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On numerical calculation of probabilities according to Dirichlet distribution
Abstract
The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation
of the probability values according to the underlying multivariate probability distribution. In addition, when we are using
a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives
may also be necessary.
In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative
distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions
we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There
will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of
these formulae is that they involve only lower dimensional cumulative distribution function calculations. Numerical test results
will also be presented.
... Here, we note the similarity of the region the random variables in (29) are defined to a Dirichlet distribution [17]. In [18], the high complexity involved in evaluation of a high dimensional multivariate CDF is addressed in a Dirichlet distribution context. Using the inclusion-exclusion principle [12] with an earlier result on multivariate Dirichlet distribution [19], the authors develop a recursive technique that allows the high dimensional multivariate CDF expression to be represented by marginal CDFs of individual variables. ...
... Using the inclusion-exclusion principle [12] with an earlier result on multivariate Dirichlet distribution [19], the authors develop a recursive technique that allows the high dimensional multivariate CDF expression to be represented by marginal CDFs of individual variables. In the following, we apply the technique described in [18] with some manipulation for the calculation of F zn (t 1 , t 2 , . . . , t n ). ...
... , t n ) for any n with n ∈ {3, . . . , M } in a recursive manner as given in (34) on the next page [17], [18]. Theoretically, one can calculate F zn (t 1 , t 2 , . . . ...
Simultaneous multiuser beamforming in multiantenna downlink channels can
entail dirty paper (DP) precoding (optimal and high complexity) or linear
precoding (suboptimal and low complexity) approaches. The system performance is
typically characterized by the sum capacity with homogenous users with perfect
channel state information at the transmitter. The sum capacity performance
analysis requires the exact probability distributions of the user
signal-to-noise ratios (SNRs) or signal-to-interference plus noise ratios
(SINRs). The standard techniques from order statistics can be sufficient to
obtain the probability distributions of SNRs for DP precoding due to the
removal of known interference at the transmitter. Derivation of such
probability distributions for linear precoding techniques on the other hand is
much more challenging. For example, orthogonal beamforming techniques do not
completely cancel the interference at the user locations, thereby requiring the
analysis with SINRs. In this paper, we derive the joint probability
distributions of the user SINRs for two orthogonal beamforming methods combined
with user scheduling: adaptive orthogonal beamforming and orthogonal linear
beamforming. We obtain compact and unified solutions for the joint probability
distributions of the scheduled users' SINRs. Our analytical results can be
applied for similar algorithms and are verified by computer simulations.
... For instance, if in (1) the random vector is separated, i.e., g(z, ξ) = ξ − h(z), then P (g(z, ξ) ≤ 0) = P (ξ ≤ h(z)) = F ξ (h(z)), (2) where F ξ denotes the (multivariate) distribution function of ξ. We note that for many prominent multivariate distributions (like Gaussian, t-, Gamma, Dirichlet, Exponential, log-normal, truncated normal) there exist methods for calculating the corresponding distribution function that clearly outperform a crude Monte Carlo approach (see, e.g., [7], [19], [18], [8], [13]). When it comes to calculating gradients of such distribution functions in the context of applying some optimization algorithm, then, of course, it would be desirable to carry out this calculus in a similarly efficient way as it was done for the values themselves. ...
... In some special cases it is possible indeed to analytically reduce the calculus of gradients to the calculus of function values of the same distribution. This is true, for instance, for the Dirchlet (see [8], p. 195) and for the Gaussian distribution. We cite here the corresponding result for the Gaussian distribution which will be the starting point for the investigations in this paper. ...
We provide an explicit gradient formula for linear chance constraints under a (possibly singular) multivariate Gaussian distribution. This formula allows one to reduce the calculus of gradients to the calculus of values of the same type of chance constraints (in smaller dimension and with different distribution parameters). This is an important aspect for the numerical solution of stochastic optimization problems because existing efficient codes for e.g., calculating singular Gaussian distributions or regular Gaussian probabilities of polyhedra can be employed to calculate gradients at thesame time. Moreover, the precision of gradients can be controlled by that of function values, which is a great advantage over using finite difference approximations. Finally, higher order derivatives are easily derived explicitly. The use of the obtained formula is illustrated for an example of a stochastic transportation network.
