Dirichlet distribution | Vose Software

# Dirichlet distribution

Format: Dirichlet({a})

##### Uses

The Dirichlet distribution is used in modeling the uncertainty of probabilities, prevalence of fractions where there are multiple states to consider. It is the multinomial process extension to the beta distribution for a binomial process.

## Examples

### Example 1:

You have the results of a survey conducted in the premises of a retail outlet. The age and sex of 500 randomly selected shoppers were recorded:

<25 years, male: 38 people

25 to < 40 years, male: 72 people

> 40 years, male: 134 people

<25 years, female: 57 people

25 to < 40 years, female: 126 people

> 40 years, female: 73 people

In a manner analogous to the beta distribution, we add 1 to each number of observations to get the Dirichlet parameters, and we can estimate the fraction of all shoppers to this store that are in each category as follows:

{=VoseDirichlet({38,72,134,57,126,73}+1)}

or

{=VoseDirichlet({39,73,135,58,127,74})}

or

{=VoseDirichlet(A1:A6+1)}

where A1:A6 would contain the data. Note that the VoseDirichlet function is entered as an array function in Excel (in this case covering six cells), and then returns the uncertainty about the fraction of all shoppers that are in each of the six group.

## Example 2:

A review of 1000 companies that were S&P AAA rated last year in your sector shows their rating one year later:

AAA: 908

AA: 83

A: 7

BBB or below: 2

If we assume that the market has similar volatilities to last year, we can estimate the probability that a company rated AAA now will be in each state next year as:

{=VoseDirichlet({909,84,8,3})}

The Dirichlet then returns the uncertainty about these probabilities.

## ModelRisk functions added to Microsoft Excel for the Dirichlet distribution

VoseDirichlet generates random values from this distribution for

VoseDirichletProb returns the probability density or cumulative distribution function for this distribution.

VoseDirichletProb10 returns the log10 of the probability density or cumulative distribution function.

## Dirichlet distribution equations

The Dirichlet distribution of order K ≥ 2 with parameters α1, ..., αK > 0 has a probability density function given by

for all x1, ..., xK–1 > 0 satisfying x1 + ... + xK–1 < 1, and where xK = 1 – x1 – ... – xK–1. The density is zero outside this open (K − 1)-dimensional simplex.

The normalizing constant is the multinomial beta function, which can be expressed in terms of the gamma function: