In this section, we show the derivation of the posterior distribution over cluster-weights, π, conditioned on the cluster labels, z. We begin with the definition of a Dirichlet process from [1]. Definition A.1 (Dirichlet Process). Let H be a measure on a measureable space, Ω. If for any finite partition, (A1, A2, · · · , AK) of the space, the measure, G, on the partition follows the following D...

Suppose that P θ (g) is a linear functional of a Dirichlet process with shape θH, where θ > 0 is the total mass and H is a fixed probability measure. This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and a posterior calculus for Gamma processes to ascertain properties of linear functionals of Dirichlet processes. In particular, in...

We investigate composition-differentiation operators acting on the Dirichlet space of unit disk. Specifically, we determine characterizations for bounded, compact, and Hilbert-Schmidt operators. In addition, particular classes inducing maps, derive an adjoint formula, compute norm, identify spectrum.

We prove a Yau’s type gradient estimate for positive f-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces compact when infinite dimensional Bakry–Emery Ricci tensor and weighted mean curvature are bounded below. As an application, we give Liouville result condition. Our results do not depend any assumption potential function f.