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Topic: Dirichlet distribution

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Multinomial distribution

Conjugate prior

Dirichlet character

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	Dirichlet distribution - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Dirichlet_distribution (475 words)

	Johann Peter Gustav Lejeune Dirichlet - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Dirichlet (344 words)

	How to put error bars on histograms
		The conjugate prior/posterior to the multinomial distribution is the Dirichlet distribution.
		The probability distribution of each of the bin counts of a histogram is the marginal of a multinomial distribution, which is the binomial distribution.
		By generating samples from the posterior distribution, and accepting or rejecting them based on the smoothing weights given by Equations 12 and 13, B samples from the smoothed posterior could be obtained.
	pangea.stanford.edu /research/noble/epdu/paper (8238 words)

	Dirichlet process - MLpedia
		As draws from a Dirichlet process are discrete, an important use is as a prior in infinite mixture models.
		The generative process is therefore that a sample is drawn from a Dirichlet process, and in turn for each data point a value is drawn from this sample distribution and used as the component distribution for that data point.
		The Pitman-Yor distribution (also known as the 'two-parameter Poisson-Dirichlet process') is a generalisation of the Dirichlet process.
	www.mlpedia.org /index.php?title=Dirichlet_process (671 words)

	Normal distribution
		The normal distribution, which is also known as the Gaussian distribution, is ubiquitous in statistics.
		This section concentrates on the univariate normal distribution, as the general multivariate distribution is not needed in this thesis.
		A plot of the probability density function of the normal distribution is shown in Figure A.1.
	www.cis.hut.fi /ahonkela/dippa/node94.html (161 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		Due to the Dirichlet distribution being the conjugate prior to the multinomial distribution, the probability of the corpus is derived easily.
		LDA generates a corpus in three stages: it first chooses the length of the corpus from a Poisson distribution, then picks a latent variable Theta according to a Dirichlet distribution which it finally uses to pick a topic from a multinomial distribution and words from a multinomial distribution conditioned on the topic variable.
		First, the dimensionality k of the Dirichlet distribution (and thus the dimensionality of the topic variable z) is assumed known and fixed.
	www.cs.wisc.edu /~apirak/cs/cs838/reviews_score_15.html (7493 words)

	Bayesian smoothing through text classification
		In order to extend the Bayesian approach to sparse multinomial distributions, several authors have used the notion of maintaining uncertainty over the vocabulary from which observations are produced as well as their probabilities.
		Friedman and Singer also point out that this distribution remains well-defined in the case where the alphabet is unbounded, and show that their method gives good performance in a simple character-prediction task.
		The intuition behind this approach is that it adds to Friedman and Singer's method a second process that attempts to classify the target distribution as one of a number of known distributions, and uses the posterior probability of these distributions for full Bayesian smoothing.
	nlp.stanford.edu /courses/cs224n/2001/gruffydd/smoothing.html (3059 words)

	Dirichlet distribution
		The Dirichlet distribution is the conjugate prior of the parameters of the multinomial distribution.
		The pdfs of the Dirichlet distribution with certain parameter values are shown in Figure A.2.
		Because both the parameters of the distribution are equal, the distribution of the other component will be exactly the same.
	www.cis.hut.fi /ahonkela/dippa/node95.html (187 words)

	Gaussian Limits Associated with the Poisson--Dirichlet Distribution and the Ewens Sampling Formula (Site not responding. Last check: 2007-10-11)
		In this paper we consider large $\theta$ approximations for the stationary distribution of the neutral infinite alleles model as described by the the Poisson--Dirichlet distribution with parameter $\theta$.
		In particular, we show that if a sample of size $n$ is drawn from a population described by the Poisson--Dirichlet distribution, then the conditional probability of a particular sample configuration is asymptotically normal with mean and variance dertermined by the Ewens sampling formula.
		The asymptotic normality of the conditional sampling distribution is somewhat surprising since it is a fairly complicated function of the population frequencies.
	www.webpages.uidaho.edu /~krone/pd.html (171 words)

	Distribution of Mutual Information (Marcus Hutter)
		From the prior p(t) one can compute the posterior p(tn), from which the distribution p(In) of the mutual information can be calculated.
		[Artificial Intelligence]", keywords = "Mutual Information, Cross Entropy, Dirichlet distribution, Second order distribution, expectation and variance of mutual information.", abstract = "The mutual information of two random variables i and j with joint probabilities t_ij is commonly used in learning Bayesian nets as well as in many other fields.
		To answer questions like ``is I(n_ij/n) consistent with zero?'' or ``what is the probability that the true mutual information is much larger than the point estimate?'' one has to go beyond the point estimate.
	www.hutter1.de /ai/xentropy.htm (364 words)

