# Finding A Minimally Informative Dirichlet Prior Using Least Squares

## Abstract

In a Bayesian framework, the Dirichlet distribution is the conjugate distribution to the multinomial likelihood function, and so the analyst is required to develop a Dirichlet prior that incorporates available information. However, as it is a multiparameter distribution, choosing the Dirichlet parameters is less straightforward than choosing a prior distribution for a single parameter, such as p in the binomial distribution. In particular, one may wish to incorporate limited information into the prior, resulting in a minimally informative prior distribution that is responsive to updates with sparse data. In the case of binomial p or Poisson \lambda, the principle of maximum entropy can be employed to obtain a so-called constrained noninformative prior. However, even in the case of p, such a distribution cannot be written down in the form of a standard distribution (e.g., beta, gamma), and so a beta distribution is used as an approximation in the case of p. In the case of the multinomial model with parametric constraints, the approach of maximum entropy does not appear tractable. This paper presents an alternative approach, based on constrained minimization of a least-squares objective function, which leads to a minimally informative Dirichlet prior distribution. The alpha-factor model for common-cause failure,more »

- Authors:

- Publication Date:

- Research Org.:
- Idaho National Laboratory (INL)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1023505

- Report Number(s):
- INL/CON-11-20971

TRN: US1104565

- DOE Contract Number:
- DE-AC07-05ID14517

- Resource Type:
- Conference

- Resource Relation:
- Conference: PSA 2011,Wilmington, NC,03/13/2011,03/17/2011

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS; APPROXIMATIONS; DISTRIBUTION; ENTROPY; MINIMIZATION; SIMULATION; alpha-factor; Common-cause failures; constrained noninformative prior; Jeffreys prior

### Citation Formats

```
Dana Kelly.
```*Finding A Minimally Informative Dirichlet Prior Using Least Squares*. United States: N. p., 2011.
Web.

```
Dana Kelly.
```*Finding A Minimally Informative Dirichlet Prior Using Least Squares*. United States.

```
Dana Kelly. Tue .
"Finding A Minimally Informative Dirichlet Prior Using Least Squares". United States. https://www.osti.gov/servlets/purl/1023505.
```

```
@article{osti_1023505,
```

title = {Finding A Minimally Informative Dirichlet Prior Using Least Squares},

author = {Dana Kelly},

abstractNote = {In a Bayesian framework, the Dirichlet distribution is the conjugate distribution to the multinomial likelihood function, and so the analyst is required to develop a Dirichlet prior that incorporates available information. However, as it is a multiparameter distribution, choosing the Dirichlet parameters is less straightforward than choosing a prior distribution for a single parameter, such as p in the binomial distribution. In particular, one may wish to incorporate limited information into the prior, resulting in a minimally informative prior distribution that is responsive to updates with sparse data. In the case of binomial p or Poisson \lambda, the principle of maximum entropy can be employed to obtain a so-called constrained noninformative prior. However, even in the case of p, such a distribution cannot be written down in the form of a standard distribution (e.g., beta, gamma), and so a beta distribution is used as an approximation in the case of p. In the case of the multinomial model with parametric constraints, the approach of maximum entropy does not appear tractable. This paper presents an alternative approach, based on constrained minimization of a least-squares objective function, which leads to a minimally informative Dirichlet prior distribution. The alpha-factor model for common-cause failure, which is widely used in the United States, is the motivation for this approach, and is used to illustrate the method. In this approach to modeling common-cause failure, the alpha-factors, which are the parameters in the underlying multinomial model for common-cause failure, must be estimated from data that are often quite sparse, because common-cause failures tend to be rare, especially failures of more than two or three components, and so a prior distribution that is responsive to updates with sparse data is needed.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2011},

month = {3}

}