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Edwin L Turner - One of the best experts on this subject based on the ideXlab platform.
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the Posterior Distribution of sin i values for exoplanets with mt sin i determined from radial velocity data
The Astrophysical Journal, 2011Co-Authors: Edwin L Turner, Shirley HoAbstract:Radial velocity (RV) observations of an exoplanet system giving a value of MT sin(i) condition (i.e., give information about) not only the planet's true mass MT but also the value of sin(i) for that system (where i is the orbital inclination angle). Thus, the value of sin(i) for a system with any particular observed value of MT sin(i) cannot be assumed to be drawn randomly from a Distribution corresponding to an isotropic i Distribution, i.e., the presumptive prior Distribution. Rather, the Posterior Distribution from which it is drawn depends on the intrinsic Distribution of MT for the exoplanet population being studied. We give a simple Bayesian derivation of this relationship and apply it to several "toy models" for the intrinsic Distribution of MT , on which we have significant information from available RV data in some mass ranges but little or none in others. The results show that the effect can be an important one. For example, even for simple power-law Distributions of MT , the median value of sin(i) in an observed RV sample can vary between 0.860 and 0.023 (as compared to the 0.866 value for an isotropic i Distribution) for indices of the power law in the range between –2 and +1, respectively. Over the same range of indices, the 95% confidence interval on MT varies from 1.0001-2.405 (α = –2) to 1.13-94.34 (α = +2) times larger than MT sin(i) due to sin(i) uncertainty alone. More complex, but still simple and plausible, Distributions of MT yield more complicated and somewhat unintuitive Posterior sin(i) Distributions. In particular, if the MT Distribution contains any characteristic mass scale Mc , the Posterior sin(i) Distribution will depend on the ratio of MT sin(i) to Mc , often in a non-trivial way. Our qualitative conclusion is that RV studies of exoplanets, both individual objects and statistical samples, should regard the sin(i) factor as more than a "numerical constant of order unity" with simple and well-understood statistical properties. We argue that reports of MT sin(i) determinations should be accompanied by a statement of the corresponding confidence bounds on MT at, say, the 95% level based on an explicitly stated assumed form of the true MT Distribution in order to reflect more accurately the mass uncertainties associated with RV studies.
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the Posterior Distribution of sin i for exoplanets with m_t sin i determined from radial velocity data
arXiv: Earth and Planetary Astrophysics, 2010Co-Authors: Edwin L TurnerAbstract:Radial velocity (RV) observations of an exoplanet system giving a value of M_T sin(i) condition (ie. give information about) not only the planet's true mass M_T but also the value of sin(i) for that system (where i is the orbital inclination angle). Thus the value of sin(i) for a system with any particular observed value of M_T sin(i) cannot be assumed to be drawn randomly from a Distribution corresponding to an isotropic i Distribution, i.e. the presumptive prior Distribution . Rather, the Posterior Distribution from which it is drawn depends on the intrinsic Distribution of M_T for the exoplanet population being studied. We give a simple Bayesian derivation of this relationship and apply it to several "toy models" for the (currently unknown) intrinsic Distribution of M_T. The results show that the effect can be an important one. For example, even for simple power-law Distributions of M_T, the median value of sin(i) in an observed RV sample can vary between 0.860 and 0.023 (as compared to the 0.866 value for an isotropic i Distribution) for indices (alpha) of the power-law in the range between -2 and +1, respectively. Over the same range of indicies, the 95% confidence interval on M_T varies from 1.002-4.566 (alpha = -2) to 1.13-94.34 (alpha = +1) times larger than M_T sin(i) due to sin(i) uncertainty alone. Our qualitative conclusion is that RV studies of exoplanets, both individual objects and statistical samples, should regard the sin(i) factor as more than a "numerical constant of order unity" with simple and well understood statistical properties. We argue that reports of M_T sin(i) determinations should be accompanied by a statement of the corresponding confidence bounds on M_T at, say, the 95% level based on an explicitly stated assumed form of the true M_T Distribution in order to more accurately reflect the mass uncertainties associated with RV studies.
Qi-man Shao - One of the best experts on this subject based on the ideXlab platform.
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propriety of the Posterior Distribution and existence of the mle for regression models with covariates missing at random
Journal of the American Statistical Association, 2004Co-Authors: Ming-hui Chen, Qi-man Shao, Joseph G IbrahimAbstract:Characterizing model identifiability in the presence of missing covariate data is a very important issue in missing data problems. In this article, we characterize the propriety of the Posterior Distribution of the regression coefficients for some general classes of regression models, including the class of generalized linear models (GLM's) and parametric survival models with right-censored data. Toward this goal, we derive some very general and easy-to-check conditions for the matrix of covariates. We also derive sufficient conditions for the existence of the maximum likelihood estimates and establish novel results for checking propriety of the Posterior when the sample size is large. Several theorems are given to establish propriety of the Posterior and the existence of the maximum likelihood estimator. The conditions reduce to solving a system of linear equations, which can be carried out using software such as MAPLE, IMSL, or SAS. We assume that the missing covariates are missing at random and assume ...
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Propriety of Posterior Distribution for dichotomous quantal response models
Proceedings of the American Mathematical Society, 2000Co-Authors: Ming-hui Chen, Qi-man ShaoAbstract:In this article, we investigate the property of Posterior Distribution for dichotomous quantal response models using a uniform prior Distribution on the regression parameters. Sufficient and necessary conditions for the propriety of the Posterior Distribution with a general link function are established. In addition, the sufficient conditions for the existence of the Posterior moments and the Posterior moment generating function are also obtained. Finally, the relationship between the propriety of Posterior Distribution and the existence of the maximum likelihood estimate is examined.
