is important to have efficient samplers which are applicable to general nonconjugate DPM models, as well as conjugate and conditionally conjugate models. This paper proposes a new merge-split sampler for DPM models which has two advantages over existing methods: (1) It is applicable to general nonconjugate DPM models as well as conjugate DPM models, and (2) It is substantially more efficient computationally than existing samplers. The new sampler, called the sequentially-allocated merge-split (SAMS) sampler, borrows ideas from sequential importance sampling (Liu 1996; MacEachern et al. 1999). Splits are proposed by sequentially allocating observations to one of two split components using allocation probabilities that are conditional on previously allocated data. The SAMS sampler is computationally efficient since the proposed splits are likely to be supported by the model and can quickly be generated. While the conditional allocation of observations is similar to that of sequential importance sampling, the output from the SAMS sampler has the correct stationary distribution due to the use of the Metropolis-Hastings ratio. To assess the computational efficiency of the SAMS sampler and existing samplers for DPM models, a simulation study is presented. Three scenario are considered, each having a different model and dataset. It is shown that, for a fixed amount of computing resources, the SAMS sampler (in both its conjugate and nonconjugate forms) yields an effective sample size that is substantially better than that of existing methods. The implication is that the SAMS sampler allows its user to fit DPM models in a much shorter amount of time and to consider DPM models that were previously impractical. The remainder of the paper is organized as follows: Section 2 reviews DPM models and introduces the notation used to describe the sampling algorithms. In Section 3, the leading existing samplers for DPM models are described. Section 4 presents the conjugate and nonconjugate versions of the proposed SAMS sampler. The simulation study comparing the various samplers is described in Section 5. Finally, concluding remarks are found in Section 6. 3

5 2 The Dirichlet Process Mixture Model The DPM model assumes that observed data y = (y 1,..., y n ) is generated from the following hierarchical model: For i = 1,..., n, y i θ i p(y i θ i ) θ i F (θ i ) F (θ i ) (1) F (θ) DP(η 0 F 0 (θ)), where p(y θ) is a known parametric family of distributions (for the random variable y) indexed by θ and DP(η 0 F 0 (θ)) is the Dirichlet process (Ferguson 1973) centered about the distribution F 0 (θ) (for the random variable θ) and having mass parameter η 0 > 0. The notation follows Neal (2000) and is meant to imply the obvious independence relationships (e.g., y 1 given θ 1 is independent of the other y s, the other θ s, and F (θ)). Methods for fitting DPM models rely on the dimension reduction proposed by Blackwell and MacQueen (1973). Upon integrating with respect to the Dirichlet process, θ = (θ 1,..., θ n ) follows a general Polya urn scheme: θ i θ 1,..., θ i 1 θ 1 F 0 (θ 1 ) η 0F 0 (θ i ) + i 1 j=1 ψ θ j (θ i ), for i = 2,..., n, η 0 + i 1 where ψ µ (θ) is the point-mass distribution (for the random variable θ) at µ. Notice that (2) implies that θ 1,..., θ n may share values in common, a fact that is used below in an alternative parameterization of θ. The model is simplified by integrating out the random mixing distribution F over its prior distribution in (2). Thus, the model in (1) becomes: (2) y i θ i p(y i θ i ), for i = 1,..., n, θ p(θ) given in (2). (3) An alternative parameterization of θ is given in terms of a set partition π = {S 1,..., S q } for S 0 = {1,..., n} and a vector of model parameters φ = {φ S1,..., φ Sq }, where φ S is associ- 4

6 ated with component S. A set partition π for S 0 is a set of components (i.e., subsets) S 1,..., S q such that: (1) S π S = S 0, (2) S S = for all S S, and (3) S for all S π. The prior in (2) can be decomposed into two parts: (1) A prior on π, and (2) A conditional prior on φ given π. Equation (2) implies that the partition prior p(π) can be written as: p(π) = b S π η 0 Γ( S ), (4) where S is the number of elements of the component S, Γ(x) is the gamma function evaluated at x, and b 1 is a constant equal to n i=1 Γ(η 0 + i 1). The specification of the prior under this alternative parameterization is completed by noting that Korwar and Hollander (1973) showed that φ S1,..., φ Sq are independently drawn from F 0 (φ). Thus, θ is equivalent to (π, φ) and the model in (1) and (3) may be expressed as: n y π, φ p(y i φ S i) i=1 φ π S π F 0 (φ S ) (5) π p(π) given in (4), where S i is the component of π containing index i. Bayesian inference regarding the model parameters φ and set partition π is made via the posterior distribution p(π, φ y). Markov chain Monte Carlo (MCMC) techniques can be used to sample from the posterior distribution and sequential importance sampling (SIS) can be used to numerically integrate the posterior distribution. Fitting DPM models is usually computationally demanding, in part due to sticky model parameters φ, but mostly because inference on the set partition π requires sampling from a very large discrete state space. 2.1 Conjugacy If F (φ) is conjugate to p(y φ) in φ, a DPM model is said to be conjugate; otherwise, the model is nonconjugate. For conjugate DPM models, the challenge of sticky model parameters can be 5