... The key point is that the cumulative distribution function (cdf) of a Dirichlet distribution can be calculated via simple recursions given in Gouda and Szántai (2010). The rest of the proof relies on standard algebraic manipulations and is postponed to Appendix 1. ...
... The last argument comes from Gouda and Szántai (2010) who give explicit recursions to compute the uni-and bivariate cdf for the Dirichlet Dir(a), denoted G q (u; a) and G q, (u, v; a), respectively. Reminding that the approximate variational posterior of α is Dir(a) and using a simple property of the Dirichlet distribution ...
W-graph refers to a general class of random graph models that can be seen as a random graph limit. It is characterized by both its graphon function and its motif frequencies. In this paper, relying on an existing variational Bayes algorithm for the stochastic block models (SBMs) along with the corresponding weights for model averaging, we derive an estimate of the graphon function as an average of SBMs with increasing number of blocks. In the same framework, we derive the variational posterior frequency of any motif. A simulation study and an illustration on a social network complete our work.
... The proof relies on standard algebraic manipulations and is postponed to Appendix A.1. A key point is that the cdf F q, can be calculated via simple recursions given in Gouda and Szántai (2010). ...
... The last argument comes from Gouda and Szántai (2010) who give explicit recursions to compute the uni-and bi-variate cdf for the Dirichlet Dir(a), denoted G q (u; a) and G q, (u, v; a) respectively. Reminding that the approximate variational posterior of α is Dir(a) and using a simple property of the Dirichlet distribution ...
W-graph refers to a general class of random graph models that can be seen as
a random graph limit. It is characterized by both its graphon function and its
motif frequencies. The stochastic block model is a special case of W-graph
where the graphon function is block-wise constant. In this paper, we propose a
variational Bayes approach to estimate the W-graph as an average of stochastic
block models with increasing number of blocks. We derive a variational Bayes
algorithm and the corresponding variational weights for model averaging. In the
same framework, we derive the variational posterior frequency of any motif. A
simulation study and an illustration on a social network complete our work.
... We selected a first-choice PPI on the basis of cost-effectiveness, prioritizing the option with the lowest cost under the assumption of identical clinical effectiveness of PPIs. To estimate potential cost savings, we utilized the Dirichlet distribution, a statistical model that generates vectors of parameters that sum to 1. Thus, the proportion of each PPI was randomly generated between 0 and 1, and the sum of these proportions equaled 1 [24]. Using these proportions of each PPI, potential cost savings were calculated by subtracting the estimated total cost from the current expenditure. ...
Given that Korea lacks measures to promote cost-effective prescribing, this study investigated the efficiency of prescribers’ decision-making when faced with the choice of 398 products containing 15 proton pump inhibitors (PPI).
This study aimed to explore PPI prescribing patterns in ambulatory care over 10 years and elucidate prescribers’ practices. Further analysis was conducted to estimate the achievable potential savings, assuming that the recommendation of first-choice drugs is based on the rational use of medicines.
Retrospective PPI prescribing data from the ambulatory sector pertaining to medical services provided to the entire South Korean population from 2013 to 2022 were extracted from the Korean National Health Information Database; the data were analyzed to identify annual trends in PPI spending and utilization volume according to defined daily doses (DDDs). Unit price per DDD was calculated, and a preferred cost-effective choice with identical efficacy and safety profile was selected among PPIs. The potential savings were then simulated using Dirichlet distribution.
In ambulatory care, PPI drug costs increased by 13.0% per annum over 10 years. The preferred substances were esomeprazole (31.6%), rabeprazole (23.2%), and tegoprazan (16.5%), which are among the most expensive PPIs. However, the most cost-effective substance was dexlansoprazole (South Korean won [KRW] 522.7; standard deviation [SD] = 72.2; and median = 583). If dexlansoprazole is recommended as the first-choice PPI in the Korean context, the estimated cost savings would be 404.24 billion KRW, equivalent to 47.0% of the current PPI spending in ambulatory care.
This study identified opportunities for substantial cost savings related to PPI prescribing. Guided decision-making toward cost-effective PPI prescribing would increase the efficiency of prescribing and the sustainability of the healthcare system.
... In the classical textbook [28], convexity of chance constraints is ensured for a variety of distributions. Differentiability properties including gradient formulae for Gaussian distributions are analyzed in [41, 16] and [40], whereas gradient formulae for other distributions can be found in [31,28,37,15]. † EDF R&D. ...
