The `Stick`

function provides the utility of truncated
stick-breaking regarding the vector
\(\theta\). Stick-breaking is commonly referred to as a
stick-breaking process, and is used often in a Dirichlet
process (Sethuraman, 1994). It is commonly associated with
infinite-dimensional mixtures, but in practice, the `infinite' number
is truncated to a finite number, since it is impossible to estimate an
infinite number of parameters (Ishwaran and James, 2001).

`Stick(theta)`

theta

This required argument, \(\theta\) is a vector of length \((M-1)\) regarding \(M\) mixture components.

The `Stick`

function returns a probability vector wherein each
element relates to a mixture component.

The Dirichlet process (DP) is a stochastic process used in Bayesian nonparametric modeling, most commonly in DP mixture models, otherwise known as infinite mixture models. A DP is a distribution over distributions. Each draw from a DP is itself a discrete distribution. A DP is an infinite-dimensional generalization of Dirichlet distributions. It is called a DP because it has Dirichlet-distributed, finite-dimensional, marginal distributions, just as the Gaussian process has Gaussian-distributed, finite-dimensional, marginal distributions. Distributions drawn from a DP cannot be described using a finite number of parameters, thus the classification as a nonparametric model. The truncated stick-breaking (TSB) process is associated with a truncated Dirichlet process (TDP).

An example of a TSB process is cluster analysis, where the number of
clusters is unknown and treated as mixture components. In such a
model, the TSB process calculates probability vector \(\pi\)
from \(\theta\), given a user-specified maximum number of
clusters to explore as \(C\), where \(C\) is the length of
\(\theta + 1\). Vector \(\pi\) is assigned a TSB
prior distribution (for more information, see `dStick`

).

Elsewhere, each element of \(\theta\) is constrained to the interval (0,1), and the original TSB form is beta-distributed with the \(\alpha\) parameter of the beta distribution constrained to 1 (Ishwaran and James, 2001). The \(\beta\) hyperparameter in the beta distribution is usually gamma-distributed.

A larger value for a given \(\theta_m\) is associated with a higher probability of the associated mixture component, however, the proportion changes according to the position of the element in the \(\theta\) vector.

A variety of stick-breaking processes exist. For example, rather than each \(\theta\) being beta-distributed, there have been other forms introduced such as logistic and probit, among others.

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
`dStick`

.