7 eliminated by integrating away the model parameters φ as proposed by MacEachern (1994), Liu (1994), and Neal (1992). The task then reduces to fitting the posterior distribution of the partition π, denoted p(π y). This integration technique is merely a computational device used for model fitting and the full posterior distribution p(π, φ y) can easily be obtained since: p(π, φ y) = p(φ π, y)p(π y) (6) and p(φ π, y) is readily available in conjugate models. Thus, the typical approach utilizing this integration strategy draws B set partitions π 1,..., π B from p(π y) through an MCMC sampler and then samples the vectors of model parameters φ 1,..., φ B from p(φ π, y) using straight Monte Carlo. To implement the integration strategy, the partition likelihood p(y π) must be obtained. It is given as a product over components in π = {S 1,..., S q }: p(y π) = S π p(y S ) (7) where: p(y S ) = p(y k φ)df 0 (φ) (8) k S = p(y k φ)p(φ y 1,..., y k 1 )dφ, (9) k S where p(φ y 1,..., y k 1 ) is the density of the posterior distribution of a model parameter φ based on the prior F 0 (φ) and the data preceding the index k in the product above. If F 0 (φ) is conjugate to p(y φ) in φ, the integral in (9) may be evaluated analytically. Combining the partition likelihood and the partition prior, Bayes rule gives the partition posterior as: p(π y) p(y π) p(π), (10) where p(y π) and p(π) are given in (7) and (4), respectively. 6

8 3 Existing Samplers Using the notation from the previous section, this section describes three popular samplers for DPM models. Two of the three samplers are conjugate samplers; that is, they require quantities that are only available in conjugate models. Nonconjugate samplers do not make use of quantities specific to conjugate models. Although they likely are not as efficient as conjugate samplers, nonconjugate samplers can be applied to conjugate models. 3.1 Conjugate Gibbs Sampler for Conjugate DPM Models The conjugate Gibbs sampler for conjugate DPM models was introduced by MacEachern (1994) for Gaussian models and Neal (1992) for models of categorical data. This is the benchmark conjugate sampler to which other conjugate samplers are compared. One scan of the Gibbs sampler successively reallocates all indices, one at a time, according to the full conditional distribution as described below: Algorithm for Gibbs Sampler: For i in {1, 2,..., n}, Remove index i from its component S i to obtain π i, a set partition of: {1, 2,..., i 1, i + 1,..., n} containing the (potentially empty) component S i. If S i is empty, let S = S i. Otherwise, add a new empty component S to π i. Form π from π 1 by allocating index i to one of the components in π i with probability given by the full conditional distribution: Pr( i S π i, y ) w 1 (S) p(y i φ)p(φ y S )dφ for each S π 1, 7

9 where: S w m (S) = η 0 /m if S > 0, and otherwise, (11) and p(φ y S ) is the density of the posterior distribution of a component location φ based on the centering distribution F 0 (φ) and the data corresponding to the indices in S. If S is empty, p(φ y S ) reduces to the density of F 0 (φ). If S is empty, remove it from π. 3.2 Auxiliary Gibbs Sampler for Nonconjugate DPM Models Several nonconjugate samplers exists and are reviewed by Neal (2000). As with the Gibbs sampler, all of these samplers update the clustering of each index one-at-a-time, conditional on the allocation of the other indices. Being nonconjugate, however, these samplers must deal with model parameters in addition to the clustering of the observations. The Auxiliary Gibbs sampler of Neal (2000) is a nonconjugate analog to the Gibbs sampler for conjugate DPM models. Neal s simulation study shows that the Auxiliary Gibbs sampler (with a properly chosen tuning parameter) has the best computational efficiency of one-at-a-time nonconjugate samplers for DPM model. The tuning parameter is the number of auxiliary parameters in each MCMC update and Auxiliary Gibbs (m) denotes the Auxiliary Gibbs sampler with m auxiliary parameters. One scan of the Auxiliary Gibbs (m) sampler is described below: Algorithm for Auxiliary Gibbs Sampler: For i in {1, 2,..., n}, Remove index i from its component S i to obtain π i. If S i is empty, let S 1 = S i. Otherwise, add a new empty component S 1 to π i with component location φ S 1 sampled from F 0 (φ). 8