Joint chance constrained problems give rise to many algorithmic challenges. Even in the convex case, i.e., when an appropriate transformation of the probabilistic constraint is a convex function, its cutting-plane linearization is just an approximation, produced by an oracle providing subgradient and function values that can only be evaluated inexactly. As a result, the cutting-plane model may lie above the true constraint. For dealing with such upper inexact oracles, and still solving the problem up to certain precision, a special numerical algorithm must be put in place. We introduce a family of constrained bundle methods, based on the so-called improvement functions, that is shown to be convergent and encompasses many previous approaches as well as new algorithms. Depending on the oracle accuracy, we analyze to what extent the considered methods solve the joint chance constrained program. The approach is assessed on real-life energy problems, arising when dealing with stochastic hydroreservoir management.
... Hence, the evaluation of an ordered Dirichlet distribution function is usually converted to the problem of evaluating a Dirichlet distribution function. The reader is referred to [14], [15], [13], [21] and references therein for studies on computational issues about Dirichlet distributions. ...
In this paper, we present a theoretical result that applies to convex optimization problems in the presence of an uncertain stochastic parameter. We consider the min-max sample-based solution, i.e. the min-max solution computed over a finite sample of instances of the uncertain stochastic parameter, and the costs incurred by this solution in correspondence of the sampled parameter instances. Our goal is to evaluate the risks associated to the various costs, where the risk associated to a cost is the probability that the cost is exceeded when a new uncertainty instance is seen. The theoretical result proven in this paper is that the risks form a random vector whose probability distribution is always an ordered Dirichlet distribution, irrespective of the probability measure of the uncertain stochastic parameter. This evaluation characterizes completely the risks associated to the costs, and represents a full-fledged result on the reliability of the min-max sample-based solution.
... Not too high dimensional c.d.f. values of the Dirichlet distribution can be calculated by an implicit recursive algorithm developed recently by A. Gouda and T. Szántai (seeGouda and Szántai (2010)).InTables 1-9we give the results of three different 20 dimensional c.d.f. calculations. ...
Let A
1,…,A
n
be arbitrary events. The underlying problem is to give lower and upper bounds on the probability P(A
1∪⋯∪A
n
) based on , 1≤i
1<⋯<i
k
≤n, where k=1,…,d, and d≤n (usually d≪n) is a certain integer, called the order of the problem or the bound. Most bounding techniques fall in one of the following two main categories: those that use (hyper)graph structures and the ones based on binomial moment problems. In this paper we compare bounds from the two categories with each other, in particular the bounds yielded by univariate and multivariate moment problems are compared with Bukszár’s hypermultitree bounds. In the comparison we considered several numerical examples, most of which have important practical applications, e.g., the approximation of the values of multivariate cumulative distribution functions or the calculation of network reliability. We compare the bounds based on how close they are to the real value and the time required to compute them, however, the problems arising in the implementations of the methods as well as the limitations of the usability of the bounds are also illustrated.
Generative Adversarial Networks (GANs) are deep learning models that learn and generate new samples similar to existing ones. Traditionally, GANs are trained in centralized data centers, raising data privacy concerns due to the need for clients to upload their data. To address this, Federated Learning (FL) integrates with GANs, allowing collaborative training without sharing local data. However, this integration is complex because GANs involve two interdependent models-the generator and the discriminator-while FL typically handles a single model over distributed datasets. In this paper, we propose a novel asynchronous FL framework for GANs, called AsyncFedGAN, designed to efficiently and distributively train both models tailored for molecule generation. AsyncFedGAN addresses the challenges of training interactive models, resolves the straggler issue in synchronous FL, reduces model staleness in asynchronous FL, and lowers client energy consumption. Our extensive simulations for molecular discovery show that AsyncFedGAN achieves convergence with proper settings, outperforms baseline methods, and balances model performance with client energy usage.
