 Dirichlet distribution

In probability and statistics, the Dirichlet distribution (after Johann Peter Gustav Lejeune Dirichlet), often denoted , is a family of continuous multivariate probability distributions parametrized by a vector of positive reals. It is the multivariate generalization of the beta distribution. Dirichlet distributions are very often used as prior distributions in Bayesian statistics, and in fact the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution. That is, its probability density function returns the belief that the probabilities of K rival events are x_{i} given that each event has been observed α_{i} − 1 times.
The infinitedimensional generalization of the Dirichlet distribution is the Dirichlet process.
Contents
Probability density function
The Dirichlet distribution of order K ≥ 2 with parameters α_{1}, ..., α_{K} > 0 has a probability density function with respect to Lebesgue measure on the Euclidean space R^{K–1} given by
for all x_{1}, ..., x_{K–1} > 0 satisfying x_{1} + ... + x_{K–1} < 1, where x_{K} is an abbreviation for 1 – x_{1} – ... – x_{K–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:
Support
The support of the Dirichlet distribution is the set of Kdimensional vectors whose entries are real numbers in the interval (0,1); furthermore, , i.e. the sum of the coordinates is 1. These can be viewed as the probabilities of a Kway categorical event. Another way to express this is that the domain of the Dirichlet distribution is itself a probability distribution, specifically a Kdimensional discrete distribution. Note that the technical term for the set of points in the support of a Kdimensional Dirichlet distribution is the open standard K − 1simplex, which is a generalization of a triangle, embedded in the nexthigher dimension. For example, with K = 3, the support looks like an equilateral triangle embedded in a downwardangle fashion in threedimensional space, with vertices at (1,0,0),(0,1,0) and (0,0,1), i.e. touching each of the coordinate axes at a point 1 unit away from the origin.
A very common special case is the symmetric Dirichlet distribution, where all of the elements making up the parameter vector have the same value. Symmetric Dirichlet distributions are often used when a Dirichlet prior is called for, since there typically is no prior knowledge favoring one component over another. Since all elements of the parameter vector have the same value, the distribution alternatively can be parametrized by a single scalar value α, called the concentration parameter. When this value is 1, the symmetric Dirichlet distribution is equivalent to a uniform distribution over the open standard standard K − 1simplex, i.e. it is uniform over all points in its support. Values of the concentration parameter above 1 prefer variates that are dense, evenlydistributed distributions, i.e. all probabilities returned are similar to each other. Values of the concentration parameter below 1 prefer sparse distributions, i.e. most of the probabilities returned will be close to 0, and the vast majority of the mass will be concentrated in a few of the probabilities.
Properties
Moments
Let , meaning that the first K – 1 components have the above density and
Define . Then^{[1]}
Furthermore, if
(Note that the matrix so defined is singular.).
Mode
The mode of the distribution is^{[citation needed]} the vector (x_{1}, ..., x_{K}) with
Marginal distributions
The marginal distributions are Beta distributions^{[2]}:
Conjugate to multinomial
The Dirichlet distribution is conjugate to the multinomial distribution in the following sense: if
where β_{i} is the number of occurrences of i in a sample of n points from the discrete distribution on {1, ..., K} defined by X, then^{[citation needed]}
This relationship is used in Bayesian statistics to estimate the hidden parameters, X, of a categorical distribution (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.
Entropy
If X is a Dir(α) random variable, then the exponential family differential identities can be used to get an analytic expression for the expectation of log(X_{i}) and its associated covariance matrix:^{[citation needed]}
and
where ψ is the digamma function, ψ' is the trigamma function, and δ_{ij} is the Kronecker delta. The formula for yields the following formula for the information entropy of X:^{[citation needed]}
Aggregation
If then, if the random variables with subscripts i and j are dropped from the vector and replaced by their sum,
This aggregation property may be used to derive the marginal distribution of X_{i} mentioned above.
Neutrality
Main article: Neutral vectorIf , then the vector X is said to be neutral^{[3]} in the sense that X_{K} is independent of X^{( − K)}^{[citation needed]} where
and similarly for removing any of . Observe that any permutation of X is also neutral (a property not possessed by samples drawn from a generalized Dirichlet distribution^{[citation needed]}).
Related distributions
If, for
then^{[citation needed]}
and
Although the X_{i}s are not independent from one another, they can be seen to be generated from a set of K independent gamma random variables (see ^{[4]} for proof) . Unfortunately, since the sum V is lost in forming X, 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.
Applications
Multinomial opinions^{[clarification needed]} in subjective logic are equivalent to Dirichlet distributions.
