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label.switching: An R Package for Dealing with the Label Switching Problem in MCMC Outputs

Panagiotis Papastamoulis

arXiv:1503.02271v1stat.CO

TL;DR

Permutation invariance in mixture and hidden Markov models can make posterior MCMC samples non-identifiable and complicate inference. The package supplies relabelling approaches, including pivotal and probabilistic methods; in multivariate-mixture simulations, algorithms generally agreed with ground truth and one another, excluding aic.

  • Problem

    Permutation invariance of the likelihood and posterior can make MCMC samples non-identifiable, complicating inference for functions that depend on component labels.

    Source: §1 · p. 2, §2.1 · p. 4

  • Method

    The label.switching package provides relabelling methods, including pivot-based reordering and a probabilistic EM-type algorithm that treats each draw’s permutation as uncertain.

    Source: §3.3 · p. 6, §3.5 · p. 7, §6 · p. 22

  • Results

    In the multivariate-mixture simulation, relabelling algorithms excluding aic strongly agreed with the ground truth and with one another; aic’s ordering constraint performed poorly, while user-defined permutations performed reasonably.

    Source: §5.3 · p. 21

  • Takeaways & Limitations

    For large K, the authors suggest considering methods optimized with lpSolve for the assignment problem.

    Source: §6 · p. 22

    The multivariate-normal-mixture setup assumes independent component-specific parameters and a normal-Wishart prior.

    Source: §5.3 · p. 17

Abstract

from arXiv · show

Label switching is a well-known and fundamental problem in Bayesian estimation of mixture or hidden Markov models. In case that the prior distribution of the model parameters is the same for all states, then both the likelihood and posterior distribution are invariant to permutations of the parameters. This property makes Markov chain Monte Carlo (MCMC) samples simulated from the posterior distribution non-identifiable. In this paper, the \pkg{label.switching} package is introduced. It contains one probabilistic and seven deterministic relabelling algorithms in order to post-process a given MCMC sample, provided by the user. Each method returns a set of permutations that can be used to reorder the MCMC output. Then, any parametric function of interest can be inferred using the reordered MCMC sample. A set of user-defined permutations is also accepted, allowing the researcher to benchmark new relabelling methods against the available ones

1. Introduction

The introduction explains how label switching complicates Bayesian inference for mixture and hidden Markov models, then presents a package of relabelling algorithms and tools for comparing them.

1. Introduction

  • Mixture and hidden Markov models represent varied phenomena, including unobserved population heterogeneity and approximations of unknown distributions.Latent variables can decompose these models into simpler structures; data augmentation is used in both EM and Gibbs sampling.Source: §1 · p. 1
  • Permutation-invariant likelihoods can make MCMC samples non-identifiable, complicating posterior inference for mixture and hidden Markov models.The introduction notes that label switching is also a necessary condition for chain convergence to the target distribution.Source: §1 · p. 2
  • Identifiability constraints may be unavailable or fail to respect posterior geometry, motivating a range of relabelling algorithms.The introduction describes these issues as reasons that multiple algorithms have been proposed.Source: §1 · p. 2
  • The label.switching package provides eight relabelling methods, including ordering constraints and probabilistic, pivotal, ECR, and data-based algorithms.The package combines methods proposed in the literature.Source: §1 · p. 2
  • Users can compare the methods’ single best clusterings and supply permutations from other procedures to benchmark novel relabelling methods.The package reports method outputs in a comparable form and accepts user-defined permutation sets.Source: §1 · p. 2
  • The paper illustrates package implementation with two real datasets and two simulated datasets from mixtures of bivariate normal distributions.The real datasets come from classic mixture and hidden Markov models.Source: §1 · p. 2

2. Mixture models

The section defines a finite mixture through latent component assignments and component-specific distributions, then explains how permutation invariance produces label switching and complicates inference.

2. Mixture models

  • Each observation follows a component-specific distribution conditional on its latent state, and the latent states are independent draws governed by mixture weights.The model defines a sample of observations and a vector of component parameters.Source: §2 · pp. 2–3
  • Classification probabilities express the conditional probability that observation i belongs to component k.The section introduces observed and complete likelihoods alongside this quantity.Source: §2 · p. 3

2.1. Label switching phenomenon

  • A label permutation reorders component-specific parameters, weights, and latent allocations together.The section denotes the set of permutations of {1, . . . , K} by T_K.Source: §2.1 · p. 3
  • When the component-parameter and weight priors are permutation invariant, permuting labels leaves the posterior unchanged and component-specific marginal densities coincide.The section gives a common prior choice in which component parameters share a distribution and mixture weights have a non-informative Dirichlet prior.Source: §2.1 · pp. 3–4
  • MCMC draws can switch among symmetric high-density areas, making samples non-identifiable and label-dependent inference difficult; relabelling applies permutations to each draw.The labels range over K components, yielding K! symmetric posterior areas.Source: §2.1 · p. 4

3. Algorithms

The package provides seven deterministic and one probabilistic relabelling algorithms for MCMC samples, using parameter values, allocations, or classification probabilities to resolve label switching. The methods differ in their matching strategies and input requirements.