This work considers clustering nodes of a largely incomplete graph. Under the problem setting, only a small amount of queries about the edges can be made, but the entire graph is not observable. This problem finds applications in large-scale data clustering using limited annotations, community detection under restricted survey resources, and graph topology inference under hidden/removed node interactions. Prior works tackled this problem from various perspectives, e.g., convex programming-based low-rank matrix completion and active query-based clique finding. Nonetheless, many existing methods are designed for estimating the single-cluster membership of the nodes, but nodes may often have mixed (i.e., multi-cluster) membership in practice. Some query and computational paradigms, e.g., the random query patterns and nuclear norm-based optimization advocated in the convex approaches, may give rise to scalability and implementation challenges. This work aims at learning mixed membership of nodes using queried edges. The proposed method is developed together with a systematic query principle that} can be controlled and adjusted by the system designers to accommodate implementation challenges---e.g., to avoid querying edges that are physically hard to acquire. Our framework also features a lightweight and scalable algorithm with membership learning guarantees. Real-data experiments on crowdclustering and community detection are used to showcase the effectiveness of our method
The paper extends the traditional approach to measuring market concentration by embracing an element of stochasticity that should reflect the analyst’s uncertainty associated with the future development regarding concentration on the market. Whereas conventional practice relies on deterministic assessments of a market concentration measure with the use of current market shares, this says nothing about possible changes that may happen even in a near future. The paper proposes to model the analyst’s beliefs by dint of a suitable joint probability distribution for future market shares and demonstrates how this analytic framework may be employed for regulatory purposes. A total of four candidates for the joint probability distribution of market shares are considered—the Dirichlet distribution, the conditional normal distribution, the Gaussian copula with conditional beta marginals and the predictive distribution arising from the market share attraction model—and it is shown how their hyperparameters can be elicited so that a minimum burden is placed on the analyst. The proposed procedure for stochastic sensitivity analysis of concentration measures is demonstrated in a case study oriented on the Slovak banking sector.
We consider convex optimization problems in the presence of stochastic uncertainty. The min-max sample-based solution is the solution obtained by minimizing the max of the cost functions corresponding to a finite sample of the uncertainty parameter. The empirical costs are instead the cost values that the solution incurs for the various parameter realizations that have been sampled. Our goal is to evaluate the risks associated with the empirical costs, where the risk associated with a cost is the probability that the cost is exceeded when a new realization of the uncertainty parameter is seen. This task is accomplished without resorting to uncertainty realizations other than those used in optimization. The theoretical result proved in this paper is that these risks form a random vector whose probability distribution is an ordered Dirichlet distribution, irrespective of the probability measure of the stochastic uncertainty parameter. This result provides a distribution-free characterization of the risks associated with the empirical costs that can be used in a variety of application problems.
We provide an explicit gradient formula for linear chance constraints under a (possibly singular) multivariate Gaussian distribution. This formula allows one to reduce the calculus of gradients to the calculus of values of the same type of chance constraints (in smaller dimension and with different distribution parameters). This is an important aspect for the numerical solution of stochastic optimization problems because existing efficient codes for, e.g., calculating singular Gaussian distributions or regular Gaussian probabilities of polyhedra can be employed to calculate gradients at the same time. Moreover, the precision of gradients can be controlled by that of function values, which is a great advantage over using finite difference approximations. Finally, higher order derivatives are easily derived explicitly. The use of the obtained formula is illustrated for an example of a transportation network with stochastic demands.
The Dirichlet distribution is one of the most important multivariate probability distributions with wide range of applications in various areas of statistics, probabilistic modelling, engineering and geosciences. Despite growing interest in the distribution, little appeared to be known of the Dirichlet distribution in the community of geodesy and geophysics. With these thoughts in mind, the present paper is an application-driven short and simplified introduction to the fundamental issues of the Dirichlet distribution and gives some useful representations of and bounds on the Dirichlet distribution function. A new polynomial representation for the bivariate Dirichlet distribution is established. In addition, we also shed a certain light on the astronomical, geodetic and geophysical background of the Dirichlet integral in historical context. The potential possibility of geodetic and geophysical applications of the Dirichlet distribution is briefly described within the framework of recent developments and trends of statistical science and applied probability.
In this paper, we deal with a cascaded reservoir optimization problem with
uncertainty on inflows in a joint chance constrained programming setting. In particular,
we will consider inflows with a persistency effect, following a causal time series
model with Gaussian innovations. We present an iterative algorithm for solving
similarly structured joint chance constrained programming problems that requires a
Slater point and the computation of gradients. Several alternatives to the joint chance
constraint problem are presented. In particular, we present an individual chance constraint
problem and a robust model. We illustrate the interest of joint chance constrained
programming by comparing results obtained on a realistic hydro valley with
those obtained from the alternative models. Despite the fact that the alternative models
often require less hypothesis on the law of the inflows, we show that they yield
conservative and costly solutions. The simpler models, such as the individual chance
constraint one, are shown to yield insufficient robustness and are therefore not useful.
