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).

1 | ```
Stick(theta)
``` |

`theta` |
This required argument, |

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.

The `Stick`

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

Statisticat, LLC. software@bayesian-inference.com

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`

.

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