StickBreaking function

The Stick Breaking representation of the Dirichlet process.

The Stick Breaking representation of the Dirichlet process.

A Dirichlet process can be represented using a stick breaking construction [REMOVE_ME]G=i=1npiiδθi[REMOVEME2] G = \sum _{i=1} ^n pi _i \delta _{\theta _i} [REMOVE_ME_2], where πk=βkk=1n1(1βk)\pi _k = \beta _k \prod _{k=1} ^{n-1} (1- \beta _k ) are the stick breaking weights. The atoms δθi\delta _{\theta _i} are drawn from G0G_0 the base measure of the Dirichlet Process. The βkBeta(1,α)\beta _k \sim \mathrm{Beta} (1, \alpha). In theory nn should be infinite, but we chose some value of NN to truncate the series. For more details see reference.

StickBreaking(alpha, N) piDirichlet(betas)

Arguments

  • alpha: Concentration parameter of the Dirichlet Process.
  • N: Truncation value.
  • betas: Draws from the Beta distribution.

Returns

Vector of stick breaking probabilities.

Description

A Dirichlet process can be represented using a stick breaking construction

G=i=1npiiδθi G = \sum _{i=1} ^n pi _i \delta _{\theta _i}

, where πk=βkk=1n1(1βk)\pi _k = \beta _k \prod _{k=1} ^{n-1} (1- \beta _k ) are the stick breaking weights. The atoms δθi\delta _{\theta _i} are drawn from G0G_0 the base measure of the Dirichlet Process. The βkBeta(1,α)\beta _k \sim \mathrm{Beta} (1, \alpha). In theory nn should be infinite, but we chose some value of NN to truncate the series. For more details see reference.

Functions

  • piDirichlet(): Function for calculating stick lengths.

References

Ishwaran, H., & James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association, 96(453), 161-173.