We therefore conclude that Joint Chance Constrained programming appears as an approach
offering a good trade-off between cost and robustness and can be tractable for
complex realistic models.
The concept of (h, m) -hypermultitree (which is a special hypergraph) is introduced to present Bonferroni-type inequalities which are equalities for some families of sets. The main theorem is a common generalization of the earlier results of Hunter, Worsley, Tomescu and recent results of Prékopa and the author. The new bounds are based on significantly fewer probabilities than the same order Bonferroni bounds. This and some other properties explain the very high efficiency of the new bounds in many applications, e.g., estimate the values of multivariate normal distribution functions, which is demonstrated in the end of the paper.
The models discussed in the present paper are generalizations of the models introduced previously by A. Prékopa [6] and M.
Ziermann [13]. In the mentioned papers the initial stock level of one basic commodity is determined provided that the delivery
and demand process allow certain homogeneity (in time) assumptions if they are random. Here we are dealing with more than
one basic commodity and drop the time homogeneity assumption. Only the delivery processes will be assumed to be random. They
will be supposed to be stochastically independent. The first model discussed in this paper was already introduced in [9].
All these models are stochastic programming models and algorithms are used to determine the initial stock levels rather than
simple formulas. We have to solve nonlinear programming problems where one of the constraints is probabilistic. The function
and gradient values of the corresponding constraining function are determined by simulation. A numerical example is detailed.
Formulae are derived for the characteristic function of the inverted Dirichlet distribution and hence the multivariate F. The analysis involves a new function with multiple arguments that extends the confluent hypergeometric function of the second kind. This function and its properties are studied in the paper and a simple integral representation is given which is useful for numerical work. A special case connected with the multivariate t distribution is also explored.
This paper compares acceptance-rejection sampling and methods of De'ak, Genz and Schervish for the numerical computation of multivariate normal probabilities. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz (1992). The methods allow moderately accurate multivariate normal probabilities to be quickly computed for problems with a many as ten variables. 1 Introduction A common problem in many statistics computations is computing the multivariate normal distribution function P (b) = 1 p jSigmaj(2) m Z b1 Gamma1 Z b2 Gamma1 ::: Z bm Gamma1 e Gamma 1 2 x t Sigma Gamma1 x dx; where x = (x 1 ; x 2 ; :::; xm ) t , Sigma is an m Theta m symmetric positive definite covariance matrix and b i ! 1 for all i. There is reliable and efficient software available for computing P (b) for m = 1 and m = 2 (for m = 2 see Donnelly, 1973 and Drezner and Wesolowsky, 1990), so assume m ? 2. Perhaps the simpl...
The numerical computation of a multivariate normal probability is often a difficult problem. This article describes a transformation that simplifies the problem and places it into a form that allows efficient calculation using standard numerical multiple integration algorithms. Test results are presented that compare implementations of two algorithms that use the transformation, with currently available software. KEY WORDS: multivariate normal distribution, Monte-Carlo, adaptive integration. 1 Introduction A problem that arises in many statistics applications is that of computing the multivariate normal distribution function F (a; b) = 1 p jSigmaj(2) m Z b 1 a 1 Z b 2 a 2 ::: Z bm am e Gamma 1 2 ` t Sigma Gamma1 ` d`; where ` = (`1 ; `2 ; :::; `m) t and Sigma is an m Theta m symmetric positive definite covariance matrix. If, for some i, a i is Gamma1 and b i is 1, an appropriate transformation allows the ith variable to be integrated explicitly and reduc...
Stochastic programming - the science that provides us with tools to design and control stochastic systems with the aid of mathematical programming techniques - lies at the intersection of statistics and mathematical programming. The book Stochastic Programming is a comprehensive introduction to the field and its basic mathematical tools. While the mathematics is of a high level, the developed models offer powerful applications, as revealed by the large number of examples presented. The material ranges form basic linear programming to algorithmic solutions of sophisticated systems problems and applications in water resources and power systems, shipbuilding, inventory control, etc.
