Dirichlet Distribution — Dir • joker

The Dirichlet distribution is an absolute continuous probability, specifically a multivariate generalization of the beta distribution, parameterized by a vector $\boldsymbol{\alpha} = (\alpha_1, \alpha_2, ..., \alpha_k)$ with $\alpha_i > 0$.

Usage

Dir(alpha = c(1, 1))

ddir(x, alpha, log = FALSE)

rdir(n, alpha)

# S4 method for class 'Dir,numeric'
d(distr, x, log = FALSE)

# S4 method for class 'Dir,matrix'
d(distr, x)

# S4 method for class 'Dir,numeric'
r(distr, n)

# S4 method for class 'Dir'
mean(x)

# S4 method for class 'Dir'
mode(x)

# S4 method for class 'Dir'
var(x)

# S4 method for class 'Dir'
entro(x)

# S4 method for class 'Dir'
finf(x)

lldir(x, alpha)

# S4 method for class 'Dir,matrix'
ll(distr, x)

edir(x, type = "mle", ...)

# S4 method for class 'Dir,matrix'
mle(
  distr,
  x,
  par0 = "same",
  method = "L-BFGS-B",
  lower = 1e-05,
  upper = Inf,
  na.rm = FALSE
)

# S4 method for class 'Dir,matrix'
me(distr, x, na.rm = FALSE)

# S4 method for class 'Dir,matrix'
same(distr, x, na.rm = FALSE)

vdir(alpha, type = "mle")

# S4 method for class 'Dir'
avar_mle(distr)

# S4 method for class 'Dir'
avar_me(distr)

# S4 method for class 'Dir'
avar_same(distr)

Arguments

alpha: numeric. The non-negative distribution parameter vector.
x: For the density function, x is a numeric vector of quantiles. For the moments functions, x is an object of class Dir. For the log-likelihood and the estimation functions, x is the sample of observations.
log: logical. Should the logarithm of the probability be returned?
n: number of observations. If length(n) > 1, the length is taken to be the number required.
distr: an object of class Dir.
type: character, case ignored. The estimator type (mle, me, or same).
...: extra arguments.
par0, method, lower, upper: arguments passed to optim for the mle optimization.
na.rm: logical. Should the NA values be removed?

Value

Each type of function returns a different type of object:

Distribution Functions: When supplied with one argument (distr), the d(), p(), q(), r(), ll() functions return the density, cumulative probability, quantile, random sample generator, and log-likelihood functions, respectively. When supplied with both arguments (distr and x), they evaluate the aforementioned functions directly.
Moments: Returns a numeric, either vector or matrix depending on the moment and the distribution. The moments() function returns a list with all the available methods.
Estimation: Returns a list, the estimators of the unknown parameters. Note that in distribution families like the binomial, multinomial, and negative binomial, the size is not returned, since it is considered known.
Variance: Returns a named matrix. The asymptotic covariance matrix of the estimator.

Details

The probability density function (PDF) of the Dirichlet distribution is given by: $$ f(x_1, ..., x_k; \alpha_1, ..., \alpha_k) = \frac{1}{B(\boldsymbol{\alpha})} \prod_{i=1}^k x_i^{\alpha_i - 1}, $$ where $B(\boldsymbol{\alpha})$ is the multivariate Beta function: $$ B(\boldsymbol{\alpha}) = \frac{\prod_{i=1}^k \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^k \alpha_i\right)} $$ and $\sum_{i=1}^k x_i = 1$, $x_i > 0$.

References

Oikonomidis, I. & Trevezas, S. (2025), Moment-Type Estimators for the Dirichlet and the Multivariate Gamma Distributions, arXiv, https://arxiv.org/abs/2311.15025

Examples

# -----------------------------------------------------
# Dir Distribution Example
# -----------------------------------------------------

# Create the distribution
a <- c(0.5, 2, 5)
D <- Dir(a)

# ------------------
# dpqr Functions
# ------------------

d(D, c(0.3, 0.2, 0.5)) # density function
#> [1] 1.003913
x <- r(D, 100) # random generator function

