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.