Audience: Students and researchers who need to solve practical and theoretical problems in operations research, mathematics, statistics, engineering, economics, insurance, finance, biology and environmental protection.
In this article, we present a concise review of developments on various continuous multivariate distributions. We first present some basic definitions and notations. Then, we present several important continuous multivariate distributions and list their significant properties and characteristics.
In this paper we will deal with the probabilistic constrained stochastic programming problems of the form P\left( {\begin{array}{*{20}{c}}
{{d_{11}}{x_1}}& + & \ldots & + &{{d_{1n}}{x_n}}& \geqslant &{{\beta _1}} \\
\vdots &{}&{}&{}& \vdots &{}& \vdots \\
{{d_{s1}}{x_1}}& + & \ldots & + &{{d_{sn}}{x_n}}& \geqslant &{{\beta _s}}
\end{array}} \right) \geqslant p
\begin{array}{*{20}{c}}
{{a_{11}}{x_1}}& + & \ldots & + &{{a_{1n}}{x_n}}& = &{{b_1}} \\
\vdots &{}&{}&{}& \vdots &{}& \vdots \\
{{a_{m1}}{x_1}}& + & \ldots & + &{{a_{mn}}{x_n}}& = &{{b_m}}
\end{array}
\begin{array}{*{20}{c}}
{{x_1} \geqslant 0}&,& \ldots &,&{{x_n} \geqslant 0}
\end{array}
where β1,...βs have some of joint probability distribution with expected values
variances
and correlation matrix
R = \left( {\begin{array}{*{20}{c}}
1&{{r_{12}}}& \ldots &{{r_{1s}}} \\
{{r_{12}}}&1& \ldots &{{r_{2s}}} \\
\vdots & \vdots & \ddots & \vdots \\
{{r_{1s}}}&{{r_{2s}}}& \ldots &1
\end{array}} \right).
Distributions having beta conditional distributions arise in connection with the generation of distributions for random proportions which do not necessarily possess neutrality properties (see [3]). The family of bivariate distributions for which both conditional distributions are beta is evaluated, and some multivariate extensions are briefly discussed. The bivariate Dirichlet distribution is characterized by the assumptions that both conditional distributions and one marginal distribution are beta.
IntroductionHistorical RemarksMultivariate Generalization of Pearson SystemSeries Expansions and Multivariate Central Limit TheoremsTranslation SystemsMultivariate Linear Exponential-Type DistributionsSarmanov's DistributionsMultivariate Linnik's DistributionsMultivariate Kagan's DistributionsGeneration of Multivariate Nonnormal Random VariablesFréchet BoundsFréchet, Plackett and Mardia's SystemsFarlie-Gumbel-Morgenstern DistributionsMultivariate Phase-Type DistributionsChebyshev-Type and Bonferroni InequalitiesSingular DistributionsDistributions with Almost-Lack of MemoryDistributions with Specified ConditionalsDistributions with Given MarginalsMeasures of Multivariate Skewness and KurtosisBibliography
An efficient Monte Carlo simulation algorithm is developed for estimating the probability content of rectangular domains in the multinormal probability space. The algorithm makes use of the properties of the multinormal distribution, as well as the concept of importance sampling. Accurate estimates of the probability are obtained with a relatively small number of simulations, regardless of its magnitude. The algorithm also allows easy computation of the sensitivities of the probability with respect to distribution parameters or the boundaries of the domain. Application of the algorithm to structural system reliability is demonstrated through a simple example.
In this paper will be described and compared several simulation algorithms for computing the cumulative distribution function values of Dirichlet distributions. New sampling techniques as Sequential Conditioned Sampling (SCS) and Sequential Conditioned Importance Sampling (SCIS) will be introduced. On the base of an interesting property of the Dirichlet distribution new versions of the SCS and SCIS algorithms will be developed, called SCSA, SCSB and SCISA, SCISB, respectively. SCIS and the modified algorithms need more CPU time but they give significant variance reduction. Their resultant efficiency will be compared to the simple “hit-or-miss Monte Carlo” simulation method with conventional sampling of the Dirichlet distributed random vectors what we call Crude Monte Carlo (CMC) simulation method. Numerical test results will also be presented.
This article describes a reduction formula for the computation of multivariate normal probabilities first developed by Plackett. The method is recursive and, despite very promising results, seems not to have been implemented before. Extensive numerical results are given to compare the effectiveness of the approach to other methods from the literature.
