Stick {LaplacesDemon} | R Documentation |

## Truncated Stick-Breaking

### Description

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).

### Usage

```
Stick(theta)
```

### Arguments

`theta` |
This required argument, |

### Details

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.

### Value

The `Stick`

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

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### 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.

### See Also

`ddirichlet`

,
`dmvpolya`

, and
`dStick`

.

*LaplacesDemon*version 16.1.6 Index]