3. Algorithms

  • The package includes seven deterministic and one probabilistic relabelling method, with input requirements varying across algorithms.The algorithms use component parameters, allocations, or classification probabilities; their required inputs are listed in Table 1.Source: §3 · p. 4, Table 1 · p. 5

3.1. Ordering constraints

  • Ordering constraints permute each draw to order a chosen parameter, but work well only when that constraint separates symmetric posterior modes.The paper notes that this condition is rarely met.Source: §3.1 · p. 4

3.2. Stephens’ method

  • Stephens’ method iteratively minimizes Kullback-Leibler divergence between each draw’s classification probabilities and their run-wide average.It can find the correct relabelling efficiently but requires storing the m × n × K classification-probability array.Source: §3.2 · p. 5

3.3. Pivotal reordering algorithm

  • Pivotal reordering chooses each draw’s permutation to maximize similarity to a user-defined pivot parameter vector.This is equivalent to minimizing Euclidean distance between the permuted parameters and the pivot.Source: §3.3 · p. 6

3.4. ECR algorithms

  • ECR matches each draw’s allocations to a representative allocation, with default and iterative versions differing in how they obtain that representative.Default ECR uses a selected high-posterior-density pivot; iterative version 1 uses allocations and a random initial pivot, while version 2 uses classification probabilities and also requires storing them.Source: §3.4 · p. 6

3.5. Probabilistic relabelling algorithm

  • The probabilistic SJW method treats each draw’s permutation as missing data and uses an EM-type algorithm to update parameter estimates.It computes probabilities for K! permutations per iteration and is inefficient as the number of components grows.Source: §3.5 · p. 7

3.6. Data-based relabelling

  • Data-based relabelling estimates cluster centers and dispersions, then selects each draw’s permutation by minimizing a k-means-type loss against the observed data.The estimates use observed data and simulated allocations.Source: §3.6 · pp. 7–8

4. Implementation in R

The main function runs selected relabelling methods and returns permutations for reordering MCMC output. It also provides outputs for comparing methods and accepts user-supplied permutations.

4. Implementation in R

  • The main function runs any selected combination of available relabelling methods and can also accept user-defined permutations.Each method returns permutations that can be applied to reorder the MCMC sample.Source: §4 · p. 8, §4.1 · pp. 8–9

4.1. Structure of main function

  • The function returns each method’s permutations, best clustering, reordering CPU time, and a similarity matrix between methods’ best-clustering allocations.The similarity matrix uses matching allocations and can compare results with a supplied ground truth.Source: §4.1 · p. 10
  • Methods differ in required inputs: some need only raw MCMC output or observed data, while SJW also requires a user-supplied complete log-likelihood function.Stephens and ECR-ITERATIVE-2 require classification probabilities; PRA and SJW require additional inputs.Source: §4.1 · pp. 9–10
  • The supplementary permute.mcmc function applies returned permutations to reorder the stored MCMC sample.Source: §4.1 · pp. 10–11

5. Examples

The examples use fishery data to demonstrate relabelling a five-component normal-mixture MCMC sample. The raw draws exhibit label switching, which a simple ordering constraint does not resolve.

5.1. Mixture of normal distributions: fishery data

  • The fishery example uses 256 snapper length measurements and considers a five-component normal mixture.The paper describes the five-component scenario as more challenging than the previously considered range of three to five components.Source: §5.1 · p. 11
  • The raw MCMC draws show label switching, and ordering the component means does not isolate one symmetric high-posterior-density area.Source: §5.1 · p. 12
  • For methods requiring classification probabilities, the example computes component-membership probabilities from mixture weights and normal densities at each iteration.The MCMC parameters are arranged as means, variances, and weights, with allocations stored separately.Source: §5.1 · p. 12

STEPHENS

Across the fishery, fetal-lamb, and simulated multivariate examples, relabelling methods differ in how they represent posterior structure; in the simulations, most methods agree with true allocations, unlike AIC.

STEPHENS

  • For fishery means, Stephens and ECR variants produce similar reordered traces, while PRA and Data-based resemble mean-ordering constraints and SJW behaves similarly.Source: Figure 2 · p. 13, STEPHENS · pp. 14–15
  • The coloured trace exhibits multimodal behaviour, with a main mode at 12 and a minor mode between 5 and 6.Source: STEPHENS · p. 15

5.2. Poisson hidden Markov model: fetal lamb data

  • For fetal lamb data, almost all methods indicate λ3 has much larger variance than the other components; PRA and AIC, and less so Data-based, depart from posterior topology.Source: §5.2 · pp. 16–17, Figure 3 · p. 18
  • Most methods assign no observations to state 3, suggesting the four-state fetal-lamb model may be overparameterized.Source: §5.2 · p. 17

5.3. Multivariate normal mixtures

  • In both multivariate datasets, all relabellers except AIC strongly agreed with ground truth and one another; agreement was absolute for the first dataset.Source: Table 5 · p. 19, §5.3 · p. 21
  • PRA and SJW were omitted for the K = 9 dataset because they require computing K! permutations for each MCMC iteration.Source: §5.3 · p. 19
  • AIC’s µ1 ordering performed poorly, while user permutations based on µ1k−2µ2k performed reasonably.Source: Table 5 · p. 19, Figure 4 · p. 20, §5.3 · p. 21

6. Concluding remarks

The package integrates eight relabelling algorithms for mixture and hidden Markov model MCMC outputs, with options for larger component counts and user-supplied comparisons. The paper presents it as an integrated post-processing tool for a field without previous integrated software efforts.

6. Concluding remarks

  • For large K, the paper recommends lpSolve-optimized methods: dataBased, ecr, ecr.iterative.1, ecr.iterative.2, and stephens.When K is small, all algorithms can be applied.Source: §6 · p. 22
  • The USER-PERM option accepts user-provided permutations for direct comparison with the package’s methods.The methods’ single best clusterings are also reported in a comparable form.Source: §6 · p. 22
  • For trans-dimensional MCMC, partition draws by sampled K and use the subset for the desired conditional inference.With Bayes factor approaches, separate MCMC samples for each K can be used directly.Source: §6 · p. 22
  • The authors describe the package as the first integrated software effort for label-switching methods.They frame it as a post-processing tool for mixture and hidden Markov model inference.Source: §6 · p. 22

Journal of Statistical Software http://www.jstatsoft.org/

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