The second order probability technique is a method where an incompletely specified (finite-argument) probability function is modeled as a random vector over the associated natural simplex of all possible probability functions of a fixed number of common arguments. The application of this technique in a general context using the Dirichlet family generalization of the uniform distribution over a simplex, in updating an event upon a conditional probability statement when no such disjointness as in the Judy Benjamin (JB) problem holds and sought with a nonuniform prior, is analyzed.
In this paper we obtain a multivariate generalization of the inverted beta distribution. Properties of this distribution and its connection with the multivariate Student-t distribution are discussed. Two asymptotic formulae for approximating the related probability integral are developed and illustrated with numerical examples. Application of this distribution to a problem in Bayesian inference is given.
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This article describes and compares several numerical methods for finding multivariate probabilities over a rectangle. A large computational study shows how the computation times depend on the problem dimensions, the correlation structure, the magnitude of the sought probability, and the required accuracy. No method is uniformly best for all problems and—unlike previous work—this article gives some guidelines to help establish the most promising method a priori. Numerical tests were conducted on approximately 3,000 problems generated randomly in up to 20 dimensions. Our findings indicate that direct integration methods give acceptable results for up to 12-dimensional problems, provided that the probability mass of the rectangle is not too large (less than about 0.9). For problems with small probabilities (less than 0.3) a crude Monte Carlo method gives reasonable results quickly, while bounding procedures perform best on problems with large probabilities (> 0.9). For larger problems numerical integration with quasirandom Korobov points may be considered, as may a decomposition method due to Deák. The best method found four-digit accurate probabilities for every 20-dimensional problem in less than six minutes on a 533MHz Pentium III computer.
A numerically feasible algorithm is proposed for maximum likelihood estimation of the parameters of the Dirichlet distribution. The performance of the proposed method is compared with the method of moments using bias ratio and squared errors by Monte Carlo simulation. For these criteria, it is found that even in small samples maximum likelihood estimation has advantages over the method of moments.
The purpose of this paper is to extend results obtained by Sahai and Anderson (1973) and Hurlburt and Spiegel (1976) computationally as well as theoretically. We apply results given in Yassaee (1974), (1976) and apply our computer program to show that one can evaluate the probability of type one error for conditional tests in linear model, in general. We show that one can compute the probability integral of Dirichlet distribution by the use of our program which computes the probability integral of inverted Dirichlet distribution.
In this paper the important role of the Dibichlet distribution for the Pert problem is motivated. With the help of an inequality for the Dibichlet distribution, the upper bound of the completion time will be given at prescribed probability level.
In this paper we derive a k-variate inverted Dirichlet distribution, D ('v1,v2, ···, vk; vk+1) by a theorem related to k+1 independent random variables having gamma distributions with different parameters v1,v2, ···, vk+1 but with the same scale parameter, say 1. A number of results for a k-variate inverted Dirichlet distribution are proved and it is shown that one may deduce a k-dimensional multivariate logistic distribution from a k-variate inverted Dirichlet distribution.Dans cet article, nous obtenons, par un théorème concernant k+1 variables aléatiores à distributions gamma, une distribution de Districhlet inversée à K variables. Un certain nombre de résultats pour cette distribution sont démontrés et il est établi que l'on peut obtenir une distribution logistique é plusieurs variables à k dimension à partir d'une distribution de Districhlet inversée à k variables.
In this paper we describe two different methods for the calculation of the bivariate gamma probability distribution function.
One of them is based on a direct numerical integration and the other on a series expansion in terms of Laguerre polynomials.
In the multivariate case we propose a Monte Carlo method. Our method can be used for other types of multivariate probability
distributions too. In the special case of the multivariate normal distribution the computer experiments show that our method
has the same efficiency as other known methods. In the last paragraph we briefly describe the possible applications of the
proposed algorithms in stochastic programming.
An m-dimensional multinormal integral can be approximated as a product of m univariate normal integrals that are easy to compute. The paper compares the accuracy of this simple approximation against the benchmark results obtained from an efficient simulation method that utilizes the importance of sampling principle. Examples considered in the paper illustrate that the accuracy of proposed approximation is comparable to that of the simulation method.
