Mass Point, Dirac Delta in Dirichlet Processes

When dealing with Dirichlet Processes, according to [Teh, 2007], a DP is defined as by a base Probability H and a scale factor "alpha"

According to the Stick Breaking Construction, the random draws G from a DP:

G~DP(alpha,H)

are given by:

G=sum(pi_k*delta_theta_k) over k from 1 to infinity

pi_k are ordered draws from a Beta Distribution given the length of an unitary stick

delta_theta_k is a point mass centered in "theta_k" (theta_k are random draws from the base distribution)

I have pretty much a clear understanding of all the variables, but I do not know what do they mean by "mass point" is it the probability densi开发者_运维知识库ty of that draw, or is it something else.

It would be great if you could point me in any direction, only a reference would be amazing.

Thanks

G is a probability distribution over probability distributions. These (sub) probability distributions are over some domain, let's call it BigTheta.

Each theta_k is a draw from a distribution over BigTheta, so it is some element of BigTheta.

Each delta_theta_k is a probability distribution over BigTheta, defined to be delta_theta_k(theta_k) = 1 and delta_theta_k(anything else) = 0. This is what they call the 'point mass' distribution, because all the mass of the distribution is over a single point of the domain.

G is a probability distribution over probability distributions over BigTheta, defined as: for some distribution over BigTheta called f (which is parameterised by theta), G(f(theta)) = sum (pi_k * delta_theta_k(theta)).

I hope that helps, I think you generally have the right idea it's just the notation can get a little complicated (and SO isn't the best for this kind of notation). It's generally helpful whenever you encounter a symbol to think about what type of function it is, i.e. what is it defined over.