	Maximum-Likelihood and Markov Chain Monte Carlo Approaches to Estimate Inbreeding and Effective Size From Allele ...
		Histogram and kernel density estimate of MCMC drawings in the posterior distribution of the inbreeding coefficient, for a French snail population.
		The parameters of the prior distribution are equal to those presented in Table 1.
		Dirichlet distribution is not appropriate, since the distribution
	www.genetics.org /cgi/content/full/164/3/1189 (6162 words)

	Likelihoods
		The negative binomial distribution can be used to model count data that are overdispersed relative to the Poisson distribution; see McCullagh and Nelder (1985, §6.2.3 in the second edition) and Graves and Picard (2002).
		The Dirichlet distribution takes two arguments: the vectors of ``data'' (the probabilities) and the exponents in the Dirichlet prior.
		For sampling from finite populations, the hypergeometric distribution is available.
	www.stat.lanl.gov /yadas/node11.html (643 words)

	Estimating Lorenz Curves Using a Dirichlet Distribution
		The Lorenz curve relates the cumulative proportion of income to the cumulative proportion of population.
		When a particular functional form of the Lorenz curve is specified it is typically estimated by linear or nonlinear least squares assuming that the error terms are independently and normally distributed.
		This paper proposes and applies a new methodology which recognizes the cumulative proportional nature of the Lorenz curve data by assuming that the proportion of income is distributed as a Dirichlet distribution.
	ideas.repec.org /p/ecm/wc2000/1215.html (594 words)

	Estimating Lorenz Curves Using a Dirichlet Distribution
		When a particular functional form of the Lorenz curve is specified it is typically estimated by linear or nonlinear least squares, estimation techniques that have good properties when the error terms are independently and normally distributed.
		This paper proposes and applies a new methodology that recognises the cumulative proportional nature of the Lorenz curve data by assuming that the income proportions are distributed as a Dirichlet distribution.
		Maximum likelihood estimates under the Dirichlet distribution assumption provide better-fitting Lorenz curves than nonlinear least squares and another estimation technique that has appeared in the literature.
	ideas.repec.org /p/mlb/wpaper/802.html (665 words)

	Three Categorical Models
		We assume that the form of the data is such that each observation is a vector of categorical counts.
		That model is a hierarchical model where each vector of observed categorical counts has a multinomial distribution with a particular probability vector, while the probability vectors have a standard Dirichlet distribution (Johnson and Kotz (1972), pp.
		The third model is like the second model except that the probability vectors do not have a standard Dirichlet distribution, but have the same distribution as logistically transformed multivariate Gaussian random variables.
	www.hafro.is /dst2/report2/node81.html (773 words)

	Fastfit Matlab toolbox (Site not responding. Last check: 2007-10-11)
		This toolbox implements efficient maximum-likelihood estimation of various distributions.
		% Version 1.2 19-May-04 % By Thomas P. Minka % % Dirichlet % dirichlet_sample - Sample from Dirichlet distribution.
		Dirichlet-multinomial % polya_sample - Sample from Dirichlet-multinomial (Polya) distribution.
	research.microsoft.com /~minka/software/fastfit (106 words)

	Abstract: Simulation from a Normally Weighted Dirichlet Distribution (Site not responding. Last check: 2007-10-11)
		Analysis of a class of non-neutral population genetics models requires the simulation of samples, computation of sample statistics and evaluation of likelihood surfaces for a multivariate density function that is the product of a Normal density and a Dirichlet density, defined over an m-dimensional simplex.
		A method will be described for the accurate computation of the normalizing constant for the distribution function.
		Several methods will be considered for computing samples from the distribution.
	www.galaxy.gmu.edu /interface/I03/I2003WebPage/abstracts/03142.html (108 words)

	EconPapers: DIRIFIT: Stata module to fit a Dirichlet distribution
		Abstract: dirifit fits by maximum likelihood a Dirichlet distribution to a set of variables.
		Keywords: maximum likelihood; Dirichlet distribution; proportions (search for similar items in EconPapers)
		Note: This module may be installed from within Stata by typing "ssc install dirifit".
	econpapers.repec.org /software/bocbocode/s456725.htm (222 words)

	GNU Scientific Library -- Reference Manual - The Dirichlet Distribution
		random variates from a Dirichlet distribution of order K-1.
		values from gamma distributions with parameters a=alpha_i, b=1, and renormalizing.
		, \theta_K) for a Dirichlet distribution with parameters
	www.network-theory.co.uk /docs/gslref/TheDirichletDistribution.html (173 words)

	Estimating a Dirichlet distribution (Site not responding. Last check: 2007-10-11)
		The Dirichlet distribution and its compound variant, the Dirichlet-multinomial, are two of the most basic models for proportional data, such as the mix of vocabulary words in a text document.
		Yet the maximum-likelihood estimate of these distributions is not available in closed-form.
		Last modified: Wed Aug 17 18:10:52 GMT 2005
	research.microsoft.com /~minka/papers/dirichlet (89 words)

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