Complexity bounds for Dirichlet process slice samplers

Slice sampling is a standard Monte Carlo technique for Dirichlet process (DP)-based models, widely used in posterior simulation. However, formal assessments of the scalability of posterior slice samplers have remained largely unexplored, primarily because the computational cost of a slice-sampling iteration is random and potentially unbounded. In this work, we obtain high-probability bounds on the computational complexity of DP slice samplers. Our main results show that, uniformly across posterior cluster-growth regimes, the overhead induced by slice variables, relatively to the number of clusters supported by the posterior, is $O_{\mathbb P}(\log n)$. As a consequence, even in worst-case configurations, superlinear blow-ups in per-iteration computational cost occur with vanishing probability. Our analysis applies broadly to DP-based models without any likelihood-specific assumptions, still providing complexity guarantees for posterior sampling on arbitrary datasets. These results establish a theoretical foundation for assessing the practical scalability of slice sampling in DP-based models.

💡 Research Summary

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This paper provides a rigorous probabilistic analysis of the computational complexity of slice samplers for Dirichlet process (DP) based models. Slice sampling is a popular Monte‑Carlo technique that introduces auxiliary slice variables to restrict inference to a finite set of components while preserving exactness of the posterior. The main difficulty, however, is that the number of components that must be instantiated at each iteration, denoted (K_n), depends on the minimum slice value (u_{\min}) and can in principle be arbitrarily large, leading to an unbounded per‑iteration cost.

The authors adopt a high‑probability perspective rather than seeking deterministic worst‑case bounds, which are unattainable for slice samplers. They first recall a classical result (Muliere & Tardella, 1998) stating that, under the prior, (K_n-1\mid u_{\min}) follows a Poisson distribution with mean (\alpha\log(1/u_{\min})). Building on this, they analyze the posterior behavior of (u_{\min}) and derive tail bounds that hold uniformly over any dataset and any cluster‑growth regime. The key theorem shows that for any confidence level (1-\delta) there exists a constant (C=C(\alpha,\delta)) such that \