dist.Stick {LaplacesDemon} R Documentation

## Truncated Stick-Breaking Prior Distribution

### Description

These functions provide the density and random number generation of the original, truncated stick-breaking (TSB) prior distribution given \theta and \gamma, as per Ishwaran and James (2001).

### Usage

dStick(theta, gamma, log=FALSE)
rStick(M, gamma)


### Arguments

 M This accepts an integer that is equal to one less than the number of truncated number of possible mixture components (M=1). Unlike most random deviate functions, this is not the number of random deviates to return. theta This is \theta, a vector of length M-1, where M is the truncated number of possible mixture components. gamma This is \gamma, a scalar, and is usually gamma-distributed. log Logical. If log=TRUE, then the logarithm of the density is returned.

### Details

• Application: Discrete Multivariate

• Density: p(\pi) = \frac{(1-\theta)^{\beta-1}}{\mathrm{B}(1,\beta)}

• Inventor: Sethuraman, J. (1994)

• Notation 1: \pi \sim \mathrm{Stick}(\theta,\gamma)

• Notation 2: \pi \sim \mathrm{GEM}(\theta,\gamma)

• Notation 3: p(\pi) = \mathrm{Stick}(\pi | \theta, \gamma)

• Notation 4: p(\pi) = \mathrm{GEM}(\pi | \theta, \gamma)

• Parameter 1: shape parameter \theta \in (0,1)

• Parameter 2: shape parameter \gamma > 0

• Mean: E(\pi) = \frac{1}{1+\gamma}

• Variance: var(\pi) = \frac{\gamma}{(1+\gamma)^2 (\gamma+2)}

• Mode: mode(\pi) = 0

The original truncated stick-breaking (TSB) prior distribution assigns each \theta to be beta-distributed with parameters \alpha=1 and \beta=\gamma (Ishwaran and James, 2001). This distribution is commonly used in truncated Dirichlet processes (TDPs).

### Value

dStick gives the density and rStick generates a random deviate vector of length M.

### References

Ishwaran, H. and James, L. (2001). "Gibbs Sampling Methods for Stick Breaking Priors". Journal of the American Statistical Association, 96(453), p. 161–173.

Sethuraman, J. (1994). "A Constructive Definition of Dirichlet Priors". Statistica Sinica, 4, p. 639–650.

ddirichlet, dmvpolya, and Stick.
library(LaplacesDemon)