10 If m > 1, add new empty components S 2,..., S m to π i with component locations φ S 2,..., φ S m sampled independently from F 0 (φ). Allocate index i to one of the components in π i with probability given by the full conditional distribution: Pr( i S φ, π i, y ) = b w m (S) p(y i φ S ) for each S π 1, where w m (S) is defined in (11). Remove from π all S 1,..., S m that are empty. For each S in π: Perform an MCMC update for the component location φ S given the data y S in component S. For example, one could sample from p(φ S y S ). This quantity, however, will likely not be available for nonconjugate models, in which case another MCMC update which leaves its distribution invariant should be used. 3.3 Restricted Gibbs Split-Merge Sampler for Conjugate DPM Models The conjugate Gibbs and Auxiliary Gibbs samplers effective deal with sticky model parameters, but can get stuck in particular clustering configurations due to the one-at-a-time nature of their updates. Jain and Neal (2004) propose a merge-split sampler for conjugate DPM models. This is the state-of-the-art sampler for conjugate DPM models which they show can greatly improve the rate of convergence to the posterior distribution. Being a merge-split algorithm, their method updates groups of indices simultaneously and thus is able to move between high-probability modes by jumping over depressions of low probability. Jain and Neal (2004) refer to their method as the restricted Gibbs sampling split-merge procedure, which is abbreviated RGSM in this paper. As the name implies, the RGSM sampler involves a modified Gibbs sampler. Instead of proposing naive random splits of components which are unlikely to be supported by the model, the RGSM proposes splits that are more probable by 9

11 improving upon a random split through a series of t restricted Gibbs scans. The scans are restricted in the sense that an index involved in a split can only move between the two split components. The number of scans t to perform is a tuning parameter, thus RGSM is in fact a class of samplers with RGSM(t) being a particular case. The algorithm for one iteration of the RGSM(t) sampler is described below: Algorithm for RGSM(t) Sampler: Uniformly select a pair of distinct indices i and j. If i and j belong to the same component in π, say S, propose π by splitting S: Remove indices i and j from S and form singleton sets S i = {i} and S j = {j}. Make a naive split by allocating each remaining index in S to either S i or S j with equal probability. Perform t restricted Gibbs scans of the indices in S. A restricted Gibbs scan differs from a regular conjugate Gibbs scan in that: Instead of considering all the indices {1, 2,..., n}, only the indices in S the original companions of i and j, but not including i and j are considered, and When reallocating an index, it can only be placed in one of two components: S i and S j. An index cannot be placed in any other components and a singleton component cannot be formed. Perform one final restricted Gibbs scan, this time keeping track of the Gibbs sampling transition probabilities for use in computing the Metropolis-Hastings ratio. Let π be the set partition after the t + 1 restricted Gibbs scans. Compute the Metropolis-Hastings ratio and accept π as the new state of π with probability given by this ratio. Otherwise, i and j belong to different components in π, say S i and S j. Propose π by merging S i and S j : 10

12 Form a merged component S = S i S j. Propose the following merged set partition: π = π {S} \ {S i, S j }. In words, π differs from π in that the two components containing indices i and j are merged into one component. Compute the Metropolis-Hastings ratio and accept π as the current state π with probability given by this ratio. Notice that the RGSM(t) sampler requires the specification of t, the number of intermediate restricted Gibbs scans to perform before the final restricted Gibbs scan. This tuning parameter can be beneficial in that it allows flexibility in tailoring the algorithm for a particular situation. On the other hand, a practitioner may have little insight as to what a good value might be for the tuning parameter. In this case, experimentation is called for. A large number of restricted scans makes the split proposal more reasonable and therefore more likely to be accepted. The marginal benefit, however, of another restricted scan decreases and each additional scan takes time that might be better spent on another update. Based on their example, Jain and Neal (2004) recommend four to six restricted Gibbs scans. While the RGSM(t) sampling algorithm can theoretically stand alone in sampling from the posterior distribution, Jain and Neal (2004) show that convergence is greatly improved by combining the RGSM(t) sampler with the conjugate Gibbs sampler. They propose cycling between the two samplers by attempting x RGSM(t) updates and performing y Gibbs scans, where x and y are integers chosen by the practitioner. As a result, the conjugate Gibbs sampler fine tunes the current state while RGSM(t) potentially makes very large updates. 11

