A saddlepoint approximation to the distribution of the sum of independent non‐identically beta random variables

Nadarajah, Saralees; Jiang, Xiao; Chu, Jeffrey

doi:10.1111/stan.12051

“…From the equation listed above, the approximation of prevalence involves the sum of multiple variables from different beta distributions, since * follows a beta distribution. Empirically, we use a normal distribution to approximate the distribution of the sum of beta variables 35 . Therefore, the posterior of the prevalence would follows a chi-square distribution with the degree of freedom being 1 and a non-central parameter.…”

Section: Posterior Estimation Of Disease Prevalencementioning

confidence: 99%

Estimating prevalence for limb-girdle muscular dystrophy based on public sequencing databases

Liu

¹

,

Pajusalu

²

,

Lake

³

et al. 2019

Genetics in Medicine

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Purpose: Limb Girdle Muscular Dystrophies (LGMD) are a genetically heterogeneous category of autosomal inherited muscle diseases. Many genes causing LGMD have been identified, and clinical trials are beginning for treatment of some genetic subtypes. However, even with the gene-level mechanisms known, it is still difficult to get a reliable and generalizable prevalence estimation for each subtype due to the limited amount of epidemiology data and the low incidence of LGMDs. Methods: Taking advantage of recently published whole exome and genome sequencing data from the general population, we used a Bayesian method to develop a reliable disease prevalence estimator. Results: This method was applied to nine recessive LGMD subtypes. The estimated disease prevalence calculated by this method were largely comparable to published estimates from epidemiological studies, however highlighted instances of possible under-diagnosis for LGMD2B and 2L. Conclusion: The increasing size of aggregated population variant databases will allow for robust and reproducible prevalence estimates of recessive disease, which is critical for the strategic design and prioritization of clinical trials..

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“…From the equation listed above, the approximation of prevalence involves the sum of multiple variables from different beta distributions, since * follows a beta distribution. Empirically, we use a normal distribution to approximate the distribution of the sum of beta variables 35 Therefore, the posterior of the prevalence would follows a chi-square distribution with the degree of freedom being 1 and a non-central parameter. Using to denote the approximation term for disease prevalence ( * ) @ , we can get…”

Section: Posterior Estimation Of Disease Prevalencementioning

confidence: 99%

Estimating prevalence for limb-girdle muscular dystrophy based on public sequencing databases

Liu

¹

,

Pajusalu

²

,

Lake

³

et al. 2018

Preprint

View full text Add to dashboard Cite

Purpose: Limb Girdle Muscular Dystrophies (LGMD) are a genetically heterogeneous category of autosomal inherited muscle diseases. Many genes causing LGMD have been identified, and clinical trials are beginning for treatment of some genetic subtypes. However, even with the gene-level mechanisms known, it is still difficult to get a reliable and generalizable prevalence estimation for each subtype due to the limited amount of epidemiology data and the low incidence of LGMDs. Methods: Taking advantage of recently published whole exome and genome sequencing data from the general population, we used a Bayesian method to develop a reliable disease prevalence estimator. Results: This method was applied to nine recessive LGMD subtypes. The estimated disease prevalence calculated by this method were largely comparable to published estimates from epidemiological studies, however highlighted instances of possible under-diagnosis for LGMD2B and 2L. Conclusion: The increasing size of aggregated population variant databases will allow for robust and reproducible prevalence estimates of recessive disease, which is critical for the strategic design and prioritization of clinical trials.

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“…( 7) or ( 8), then the pdf of K = N i K i can be determined from statistical theory. No such distribution for beta-distributed variates exists for N > 2 [17]; however, for the gamma distribution, this is pdf…”

mentioning

confidence: 99%

Fluctuations in Hertz chains at equilibrium

Przedborski

¹

,

Sen

²

,

Harroun

³

2017

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We examine the long-term behaviour of non-integrable, energy-conserved, 1D systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. Existing dynamical studies showed the absence of energy equipartitioning in such systems, hence their long-term dynamics was described as quasi-equilibrium. Here we show that these systems do in fact reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. As a byproduct, we show how fluctuations of system quantities, and thus the distribution functions, are influenced by the Hertz potential. In particular, the variance of the system's kinetic energy probability density function is reduced by a factor related to the contact potential.Recently, there has been broad interest in 1D systems of macroscopic grains held between stationary walls and interacting via a power-law contact potential [1][2][3][4][5][6]. A long-standing open problem is whether thermalization (equipartition) can occur in these chains of grains. Only very recently has it been shown that the related FPU chain of coupled oscillators does reach equilibrium after very long times [7]. In this paper, we show this is also true for so-called Hertz chains. In the process, we obtain wholly new approximate distribution functions for interacting particles in the microcanonical ensemble.Many power-law interacting systems are notable for supporting solitary wave (SW) propagation [3,6,8]. However, in response to singular perturbations, the breakup of SWs at the walls and from gaps between grains leads the system after a long time to an equilibrium-like, ergodic phase [2][3][4]. Unusually large [2-4] and occasionally persistent (rogue) [9] fluctuations in the system's kinetic energy are seen at late times for sufficiently strong and unique perturbations. This has been seen to impede an equal sharing of energy among all the grains in the system, hence the long-term dynamics of 1D systems of interacting grains has been described as quasi-equilibrium (QEQ) [2][3][4]. The question of whether QEQ is the final state for these systems is addressed in this letter.To the time scales previously studied, quasiequilibrium has been seen to be a general feature of the dynamics of systems with no sound propagation [3]. However, we find that at sufficiently late times, kinetic energy fluctuations relax, allowing for energy to be shared equally among all grains. Of course, energy equipartitioning happens only in an average sense in finite systems, and at any given instant each grain will not have exactly the same kinetic energy. Rather, each grain's kinetic energy fluctuates according to the same probability density function (pdf), the long tail of which determines the chance of large fluctuations.The fluctuations are quantified by treating the chain as a 1D gas of interacting spheres [10]. This requires new velocity and kinetic energy distribution functions different from hard spheres, which incorporate the interaction potential. These dis...

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A saddlepoint approximation to the distribution of the sum of independent non‐identically beta random variables

Cited by 8 publications

References 8 publications

Estimating prevalence for limb-girdle muscular dystrophy based on public sequencing databases

Estimating prevalence for limb-girdle muscular dystrophy based on public sequencing databases

Estimating prevalence for limb-girdle muscular dystrophy based on public sequencing databases

Fluctuations in Hertz chains at equilibrium

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