# alternative way to use the function
df <- d(D) ; df(x) # df is a function itself
#>   [1]  24.77300121   7.30983973   0.82018079   3.09639840   9.01982497
#>   [6]  39.84500725 330.11542526   9.42548756   4.61888811   2.27497931
#>  [11]  64.27091137  18.62530960   2.10211248   1.63802351   1.19686120
#>  [16]   8.25251246  17.90121464   0.74870277   8.72460022   0.55837485
#>  [21]   8.21813142   1.88378114  18.25077604  36.30299265 193.49162463
#>  [26]  16.48802269   7.72075754   1.92680717   1.59190279   2.49922116
#>  [31]   7.30498152 125.93965445   0.54948163   2.53622638  56.48520822
#>  [36]   0.43959178   6.73568100  56.29973185  19.69573566   2.82021054
#>  [41]  17.76917321   2.44265070 136.52489485   6.02799115   9.74419696
#>  [46] 103.22800743  62.45528918  59.23861402  16.72800114  12.04350492
#>  [51]   0.21314095  15.49143498  20.53195188   8.40650362  15.71665753
#>  [56]   3.40624205   7.31707158  16.34273035   9.23467406  66.67870933
#>  [61] 103.58033370   0.05178777   2.78600739  44.65538491  16.32873325
#>  [66]   6.18668122   7.92540146  16.90022450   7.97774674 132.13019415
#>  [71]  13.78471007   9.94941804   3.18909544   3.71704118   1.92530182
#>  [76]   6.59831401  16.47057195   0.92414964 129.28524193   1.24635003
#>  [81]  80.27388019 476.34209004   3.11768072  13.39764640  29.41641117
#>  [86]   1.99204330   9.86443108  28.78544730   0.55269678  26.26901025
#>  [91]  20.60548672   5.84967397  12.47014567  50.39362983   1.54175468
#>  [96]   9.07270050   6.03121668  12.81735187   5.22700358   2.67761296

# ------------------
# Moments
# ------------------

mean(D) # Expectation
#> [1] 0.06666667 0.26666667 0.66666667
mode(D) # Mode
#> [1] -0.1111111  0.2222222  0.8888889
var(D) # Variance
#>              [,1]         [,2]         [,3]
#> [1,]  0.007320261 -0.002091503 -0.005228758
#> [2,] -0.002091503  0.023006536 -0.020915032
#> [3,] -0.005228758 -0.020915032  0.026143791
entro(D) # Entropy
#> [1] -2.452547
finf(D) # Fisher Information Matrix
#>            alpha1     alpha2      alpha3
#> alpha1  4.7921863 -0.1426159 -0.14261590
#> alpha2 -0.1426159  0.5023182 -0.14261590
#> alpha3 -0.1426159 -0.1426159  0.07870706

# List of all available moments
mom <- moments(D)
mom$mean # expectation
#> [1] 0.06666667 0.26666667 0.66666667

# ------------------
# Point Estimation
# ------------------

ll(D, x)
#> [1] 219.6185
lldir(x, a)
#> [1] 219.6185

edir(x, type = "mle")
#> $alpha
#> [1] 0.5942496 1.9223128 5.0267809
#> 
edir(x, type = "me")
#> $alpha
#> [1] 0.8187913 2.2631115 5.9893682
#> 

mle(D, x)
#> $alpha
#> [1] 0.5942496 1.9223128 5.0267809
#> 
me(D, x)
#> $alpha
#> [1] 0.8187913 2.2631115 5.9893682
#> 
e(D, x, type = "mle")
#> $alpha
#> [1] 0.5942496 1.9223128 5.0267809
#> 

mle("dir", x) # the distr argument can be a character
#> $alpha
#> [1] 0.5942496 1.9223128 5.0267809
#> 

# ------------------
# Estimator Variance
# ------------------

vdir(a, type = "mle")
#>           alpha1    alpha2    alpha3
#> alpha1 0.2581065 0.4243918  1.236676
#> alpha2 0.4243918 4.7978389  9.462596
#> alpha3 1.2366759 9.4625963 32.092246
vdir(a, type = "me")
#>           alpha1    alpha2    alpha3
#> alpha1 0.6967287  1.109928  3.213417
#> alpha2 1.1099281  6.555295 13.730859
#> alpha3 3.2134166 13.730859 43.124878

avar_mle(D)
#>           alpha1    alpha2    alpha3
#> alpha1 0.2581065 0.4243918  1.236676
#> alpha2 0.4243918 4.7978389  9.462596
#> alpha3 1.2366759 9.4625963 32.092246
avar_me(D)
#>           alpha1    alpha2    alpha3
#> alpha1 0.6967287  1.109928  3.213417
#> alpha2 1.1099281  6.555295 13.730859
#> alpha3 3.2134166 13.730859 43.124878

v(D, type = "mle")
#>           alpha1    alpha2    alpha3
#> alpha1 0.2581065 0.4243918  1.236676
#> alpha2 0.4243918 4.7978389  9.462596
#> alpha3 1.2366759 9.4625963 32.092246