13 4 Sequentially-Allocated Merge-Split Sampler This section introduces the novel sequentially-allocated merge-split (SAMS) sampler proposed in this paper. The SAMS sampler is applicable to general nonconjugate DPM models, as well as conjugate DPM models. Further, the SAMS sampler is substantially more efficient than the existing leading samplers. By using the SAMS sampler, researchers can fit existing models faster and are able to consider rich models that would otherwise be impractical to fit. In the SAMS sampler, splits are proposed by sequentially allocating observations to one of two split components using allocation probabilities that condition on previously allocated data. The SAMS sampler is computationally efficiency because the proposed splits are generated with relative few computations, yet the proposed splits are likely to be supported by the model. The method is reminiscent of sequential importance sampling (Liu 1996; MacEachern et al. 1999), but the draws from the SAMS sampler have the correct stationary distribution (whereas draws from sequential importance sampling are weighted for Monte Carlo integration). In contrast to the RGSM(t) sampler, the SAMS sampler does not have a tuning parameter that must be specified by the practitioner, making its application more automatic. As with the RGSM(t) sampler, cycling between the SAMS sampler and Gibbs sampler is recommended. 4.1 Conjugate Version of the Algorithm The SAMS sampler has both conjugate and nonconjugate versions. The conjugate sampler is able to utilize the dimension reduction technique of Section 2.1 to eliminate the model parameters and leave only the set partition. The conjugate version of the SAMS sampler, denoted SAMS(conjugate), is described below: 12

14 Algorithm for SAMS(conjugate) Sampler: Uniformly select a pair of distinct indices i and j. If i and j belong to the same component in π, say S, propose π by splitting S: Remove indices i and j from S and form singleton sets S i = {i} and S j = {j}. Letting k be successive values in a random permutation of the indices in S, add k to S i with probability: Pr( k S i S i, S j, y ) = S i p(y k φ)p(φ y S i)dφ S i p(y k φ)p(φ y S i)dφ + S j p(y k φ)p(φ y S j)dφ, where p(φ y S ) is the posterior distribution of a component location φ based on the prior F 0 (φ) and the data corresponding to the indices in S. Otherwise, add k to S j. Note that, at each allocation above, either S i or S j gains an index. As a result, S i and S j as new indices are allocated. Further, in this conjugate version of the sampler, p(φ y S i) and p(φ y S j) evolve to account for each additional index. Compute the Metropolis-Hastings ratio and accept π as the current state π with probability given by this ratio. See Section 4.3 for details. Otherwise, i and j belong to different components in π, say S i and S j. Propose π by merging S i and S j : Form a merged component S = S i S j. Propose the following set partition: π = π {S} \ {S i, S j }. In words, π differs from π in that the two components containing indices i and j are merged into one component. Compute the Metropolis-Hastings ratio and accept π as the current state π with probability given by this ratio. See Section 4.3 for details. (12) 13

15 4.2 Nonconjugate Version of the Algorithm The nonconjugate version of the SAMS sampler differs from the conjugate version in three ways. Firstly, the probability in (12) is replaced by Pr( k S i S i, S j, φ, y ) = S i p(y k φ S i) S i p(y k φ S i) + S j p(y k φ S j). (13) Secondly, when i and j belong to the same component, the model parameter φ S i of component S i should be updated. Many methods can be used to propose new values for the model parameter φ S i. Here, two methods are considered: (1) Propose new values for the model parameters φ S i by sampling from the centering distribution F 0 (φ) and (2) Propose new values for the model parameters φ S i using some random walk. Finally, when i and j belong to different components, the model parameter φ S of the merged component S is set equal to φ S j, the model parameter in the original component S j. The nonconjugate SAMS sampler based on sampling from the centering distribution F 0 (φ) is denoted SAMS(prior). The random walk version is denoted SAMS(random walk). One might expect that the random walk version to be superior to the prior version and, indeed, the simulation results in Section 5 confirm this. 4.3 Metropolis-Hastings Ratio for SAMS Sampler For the conjugate SAMS sampler, the Metropolis-Hastings ratio a(π π) gives the probability that, at the current partition π, a proposed partition π is accepted. The MH ratio for the conjugate SAMS sampler is given as: a(π π) = min [ 1, p(π y) p(π y) ] q(π π ) q(π, (14) π) where p(π y) is the density of the partition posterior distribution evaluated at π given in (10) and q(π π ) is the probability of proposing π from the state π. 14

16 For the nonconjugate SAMS samplers, the Metropolis-Hastings ratio is denoted a(π, φ π, φ) and is given as: a(π, φ π, φ) = min [ 1, p(π, φ y) p(π, φ y) ] q(π, φ π, φ ) q(π, φ, (15) π, φ) where p(π, φ y) is the density of the joint posterior distribution evaluated at (π, φ) given in (6) and q(π, φ π, φ ) is the probability of proposing (π, φ) from the state (π, φ ). In practice, only parts of the posterior distribution need to be evaluated since contributions from components not involved in the split or merge will cancel. Making use of this fact reduces rounding error and greatly speeds up computations. Computing the Metropolis-Hastings ratio for the SAMS sampler requires a similar amount of bookkeeping as is required when using the RGSM(t) sampler. When the proposal involves a split, q(π π) is merely the product of the probabilities in (12) (or their complements) associated with the chosen allocations. Likewise, q(π, φ π, φ) is the product of probabilities in (13) times the proposal density used to move φ S i. Since the two split components could only be merged in one way, the reverse proposal probability is always 1. Conversely, consider the case where the proposal is a merged partition. Since the two split components could only be merged in one way, the proposal probability is 1. In the conjugate case, the reverse proposal probability q(π π ) is the product of the probabilities in (12) associated with the allocation choices that would need to be made to obtain the split partition π from the merged partition π. That is, for the purposes of calculating q(π π ), the computer merely temporarily reverses the current and proposed partitions and is forced to chose allocations which yield the split partition π from the merged partition π. The calculation for the nonconjugate case follows that of the conjugate case, with the additional evaluation of the proposal density used to move from φ S j to φ S i. 15

17 5 Comparison of SAMS Samplers to Other Samplers All (properly constructed) MCMC samplers for DPM models have the posterior distribution of interest as their stationary distribution. Experience has shown, however, that the computational efficiency of the samplers can vary widely. A DPM model that may be completely impractical to fit using one sampler can suddenly be feasible when a more efficient sampler becomes available. To compare the computational efficiency of the proposed SAMS sampler with the leading conjugate and nonconjugate samplers presented in Section 3, a Monte Carlo study was conducted. Jain and Neal (2004) note that their merge-split sampler performs better when used in combination with the conjugate Gibbs sampler. Following this observation, this simulation study compares the conjugate Gibbs sampler with hybrid samplers in which half of the CPU time was spent on conjugate Gibbs sampling and the other half was spent on the RGSM(1), RGSM(3), RGSM(5), or SAMS(conjugate) sampler. Also, the Auxiliary Gibbs (1) sampler is compared with a hybrid sampler in which half of the CPU time was spent on Auxiliary Gibbs (1) sampling and the other half was spent on the Auxiliary Gibbs (2), SAMS(prior), or SAMS(random walk) sampler. The simulation study was conducted using code written in the Ruby programming language and executed on a computer running the GNU/Linux operating system with an AMD Athlon processor and 2 GB of RAM. The software used for the comparisons is available at dahl/sams. 5.1 Accessing Computational Efficiency The computational effort required to fit a Bayesian model using MCMC can be divided into two parts: (1) Reaching a burned-in state (i.e., a state that is relatively probable under the posterior distribution) and (2) Sampling from the posterior distribution after burn-in. Jain and Neal (2004) provide a simulation study to compare their RGSM sampler to the conjugate Gibbs sampler. Their study focused on the number of MCMC updates required to burn-in. They found that their 16

18 RGSM sampler (coupled with the conjugate Gibbs sampler) is able to attain a burned-in state in substantially fewer MCMC updates. Reaching a burned-in state is essential to being able to sample from the posterior, but the vast majority of the computational effort is typically spent on sampling from the posterior. For this reason, the simulation study in this paper focuses on the computational efficiency of the various methods in sampling from the posterior distribution. Each sampler starts from a state that is already well supported by the posterior (i.e., no further burn-in is required). In particular, for a each replicate in the Monte Carlo study, every sampler uses a common burned-in state. This burned-in state is obtained by running a hybrid sampler in which half of its CPU time is spent on conjugate Gibbs sampling and the other half on SAMS(conjugate) sampling. In each example model and dataset, the effective sample size was computed for each of 20 independent burned-in states. Comparing the various samplers by letting each perform a specified number of updates is problematic. To take an extreme example, consider comparing the RGSM(1 000) sampler to the RGSM(3). Certainly the RGSM(1 000) will propose splits well supported by the posterior due to the 1,000 intermediate restricted Gibbs scans. These 1,000 scans come at a cost, however; for the same amount of CPU time used to perform one RGSM(1000) update, perhaps hundreds of RGSM(3) updates could have been performed. Hence, RGSM(3) may well be more computationally efficient. To make the samplers commensurate, each sampler is given the same amount of CPU time. Even with a fixed CPU time, care must be taken in interpreting the results since successive draws from a Markov chain are not independent. The degree to which the draws are correlated affects the amount of information that each sample carries. The autocorrelation time (Ripley 1987; Kass et al. 1998) gives the effective amount by which the sample size is reduced when measuring 17

