Collapsed Gibbs SamplingEdit
Collapsed Gibbs sampling is a practical inference technique in Bayesian statistics that specializes Gibbs sampling by integrating out certain model parameters. This collapsing step reduces the dimensionality of the sampling problem and often yields faster mixing when inferring latent structure in models with latent discrete assignments. The most common arena for collapsed Gibbs sampling is topic modeling, where it is used to infer topic assignments for words in documents under the latent Dirichlet allocation framework. This approach relies on conjugate priors to allow exact marginalization, leaving a simple conditional distribution over the remaining latent variables.
In the typical setting, documents are seen as mixtures of topics, and topics are distributions over words. The model uses Dirichlet priors for these distributions, which makes the Dirichlet–Multinomial conjugacy exact enough to collapse the document-topic proportions and the topic-word distributions. As a result, the sampler operates on word-level topic assignments rather than on all parameters at once. This is advantageous because it often yields clearer, more stable inference for the latent topic structure, and it scales well for moderately large corpora when implemented efficiently. See Latent Dirichlet Allocation and Dirichlet distribution for the foundational building blocks, and consider Gibbs sampling as the broader class of methods to which collapsed Gibbs sampling belongs.
Overview and foundations
Topic models and their goals: collecting documents into coherent topic mixtures, enabling tasks such as document classification, search, and exploratory data analysis. See Topic modeling and Natural language processing for broader context.
The role of conjugacy: Dirichlet priors over topic distributions and over word distributions within topics lead to closed-form marginal distributions when parameters are integrated out. This collapsing produces a tractable sampling problem focused on discrete assignments, typically the topic label for each word token. See Conjugate prior.
Why collapse can help: by marginalizing θ (document-topic proportions) and φ (topic-word distributions), the sampler avoids sampling high-dimensional continuous parameters and instead samples discrete topic assignments with a simpler conditional form. This often improves convergence speed and reduces variance in estimates of the latent structure. See Gibbs sampling and Markov chain Monte Carlo.
Core reference model: collapsed Gibbs sampling is most widely described in the context of a standard LDA setup, where words w and their topic assignments z are the primary objects of inference after collapsing θ and φ. See Latent Dirichlet Allocation.
The algorithm in brief
Initialization: assign a topic to every word token at random or by a heuristic. Track counts:
- n_{d,k}, the number of words in document d assigned to topic k
- n_{k,w}, the number of times word w is assigned to topic k
- n_{k}, the total number of words assigned to topic k
- α and β as hyperparameters for the Dirichlet priors on document-topic and topic-word distributions, respectively
Iterative updates: for each word token (d, n) with current topic z_{d,n} = k, remove it from the counts, then sample a new topic from the conditional distribution p(z_{d,n} = k | z_{-d,n}, w, α, β). In the collapsed setting, this conditional is proportional to: (n_{d,k}^{-n} + αk) * (n{k,w_{d,n}}^{-n} + β{w{d,n}}) / (n_k^{-n} + ∑_w β_w) where the superscripts -n indicate counts excluding the current word token.
Repeat for many iterations: after a suitable burn-in, collect samples to estimate quantities of interest, such as topic–word distributions or document–topic mixtures. See Gibbs sampling and Convergence diagnostics for general sampling considerations.
Inference outputs: the collected samples yield estimates of topics per word, topic distributions over vocabulary, and document-topic mixtures. See Perplexity as one common metric for held-out evaluation.
Practical considerations and extensions
Advantages: typically faster per-iteration than non-collapsed Gibbs sampling when conjugacy holds, often better mixing for topic assignments, and straightforward implementation for standard LDA.
Limitations: the closed-form updates rely on conjugate priors; non-conjugate variants require approximations. Collapsing can complicate parallelization, though several blocked or partly collapsed strategies exist to exploit modern multicore architectures. See Nonparametric Bayesian approaches such as the Hierarchical Dirichlet process for models that can adapt the number of topics, and note how collapsed variants interact with these extensions.
Extensions and alternatives: online variants and streaming inference adapt collapsed Gibbs ideas to growing corpora, while hybrid methods combine collapsed updates with variational approximations. See Online learning and Variational inference for related families of methods; also consider Hierarchical Dirichlet process for nonparametric topic modeling.
Diagnostics and controversies: debates in the literature concern when collapsed Gibbs sampling is preferable to variational inference, especially at very large scales where VI can be faster and more scalable, albeit sometimes at the cost of weaker posterior fidelity. Convergence diagnostics for MCMC, including collapsed schemes, rely on measures like chain mixing, autocorrelation, and predictive metrics such as held-out perplexity. See Convergence diagnostics and Perplexity for more on evaluation and reliability concerns.
Practical tips: data sparsity and hyperparameter choices for α and β significantly influence performance; careful initialization and optional partially collapsed strategies can improve robustness. See Dirichlet distribution and Conjugate prior for deeper theory behind these choices.
Applications and impact
In natural language processing and information retrieval, collapsed Gibbs sampling is a standard tool for discovering latent thematic structure in text corpora, enabling tasks from document clustering to topic-based retrieval. See Latent Dirichlet Allocation and Topic modeling.
Beyond text, collapsed sampling methods are used in other discrete latent-variable models where conjugacy can be exploited, including certain recommender-system models and biological sequence analyses. See Bayesian statistics and Markov chain Monte Carlo for broader methodological context.
Relationship to other inference approaches: collapsed Gibbs sampling sits alongside variational methods as a major family of approximate inference techniques in probabilistic modeling. See Variational inference for contrast and comparison.