# Dirichlet Process with known mean

A Dirichlet process $\operatorname{DP}(\alpha,G_0)$ can be thought of as a distribution of distributions. I would like an object similar to a Dirichlet process, but which has support only on distributions whose mean value is some fixed and known value. Precisely, I want $\mathbb{E}_G[\theta]=\mathbb{E}_{G_0}[\theta]$ to hold whenever $G$ is sampled from this Dirichlet-process-like object.

Has anyone come across something like this, or do you see a construction?

Bonus points if other moments can be constrained, too. This is to be used in a density estimation problem where the mean value of the density must be fixed.

This approach uses a straightforward of the usual Dirichlet process so that the mean $E_{G}[\theta]$ and variance $V_{G}[\theta]$ can equal specified values (for all G).