19 the expectation of a quantity. It is defined as: ρ i, i=1 where ρ i is the autocorrelation between a univariate summary of two states i lags part in the Markov chain. In practice, ρ i is estimated using the sample autocorrelations and the infinite sum is replace by a sum until the sample autocorrelations are statistically indistinguishable from zero. Several measures could be used to compute the effective sample size and thereby compare the samplers. Two measures are commonly used: (1) The size of the largest cluster in the state and (2) The number of clusters in the state. These measures, however, have major deficiencies for comparing samplers. The size of the largest cluster ignores information about the number and sizes of the other clusters in the state. A very dramatic change in the clustering may result from an update, but still not affect the size of the largest cluster. Likewise, the number of clusters ignores potentially important information: the size of the clusters. For example, a sampler may move many indices from existing clusters to form a new cluster. In terms of the number of clusters, this dramatic update is rewarded the same as the very modest update of forming a new singleton cluster. The measure used to compute the effective sample size in this simulation study is the clustering entropy: q i=1 S i ( n log Si ). n Experience has shown that entropy is more discriminatory than the size of the largest cluster or the number of clusters. Entropy captures changes both in the size and number of clusters. Small changes in the size or number of clusters lead to small changes in entropy; more dramatic clustering changes are reflected by large changes in the entropy. 18

20 5.2 Monte Carlo Three models and datasets were used in the Monte Carlo study. All three of these DPM models are conjugate, hence either a conjugate or a nonconjugate DPM sampler can be used to fit them. The three cases considered are the Tack, Galaxy, and Jain/Neal examples Tack Example: Beckett and Diaconis (1994) presented data resulting from rolling thumbtacks. The data consist of 320 observations, each counting the number of times a tack lands pointing up in nine rolls. A DPM model for this data was presented by Liu (1996). The observational component p(y θ) is Binomial(9, θ), the centering distribution F 0 (θ) is Beta(1, 1), and mass parameter η 0 is 1.0. The random walk version of the SAMS sampler updated the model parameter θ using a Gaussian proposal centered at the current state θ and having variance θ(1 θ)/n, where n is the size of the cluster. For each replicate, the samplers were given 3.5 hours to sample from the posterior Galaxy Example: Roeder (1990) analyzed the velocities at which 82 galaxies are moving away from our galaxy. (See also Richardson and Green (1997)). In the DPM model used for this simulation study, the observational component p(y θ) is Normal(θ, ), the centering distribution F 0 (θ) is Normal(20 000, ), and mass parameters η 0 is 1.0. The random walk version of the SAMS sampler updated the model parameter θ using a Gaussian proposal centered at the current state θ and having variance /n, where n is the size of the cluster. For each replicate, the samplers were given 30 minutes to sample from the posterior. 19

21 5.2.3 Jain & Neal Example: To demonstrate their proposed sampler, Jain and Neal (2004) present synthetic datasets and a multivariate-bernoulli/beta DPM model. In this simulation study, we consider their higher dimension problem (Example 2 in their paper) which demonstrated both the efficiency of their method and the problems the Gibbs sampler can encounter. The random walk version of the SAMS sampler updates each element θ i of the model parameter using a Gaussian proposal centered at the current state θ i and having variance θ i (1 θ i )/n, where n is the size of the cluster. For each replicate, the samplers were given 3.5 hours to sample from the posterior. 5.3 Results The results of the simulation study are displayed in Tables 1, 2, and 3. Each table has two panels; panels a) and b) contain results from conjugate and nonconjugate samplers, respectively. For each sampler, the mean effective sample size based on the 20 replicates is shown, as well as the standard error of this mean. The last column shows that ratio of the effective sample size of each sampler to the reference sampler (i.e., the conjugate Gibbs sampler in the case of conjugate samplers and the Auxiliary Gibbs (1) sampler for nonconjugate samplers). This ratio gives the factor by which the effective sample size is multiplied. Put another way, this ratio shows how much faster a particular sampler is than its reference. In the case of the Tack example (Table 1), the hybrid sampler based on the SAMS(conjugate) sampler leads to a sample size that is effectively 4.77 times larger than that of the conjugate Gibbs sampler. In other words, the SAMS(conjugate) fits the model almost 5 times faster than the conjugate Gibbs sampler. The RGSM samplers with tuning parameters 1, 3, and 5 are statistically better than the conjugate Gibbs, but not nearly as good as the SAMS(conjugate) sampler. For example, the SAMS(conjugate) sampler is about 4.77/1.53 = 3.12 times better than the best RGMS sampler (performing 1 restricted Gibbs scan). In the nonconjugate case, both the random walk and 20

