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Topic Modeling
Published in John Atkinson-Abutridy, Text Analytics, 2022
Formally, a Dirichlet distribution defines a probability density for an entry of k categorical (topical) values in a vector α that has the same number of characteristics as the multinomial parameter θ. A Dirichlet of parameter α or Dir(α) for a certain number of topics k is defined as: Dir(α)→p(θ|α)=Γ(∑i=1kαi)∏i=1kτ(αi)∏i=1kθαi−1
Dirichlet Process
Published in Nezameddin Faghih, Ebrahim Bonyadi, Lida Sarreshtehdari, Quality Management and Operations Research, 2021
Nezameddin Faghih, Ebrahim Bonyadi, Lida Sarreshtehdari
And we write (p1,p2,…,pn)~Dir(a1,a2,…,an), in which Γ(.) denotes the gamma function. As a matter of fact, it can be said that the Dirichlet distribution is a distribution over distributions. That means, the Dirichlet distribution covers other distributions.
Early Warning Strategy of Sparse Failures for Highly Reliable Products Based on the Bayesian Method
Published in Lirong Cui, Ilia Frenkel, Anatoly Lisnianski, Stochastic Models in Reliability Engineering, 2020
Shirong Zhou, Yincai Tang, Ancha Xu, Yongqiang Lian
The Bayesian estimator of θ requires a prior density. It is common to use a conjugate prior, to simplify the mathematical derivation. A prior is called conjugate if, when multiplied by the likelihood, the resulting posterior is in the same parametric family as the prior. It turns out that the Dirichlet distribution is the conjugate prior of a multinomial distribution. So, let the prior for parameter vector θ be θ~Dir(α),α=(α0,α1,…,αJ),
Flexible Distribution-Based Regression Models for Count Data: Application to Medical Diagnosis
Published in Cybernetics and Systems, 2020
Pantea Koochemeshkian, Nuha Zamzami, Nizar Bouguila
Dirichlet distribution (Mosimann 1962), is a generalization of the Beta distribution, offering significant flexibility and ease of use. The Dirichlet distribution has the advantage that by varying its parameters (Bouguila and Ziou 2005b) it permits multiple modes and asymmetries and can thus approximate a wide variety of shapes (Bouguila, Ziou, and Vaillancourt 2004, Bouguila and Ziou 2005a). The Dirichlet distribution is commonly used given its flexibility and its several interesting properties, such as the consistency of its estimates, and its ease of use as well as the fact that it is conjugate to the multinomial distribution. Considering the Dirichlet as a prior distribution to the multinomial results in the Dirichlet Multinomial (DM) Distribution (Wang and Zhao 2017; Bouguila, Ziou, and Vaillancourt 2003).
Responsive transport network design: minimal investment for desired travel time reduction
Published in Transportation Letters, 2022
The Dirichlet allocation method is based on Dirichlet distribution and it can be classified into Monte Carlo simulation method. The Dirichlet distribution, usually denoted as , is a family of continuous multivariate probability distributions parameterized by a vector of positive reals . The vector length is equal to the number of candidate links . That is:
Decomposition methods for solving Markov decision processes with multiple models of the parameters
Published in IISE Transactions, 2021
Lauren N. Steimle, Vinayak S. Ahluwalia, Charmee Kamdar, Brian T. Denton
To test the performance of the proposed algorithms, we present an MDP related to machine maintenance adapted from Delage and Mannor (2009). The states of the MDP represent the quality of the machine. Over the course of the planning horizon, the DM chooses between doing nothing and different levels of repairs with the goal of minimizing expected total cost. We generated test instances of this MDP of four different problem sizes in terms of the number of states, actions, and decision epochs, and used a Dirichlet distribution to sample the transition probabilities. For each problem size, we varied the number of models and variance among the transition probability parameters through the use of a Dirichlet distribution. The Dirichlet distribution is a generalization of the Beta distribution and describes a family of continuous probability distributions over a set of n discrete categories. The distribution is characterized by n parameters where with is the base measure of the distribution and c > 0 is the concentration parameter. The expected value of random variable Xi drawn from the distribution X1, X2 ,…, Xn ∼ Dirichlet is given by (Readers who wish to learn more about the Dirichlet distribution are referred to Ferguson et al. (1974)). To generate these test instances, we used the nominal rows of the transition matrix described in Delage and Mannor (2009) as the base measure of the Dirichlet distribution and varied the value of the concentration parameter. For a fixed base measure, the concentration parameter of the Dirichlet distribution controls how concentrated the parameters from the different models are to their mean value.