In structural system reliability theory, the evaluation of multivariate normal distributions is an important problem. Numerical integration of multinormal distributions with high accuracy and efficiency is known to be impractical when the number of distribution dimensions is large, typically greater than five. The paper presents a practical and effective approach to approximate a multinormal integral by a product of one-dimensional normal integrals, which are easy to evaluate. Examples considered in the paper illustrate a remarkable accuracy of the approximation in comparison with exact integration. Unlike a first-order multinormal approximation widely used in the literature, this method does not involve any iterative linearization, minimization or integration. Computational simplicity with high accuracy is the major advantage of the proposed method, which also highlights its potential for estimating reliability of structural systems.
Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function value are given in the paper. The author's variance reduction technique was based on the Bonferroni bounds involving the first two binomial moments only. The new variance reduction technique is adapted to the most refined new bounds developed in the last decade for the estimation the probability of union respectively intersection of events. Numerical test results prove the efficiency of the simulation procedures described in the paper.
Algorithms for the computation of bivariate and trivariate normal and t probabilities for rectangles are reviewed. The algorithms use numerical integration to approximate transformed probability distribution integrals. A generalization of Plackett's formula is derived for bivariate and trivariate t probabilities. New methods are described for the numerical computation of bivariate and trivariate t probabilities. Test results are provided, along with recommendations for the most efficient algorithms for single and double precision computations.
Various multiple comparison procedures involve the evaluation of multivariate normal and t integrals with non-decomposable correlation matrices. While exact methods exist for their computations, it is sometimes necessary to consider simpler and faster approximations. We consider approximations based on approximations to the correlation matrix (methods which provide no error control) as well as inequality based methods (where, by definition, the sign of the error is known). Comparisons of different methods, to assess accuracy, are given for particular multiple comparison problems which require high-dimensional integrations.
cepts of sparseness and crowdedness are introduced for these b blue cells based on a fixed number n of observations. The (Type 1) Dirichlet distribution is used to evaluate the probability laws, the cumulative distribution functions (cdf's), the moments, the joint probability law and the joint moments of the number S of sparse blue cells and the number C of crowded blue cells. The results are put in the form of moment generating functions. Applications of some of these results are also considered in Sections 7 and 8. 2. The distribution of S. A sparse blue cell is one with at most u observations in it. A crowded blue cell is one with at least v observations in it. Let S,un)= S denote the random number of sparse blue cells when there are b blue cells with common probability p, n observations, and u defines sparseness; similarly, let Czvo7) = C denote the random number of crowded blue cells with v defining crowdedness. We use the symbolism max (j, n) v (for integers v) denotes the event that the minimum frequency (based on n observations) in a specified set of j blue cells is at least v. It has already been noted elsewhere (cf. e.g., [2] and [4]) that
Let be a random vector with all and . Let , and suppose that none of the , nor vanishes almost surely. Without any further regularity assumptions, each of two conditions is shown to be necessary and sufficient for X to be distributed according to a Dirichlet distribution or a limit of such distributions. Either condition requires that certain proportions between components of X be independent of one or more other components of X.
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, estimation techniques that have good properties when the error terms are independently and normally distributed. Observations on cumulative proportions are clearly neither independent nor 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. Five Lorenz-curve specifications are used to demonstrate the technique. 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.
Rectangle probabilities of the trivariate normal distribution (Working Paper). School of Business Administration
- H Gassmann
Gassmann, H. (2000). Rectangle probabilities of the trivariate normal distribution (Working Paper). School
of Business Administration, Dalhousie University, Halifax, Canada.
Probabilistic constrained programming and distributions with given marginals Distributions with given marginals and moment problems, proceedings of the 3rd conference on distributions with given marginals and moment problems (pp. 205–210)
- T Szántai
Distributions with given marginals and moment problems, proceedings of the 3rd conference on distributions with given marginals and moment problems
- T Szántai
Szántai, T. (1997). Probabilistic constrained programming and distributions with given marginals. In V. Benes
& J. Stepan (Eds.), Distributions with given marginals and moment problems, proceedings of the 3rd
conference on distributions with given marginals and moment problems (pp. 205-210). Czech Agricultural University, Prague, Czech Republic, 2-6 September 1996. Dordrecht: Kluwer Academic.
The inverted Dirichlet distribution with applications
- G G Tiao
- I Guttman
- G. G. Tiao
On integrals of Dirichlet distribution and their applications (Preprint)
- H Yassaee