22 prior versions of the SAMS sampler are about 47% better than the Auxiliary Gibbs (1) sampler. The Galaxy example (Table 2) also show the benefit of using the SAMS sampler. The SAMS(conjugate) is 64% better than the conjugate Gibbs and the SAMS(random walk) is 80% better than the Auxiliary Gibbs (1). Notice that the prior version of the SAMS sampler is greatly inferior to the random walk version. This example also shows that the RGSM sampler can actually perform worse than the conjugate Gibbs sampler and under the best case, performs no better than the conjugate Gibbs sampler. The most striking demonstration of the benefit of the SAMS sampler is illustrated in the Jain/Neal example. As Jain and Neal (2004) report, their RGSM sampler indeed performs better than the conjugate Gibbs sampler. This simulation study shows that the effective sample size is 3.39, 3.25, and 2.79 times better than that of the conjugate Gibbs sampler. The SAMS(conjugate) sampler, however, performs 7.66 times better than the conjugate Gibbs sampler and 7.66/3.39 = 2.26 times better than the best RGSM sampler. The nonconjugate case reveals that the Auxiliary Gibbs samplers and the SAMS(prior) samplers are very inefficient samplers for this model and dataset. The SAMS(random walk) is about 34 times better than any other nonconjugate sampler for this model and dataset. 6 Conclusion This article introduces a new efficient split-merge MCMC algorithm for both conjugate and nonconjugate Dirichlet process mixture models. The proposed sampler makes split proposals by sequentially allocating indices to one of two split components using allocation probabilities that are conditional on previously allocated indices. The algorithm is computationally efficient because the proposal mechanism is both fast and well supported by the model. No tuning parameter needs to be chosen and hence the application of the sampler is automatic. While the conditional allocation of observations is reminiscent of sequential importance sampling, the output from the sampler 21

23 has the correct stationary distribution due to the use of the Metropolis-Hastings ratio. The SAMS sampler bears a resemblance to sequential importance sampling of Liu (1994) and MacEachern et al. (1999). In both sequential important sampling and the SAMS sampler, indices are allocated one at a time and previously allocated indices are used in helping to choose how to allocate the current index. Despite similarities, the algorithms differ in at least two important ways. First, sequential importance sampling uses every draw from the proposal distribution, but these draws must be adjusted by importance weights when computing Monte Carlo integrals with respect to the posterior distribution of interest. Conversely, the SAMS sampler is a Markov chain Monte Carlo algorithm whose stationary distribution is the posterior distribution of interest. Proposals are only accepted according to the probability given by the Metropolis-Hastings ratio. The second way in which the SAMS sampler and sequential importance sampler differ is in the allocation of indices. In sequential importance sampling, indices can be allocated to the full range of existing components or to a new component. In the SAMS sampler, indices involved in a split can only be allocated to one of two split components. Simulation results indicated that the proposed SAMS (sequentially-allocated merge-split) sampler performs substantially better than the RGSM (restricted Gibbs merge-split) sampler of Jain and Neal (2004) and the conjugate Gibbs sampler. Perhaps more importantly for practical applications where conjugate models may not be available, the SAMS(random walk) sampler show substantially improved performance over the Auxiliary Gibbs (1) sampler. Note that in two of the three examples, the SAMS(prior) was significantly worse than the SAMS(random walk) sampler. Therefore, practitioners are strongly encouraged to use the random walk version of the SAMS sampler for nonconjugate DPM models. 22

24 References Antoniak, C. E. (1974), Mixtures of Dirichlet Processes With Applications to Bayesian Nonparametric Problems, The Annals of Statistics, 2, Beckett, L. A. and Diaconis, P. W. (1994), Spectral Analysis for Discrete Longitudinal Data, Advances in Mathematics, 103, Blackwell, D. and MacQueen, J. B. (1973), Ferguson Distributions Via Polya Urn Schemes, The Annals of Statistics, 1, Bush, C. A. and MacEachern, S. N. (1996), A semiparametric Bayesian model for randomised block designs, Biometrika, 83, Celeux, G., Hurn, M., and Robert, C. P. (2000), Computational and inferential difficulties with mixture posterior distributions, Journal of the American Statistical Association, 95, Damien, P., Wakefield, J., and Walker, S. (1999), Gibbs Sampling for Bayesian Non-conjugate and Hierarchical Models By Using Auxiliary Variables, Journal of the Royal Statistical Society, Series B, Methodological, 61, Escobar, M. D. (1994), Estimating Normal Means With a Dirichlet Process Prior, Journal of the American Statistical Association, 89, Ferguson, T. S. (1973), A Bayesian Analysis of Some Nonparametric Problems, The Annals of Statistics, 1, (1974), Prior distributions on spaces of probability measures, The Annals of Statistics, 2, Jain, S. and Neal, R. M. (2004), A Split-Merge Markov Chain Monte Carlo Procedure for the Dirichlet Process Mixture Model, Journal of Computational and Graphical Statistics, 13, (2005), Splitting and Merging Components of a Nonconjugate Dirichlet Process Mixture Model, Tech. Rep. 0507, Dept. of Statistics, University of Toronto. Kass, R. E., Carlin, B. P., Gelman, A., and Neal, R. M. (1998), Markov Chain Monte Carlo in Practice: A Roundtable Discussion, The American Statistician, 52,