Random number generation
Gamma distribution
A fast method to sample a random vector from the Kdimensional Dirichlet distribution with parameters follows immediately from this connection. First, draw K independent random samples from gamma distributions each with density
and then set
Below is example python code to draw the sample:
params = [a1, a2, ..., ak] sample = [random.gammavariate(a,1) for a in params] sample = [v/sum(sample) for v in sample]
Marginal beta distributions
A less efficient algorithm^{[5]} relies on the univariate marginal and conditional distributions being beta and proceeds as follows. Simulate x_{1} from a distribution. Then simulate in order, as follows. For , simulate ϕ_{j} from a distribution, and let . Finally, set .
Below is example python code to draw the sample:
params = [a1, a2, ..., ak] xs = [random.betavariate(params[0], sum(params[1:]))] for j in range(1,len(params)1): phi = random.betavariate(params[j], sum(params[j+1:])) xs.append((1sum(xs)) * phi) xs.append(1sum(xs))
Intuitive interpretations of the parameters
The concentration parameter
Dirichlet distributions are very often used as prior distributions in Bayesian inference. The simplest and perhaps most common type of Dirichlet prior is the symmetric Dirichlet distribution, where all parameters are equal. This corresponds to the case where you have no prior information to favor one component over any other. As described above, the single value α to which all parameters are set is called the concentration parameter. If the sample space of the Dirichlet distribution is interpreted as a discrete probability distribution, then intuitively the concentration parameter can be thought of as determining how "concentrated" the probability mass of a sample from a Dirichlet distribution is likely to be. With a value much less than 1, the mass will be highly concentrated in a few components, and all the rest will have almost no mass. With a value much greater than 1, the mass will be dispersed almost equally among all the components. See the article on the concentration parameter for further discussion.
String cutting
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 a designated average length, but allowing some variation in the relative sizes of the pieces. The α/α_{0} values specify the mean lengths of the cut pieces of string resulting from the distribution. The variance around this mean varies inversely with α_{0}.
Pólya's urn
Consider an urn containing balls of K different colors. Initially, the urn contains α_{1} balls of color 1, α_{2} balls of color 2, and so on. Now perform N draws from the urn, where after each draw, the ball is placed back into the urn with an additional ball of the same color. In the limit as N approaches infinity, the proportions of different colored balls in the urn will be distributed as Dir(α_{1},...,α_{K}).^{[6]}
For a formal proof, note that the proportions of the different colored balls form a bounded [0,1]^{K}valued martingale, hence by the martingale convergence theorem, these proportions converge almost surely and in mean to a limiting random vector. To see that this limiting vector has the above Dirichlet distribution, check that all mixed moments agree.
Note that each draw from the urn modifies the probability of drawing a ball of any one color from the urn in the future. This modification diminishes with the number of draws, since the relative effect of adding a new ball to the urn diminishes as the urn accumulates increasing numbers of balls. This "diminishing returns" effect can also help explain how large α values yield Dirichlet distributions with most of the probability mass concentrated around a single point on the simplex.
See also
 Beta distribution
 Binomial distribution
 Categorical distribution
 Generalized Dirichlet distribution
 Latent Dirichlet allocation
 Dirichlet process
 Multinomial distribution
 Multivariate Pólya distribution
References
 ^ Eq. (49.9) on page 488 of Kotz, Balakrishnan & Johnson (2000). Continuous Multivariate Distributions. Volume 1: Models and Applications. New York: Wiley.
 ^ Ferguson, Thomas S. (1973). "A Bayesian analysis of some nonparametric problems". The Annals of Statistics 1 (2): 209–230. doi:10.1214/aos/1176342360.
 ^ Connor, Robert J.; Mosimann, James E (1969). "Concepts of Independence for Proportions with a Generalization of the Dirichlet Distribution". Journal of the American statistical association (American Statistical Association) 64 (325): 194–206. doi:10.2307/2283728. JSTOR 2283728.
 ^ Devroye, Luc (1986). NonUniform Random Variate Generation. pp. 594. http://cg.scs.carleton.ca/~luc/rnbookindex.html.
 ^ A. Gelman and J. B. Carlin and H. S. Stern and D. B. Rubin (2003). Bayesian Data Analysis (2nd ed.). pp. 582. ISBN 158488388X.
 ^ Blackwell, David; MacQueen, James B. (1973). "Ferguson distributions via Polya urn schemes". Ann. Stat. 1 (2): 353–355. doi:10.1214/aos/1176342372.
External links
 Introduction to the Dirichlet Distribution and Related Processes by Frigyik, Kapila and Gupta
 Dirichlet Distribution
 Estimating the parameters of the Dirichlet distribution by Thomas Minka
 NonUniform Random Variate Generation by Luc Devroye
 Dirichlet Random Measures, Method of Construction via Compound Poisson Random Variables, and Exchangeability Properties of the resulting Gamma Distribution
Categories: Multivariate continuous distributions
 Conjugate prior distributions
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