On a class of distributions on the simplex

Favaro, Stefano; Hadjicharalambous, Georgia; Prünster, Igor

doi:10.1016/j.jspi.2011.03.015

Cited by 27 publications

(38 citation statements)

References 32 publications

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“…, r k are non-negative integers such that r = k i=1 r i . This result extends to mixed moments the one in [10,Proposition 2] and is analogous to the one obtained in [16, Appendix A] for normalized Lévy processes. Notice that, (i) passing the expectation through the integral (second equality) is justified by Fubini's theorem; (ii) passing the derivatives through the integrals (fifth equality) is justified by the fact that x r e −ux is continuous and bounded.…”

Section: Normalized Infinitely Divisible Distributionssupporting

confidence: 83%

“…, M, with W = X 1 + · · · + X M . In [10] moments of an arbitrary order are obtained for an NID variable P i ; here we extend those results to mixed moments and to the posterior distribution of P i given the vector of counts n. Similar results are presented in [16] for the normalization of Lévy processes. By exploiting the equality…”

Section: Normalized Infinitely Divisible Distributionssupporting

confidence: 70%

“…The aim of this section is to introduce the class of Normalized Infinitely Divisible (NID) distributions and review some general properties that allow to characterize them [10]. An NID distribution is obtained by normalizing a collection of Infinitely Divisible (ID) positive random variables X 1 , .…”

Section: The Class Of Normalized Infinitely Divisible Distributionsmentioning

confidence: 99%

“…For our purpose, an interesting class is that of the Infinitely Divisible (ID) distributions [10]. There are many examples of such distributions, including the inverse Gaussian distribution, the right-skewed stable distributions, etc.…”

Section: Introductionmentioning

confidence: 99%

“…A counter-example is, for instance, the uniform distribution. The class of distributions that are obtained through a normalization of positive ID distributed variables, which is known as the class of Normalized Infinitely Divisible (NID) distributions, has been studied by Favaro et al in [10], and is described in details in Section 2. Favaro et al derive general expressions for the moments of any order of an NID distribution (similar to those presented in Section 2.2 of this paper for the mixed moments) and focus on special cases of NID that lead to explicit expressions of the PDF.…”

Section: Introductionmentioning

confidence: 99%

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New prior near-ignorance models on the simplex

Mangili

Benavoli

2015

International Journal of Approximate Reasoning

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The aim of this paper is to derive new near-ignorance models on the probability simplex, which do not directly involve the Dirichlet distribution and, thus, are alternative to the Imprecise Dirichlet Model (IDM). We focus our investigation on a particular class of distributions on the simplex which is known as the class of Normalized Infinitely Divisible (NID) distributions; it includes the Dirichlet distribution as a particular case. For this class it is possible to derive general formulae for prior and posterior predictive inferences, by exploiting the Lévy-Khintchine representation theorem. This allows us to generally characterize the near-ignorance properties of the NID class. After deriving these general properties, we focus our attention on three members of this class. We will show that one of these near-ignorance models satisfies the representation invariance principle and, for a given value of the prior strength, always provides inferences that encompass those of the IDM. The other two models do not satisfy this principle, but their imprecision depends linearly or almost linearly on the number of observed categories; we argue that this is sometimes a desirable property for a predictive model.

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Section: Normalized Infinitely Divisible Distributionssupporting

confidence: 83%

Section: Normalized Infinitely Divisible Distributionssupporting

confidence: 70%

Section: The Class Of Normalized Infinitely Divisible Distributionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New prior near-ignorance models on the simplex

Mangili

Benavoli

2015

International Journal of Approximate Reasoning

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The Double Flexible Dirichlet: A Structured Mixture Model for Compositional Data

Ascari¹,

Migliorati²,

Ongaro³

2021

Applied Modeling Techniques and Data Analysis 2

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Shrinkage priors for isotonic probability vectors and binary data modeling, with applications to dose–response modeling

Boonstra,

Owen,

Kang

2024

Pharmaceutical Statistics

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Motivated by the need to model dose–response or dose‐toxicity curves in clinical trials, we develop a new horseshoe‐based prior for Bayesian isotonic regression modeling a binary outcome against an ordered categorical predictor, where the probability of the outcome is assumed to be monotonically non‐decreasing with the predictor. The set of differences between outcome probabilities in consecutive categories of the predictor is equipped with a multivariate prior having support over simplex. The Dirichlet distribution, which can be derived from a normalized sum of independent gamma‐distributed random variables, is a natural choice of prior, but using mathematical and simulation‐based arguments, we show that the resulting posterior is prone to underflow and other numerical instabilities, even under simple data configurations. We propose an alternative prior based on horseshoe‐type shrinkage that is numerically more stable. We show that this horseshoe‐based prior is not subject to the numerical instability seen in the Dirichlet/gamma‐based prior and that the horseshoe‐based posterior can estimate the underlying true curve more efficiently than the Dirichlet‐based one. We demonstrate the use of this prior in a model predicting the occurrence of radiation‐induced lung toxicity in lung cancer patients as a function of dose delivered to normal lung tissue. Our methodology is implemented in the R package isotonicBayes and therefore suitable for use in the design of dose‐finding studies or other dose–response modeling contexts.

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On a class of distributions on the simplex

Cited by 27 publications

References 32 publications

New prior near-ignorance models on the simplex

New prior near-ignorance models on the simplex

The Double Flexible Dirichlet: A Structured Mixture Model for Compositional Data

Shrinkage priors for isotonic probability vectors and binary data modeling, with applications to dose–response modeling

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