25 Korwar, R. M. and Hollander, M. (1973), Contributions to the theory of Dirichlet processes, The Annals of Probability, 1, Liu, J. S. (1994), The collapsed Gibbs sampler in Bayesian computations with applications to a gene regulation problem, Journal of the American Statistical Association, 89, (1996), Nonparametric Hierarchical Bayes Via Sequential Imputations, The Annals of Statistics, 24, MacEachern, S. and Müller, P. (2000), Efficient MCMC schemes for robust model extensions using ecompassing Dirichlet process mixture models, in Robust Bayesian analysis, pp MacEachern, S. N. (1994), Estimating Normal Means With a Conjugate Style Dirichlet Process Prior, Communications in Statistics, Part B Simulation and Computation, 23, MacEachern, S. N., Clyde, M., and Liu, J. S. (1999), Sequential Importance Sampling for Nonparametric Bayes Models: The Next Generation, The Canadian Journal of Statistics, 27, Neal, R. M. (1992), Bayesian mixture modeling, in Maximum Entropy and Bayesian Methods: Proceedings of the 11th International Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis, pp (2000), Markov Chain Sampling Methods for Dirichlet Process Mixture Models, Journal of Computational and Graphical Statistics, 9, Quintana, F. A. and Newton, M. A. (2000), Computational Aspects of Nonparametric Bayesian Analysis With Applications to the Modeling of Multiple Binary Sequences, Journal of Computational and Graphical Statistics, 9, Richardson, S. and Green, P. J. (1997), On Bayesian Analysis of Mixtures With An Unknown Number of Components (Disc: P ) (Corr: 1998V60 P661), Journal of the Royal Statistical Society, Series B, Methodological, 59, Ripley, B. D. (1987), Stochastic simulation, John Wiley & Sons. Roeder, K. (1990), Density Estimation With Confidence Sets Exemplified By Superclusters and Voids in the Galaxies, Journal of the American Statistical Association, 85,

26 Table 1: Efficiency in Terms of the Effective Sample Size for Tack Example. a) Conjugate Samplers Standard Ratio with Mean Error Gibbs Conjugate Gibbs 2, RGSM (1) & Conj. Gibbs 3, RGSM (3) & Conj. Gibbs 2, RGSM (5) & Conj. Gibbs 2, SAMS (conjugate) & Conj. Gibbs 11, b) Nonconjugate Samplers Standard Ratio with Mean Error Aux. Gibbs (1) Auxiliary Gibbs (1) 3, Auxiliary Gibbs (2) & Aux. Gibbs (1) 2, SAMS (prior) & Aux. Gibbs (1) 4, SAMS (random walk) & Aux. Gibbs (1) 4, Table 2: Efficiency in Terms of the Effective Sample Size for Galaxy Example. a) Conjugate Samplers Standard Ratio with Mean Error Gibbs Conjugate Gibbs 2, RGSM (1) & Conj. Gibbs 3, RGSM (3) & Conj. Gibbs 2, RGSM (5) & Conj. Gibbs 2, SAMS (conjugate) & Conj. Gibbs 4, b) Nonconjugate Samplers Standard Ratio with Mean Error Aux. Gibbs (1) Auxiliary Gibbs (1) 6, Auxiliary Gibbs (2) & Aux. Gibbs (1) 6, SAMS (prior) & Aux. Gibbs (1) 4, SAMS (random walk) & Aux. Gibbs (1) 11,

27 Table 3: Efficiency in Terms of the Effective Sample Size for Jain & Neal Example. a) Conjugate Samplers Standard Ratio with Mean Error Gibbs Conjugate Gibbs 2, RGSM (1) & Conj. Gibbs 7, RGSM (3) & Conj. Gibbs 7, RGSM (5) & Conj. Gibbs 6, SAMS (conjugate) & Conj. Gibbs 17, b) Nonconjugate Samplers Standard Ratio with Mean Error Aux. Gibbs (1) Auxiliary Gibbs (1) Auxiliary Gibbs (2) & Aux. Gibbs (1) SAMS (prior) & Aux. Gibbs (1) SAMS (random walk) & Aux. Gibbs (1) 17,