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Encyclopedia > Dirichlet distribution
Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4).
Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4).

In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet), often denoted Dir(α), is a family of continuous multivariate probability distributions parametrized by the vector α of nonnegative reals. It is the multivariate generalization of the beta distribution, and conjugate prior of the multinomial distribution in Bayesian statistics. That is, its probability density function returns the belief that the probabilities of K rival events are xi given that each event has been observed αi − 1 times. Image File history File links Size of this preview: 695 × 599 pixel Image in higher resolution (885 × 763 pixel, file size: 337 KB, MIME type: image/png) Several images of probability densities of the Dirichlet distribution as functions on the 2-simplex. ... Image File history File links Size of this preview: 695 × 599 pixel Image in higher resolution (885 × 763 pixel, file size: 337 KB, MIME type: image/png) Several images of probability densities of the Dirichlet distribution as functions on the 2-simplex. ... Probability is the extent to which something is likely to happen or be the case[1]. Probability theory is used extensively in areas such as statistics, mathematics, science, philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. ... Template:Otherusescccc A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 – May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous. ... A multivariate random variable or random vector is a vector X = (X1, ..., Xn) whose components are scalar-valued random variables on the same probability space (Ω, P). ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics, the real numbers may be described informally in several different ways. ... In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where α and β are parameters that must be greater than zero and B is the beta function. ... In Bayesian probability theory, a conjugate prior is a family of prior probability distributions which has the property that the posterior probability distribution also belongs to that family. ... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ... Bayesian refers to probability and statistics -- either methods associated with the Reverend Thomas Bayes (ca. ...

Contents

Probability density function

The probability density function of the Dirichlet distribution of order K is: In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

f(x_1, dots, x_K; alpha_1, dots, alpha_K) = frac{1}{mathrm{B}(alpha)} prod_{i=1}^K x_i^{alpha_i - 1}

where x_i ge 0,, sum_{i=1}^K x_i = 1,, and alpha_i ge 0,.


The normalizing constant is the multinomial beta function, which can be expressed in terms of the gamma function: The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ... A separate article treats the beta-function (written with a hyphen) of physics. ... The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...

mathrm{B}(alpha) = frac{prod_{i=1}^K Gamma(alpha_i)}{Gammaleft(sum_{i=1}^K alpha_iright)}

Properties

Let X = (X_1, ldots, X_K)simoperatorname{Dir}(alpha) and alpha_0 = sum_{i=1}^Kalpha_i, then

mathrm{E}[X_i|alpha] = frac{alpha_i}{alpha_0},
mathrm{Var}[X_i|alpha] = frac{alpha_i (alpha_0-alpha_i)}{alpha_0^2 (alpha_0+1)},
mathrm{Cov}[X_iX_j|alpha] = frac{- alpha_i alpha_j}{alpha_0^2 (alpha_0+1)}.

The mode of the distribution is the vector (x1, ..., xK) with

x_i = frac{alpha_i - 1}{alpha_0 - K}, quad alpha_i > 1.

The Dirichlet distribution is conjugate to the multinomial distribution in the following sense: if In probability theory, the multinomial distribution is a generalization of the binomial distribution. ...

beta|X=(beta_1, ldots, beta_{K})|X sim operatorname{Mult}(X),

where βi is the number of occurrences of i in a sample of n points from the discrete distribution on {1, ..., K} defined by X, then

X | beta sim operatorname{Dir}(alpha + beta).

This relationship is used in Bayesian statistics to estimate the hidden parameters, X, of a discrete probability distribution given a collection of n samples. Intuitively, if the prior is represented as Dir(α), then Dir(α + β) is the posterior following a sequence of observations with histogram β. Bayesian refers to probability and statistics -- either methods associated with the Reverend Thomas Bayes (ca. ... A prior probability is a marginal probability, interpreted as a description of what is known about a variable in the absence of some evidence. ... In Bayesian probability theory, the posterior probability is the conditional probability of some event or proposition, taking empirical data into account. ... Example of a histogram of 100 normally distributed random values. ...


Connections to other distributions

If, for iin{1,2,ldots,K},

Y_isimoperatorname{Gamma}(textrm{shape}=alpha_i,textrm{scale}=1) independently,

then

V=sum_{i=1}^K Y_isimoperatorname{Gamma}(textrm{shape}=sum_{i=1}^Kalpha_i,textrm{scale}=1), and
(X_1,ldots,X_K) = (Y_1/V,ldots,Y_K/V)sim operatorname{Dir}(alpha_1,ldots,alpha_K).

Though the Xis are not independent from one another, they can be seen to be generated from a set of K independent gamma random variables. Unfortunately, since the sum V is lost in the process of forming X = (X1, ..., XK), it is not possible to recover the original gamma random variables from these values alone. Nevertheless, because independent random variables are simpler to work with, this reparametrization can still be useful for proofs about properties of the Dirichlet distribution. In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of exponentially distributed random variables, each of which has mean . ...


Random number generation

A method to sample a random vector x=(x_1, ldots, x_K) from the K-dimensional Dirichlet distribution with parameters (alpha_1, ldots, alpha_K) follows immediately from this connection. First, draw K independent random samples y_1, ldots, y_K from gamma distributions each with density In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of exponentially distributed random variables, each of which has mean . ...

frac{y_i^{alpha_i-1} ; e^{-y_i}}{Gamma (alpha_i)}, !

and then set

x_i = y_i/sum_{j=1}^K y_j. !

Intuitive interpretation of the parameters

One example use of the Dirichlet distribution is if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces. The α/α0 values specify the mean lengths of the cut pieces of string resulting from the distribution. The variance around this mean varies inversely with α0.


See also

In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where α and β are parameters that must be greater than zero and B is the beta function. ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... Often confused with the multinomial distribution. ... In probability theory, the multinomial distribution is a generalization of the binomial distribution. ...

References

Non-Uniform Random Variate Generation, by Luc Devroye http://cg.scs.carleton.ca/~luc/rnbookindex.html

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  Results from FactBites:
 
Dirichlet distribution - Wikipedia, the free encyclopedia (475 words)
In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet), often denoted Dir(α), is a family of continuous multivariate probability distributions parametrized by the vector α of nonnegative reals.
It is the multivariate generalization of the beta distribution, and conjugate prior of the multinomial distribution in Bayesian statistics.
This relationship is used in Bayesian statistics to estimate the hidden parameters, X, of a discrete probability distribution given a collection of n samples.
Johann Peter Gustav Lejeune Dirichlet - Wikipedia, the free encyclopedia (344 words)
His family hailed from the town of Richelet in Belgium, from which his surname "Lejeune Dirichlet" ("le jeune de Richelet", French for "the young chap from Richelet") was derived, and that was where his grandfather lived.
Dirichlet was born in Düren, where his father was the postmaster.
He married Rebecka Mendelssohn Bartholdy, who came from a distinguished family of converts from Judaism to Christianity; she was a granddaughter of the philosopher Moses Mendelssohn, daughter of Abraham Mendelssohn Bartholdy and a sister of the composer Felix Mendelssohn Bartholdy.
  More results at FactBites »


 
 

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