Lodewyk F A Wessels - One of the best experts on this subject based on the ideXlab platform.
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approximating multivariate Posterior Distribution functions from monte carlo samples for sequential bayesian inference
PLOS ONE, 2020Co-Authors: Bram Thijssen, Lodewyk F A WesselsAbstract:An important feature of Bayesian statistics is the opportunity to do sequential inference: the Posterior Distribution obtained after seeing a dataset can be used as prior for a second inference. However, when Monte Carlo sampling methods are used for inference, we only have a set of samples from the Posterior Distribution. To do sequential inference, we then either have to evaluate the second Posterior at only these locations and reweight the samples accordingly, or we can estimate a functional description of the Posterior probability Distribution from the samples and use that as prior for the second inference. Here, we investigated to what extent we can obtain an accurate joint Posterior from two datasets if the inference is done sequentially rather than jointly, under the condition that each inference step is done using Monte Carlo sampling. To test this, we evaluated the accuracy of kernel density estimates, Gaussian mixtures, mixtures of factor analyzers, vine copulas and Gaussian processes in approximating Posterior Distributions, and then tested whether these approximations can be used in sequential inference. In low dimensionality, Gaussian processes are more accurate, whereas in higher dimensionality Gaussian mixtures, mixtures of factor analyzers or vine copulas perform better. In our test cases of sequential inference, using Posterior approximations gives more accurate results than direct sample reweighting, but joint inference is still preferable over sequential inference whenever possible. Since the performance is case-specific, we provide an R package mvdens with a unified interface for the density approximation methods.
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approximating multivariate Posterior Distribution functions from monte carlo samples for sequential bayesian inference
arXiv: Computation, 2017Co-Authors: Bram Thijssen, Lodewyk F A WesselsAbstract:An important feature of Bayesian statistics is the possibility to do sequential inference: the Posterior Distribution obtained after seeing a first dataset can be used as prior for a second inference. However, when Monte Carlo sampling methods are used for the inference, we only have one set of samples from the Posterior Distribution, which is typically insufficient for accurate sequential inference. In order to do sequential inference in this case, it is necessary to estimate a functional description of the Posterior probability from the Monte Carlo samples. Here, we explore whether it is feasible to perform sequential inference based on Monte Carlo samples, in a multivariate context. To approximate the Posterior Distribution, we can use either the apparent density based on the sample positions (density estimation) or the relative Posterior probability of the samples (regression). Specifically, we evaluate the accuracy of kernel density estimation, Gaussian mixtures, vine copulas and Gaussian process regression; and we test whether they can be used for sequential Bayesian inference. Additionally, both the density estimation and the regression methods can be used to obtain a post-hoc estimate of the marginal likelihood. In low dimensionality, Gaussian processes are most accurate, whereas in higher dimensionality Gaussian mixtures or vine copulas perform better. We show that sequential inference can be computationally more efficient than joint inference, and we also illustrate the limits of this approach with a failure case. Since the performance is likely to be case-specific, we provide an R package mvdens that provides a unified interface for the density approximation methods.
Shirley Ho - One of the best experts on this subject based on the ideXlab platform.
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the Posterior Distribution of sin i values for exoplanets with mt sin i determined from radial velocity data
The Astrophysical Journal, 2011Co-Authors: Edwin L Turner, Shirley HoAbstract:Radial velocity (RV) observations of an exoplanet system giving a value of MT sin(i) condition (i.e., give information about) not only the planet's true mass MT but also the value of sin(i) for that system (where i is the orbital inclination angle). Thus, the value of sin(i) for a system with any particular observed value of MT sin(i) cannot be assumed to be drawn randomly from a Distribution corresponding to an isotropic i Distribution, i.e., the presumptive prior Distribution. Rather, the Posterior Distribution from which it is drawn depends on the intrinsic Distribution of MT for the exoplanet population being studied. We give a simple Bayesian derivation of this relationship and apply it to several "toy models" for the intrinsic Distribution of MT , on which we have significant information from available RV data in some mass ranges but little or none in others. The results show that the effect can be an important one. For example, even for simple power-law Distributions of MT , the median value of sin(i) in an observed RV sample can vary between 0.860 and 0.023 (as compared to the 0.866 value for an isotropic i Distribution) for indices of the power law in the range between –2 and +1, respectively. Over the same range of indices, the 95% confidence interval on MT varies from 1.0001-2.405 (α = –2) to 1.13-94.34 (α = +2) times larger than MT sin(i) due to sin(i) uncertainty alone. More complex, but still simple and plausible, Distributions of MT yield more complicated and somewhat unintuitive Posterior sin(i) Distributions. In particular, if the MT Distribution contains any characteristic mass scale Mc , the Posterior sin(i) Distribution will depend on the ratio of MT sin(i) to Mc , often in a non-trivial way. Our qualitative conclusion is that RV studies of exoplanets, both individual objects and statistical samples, should regard the sin(i) factor as more than a "numerical constant of order unity" with simple and well-understood statistical properties. We argue that reports of MT sin(i) determinations should be accompanied by a statement of the corresponding confidence bounds on MT at, say, the 95% level based on an explicitly stated assumed form of the true MT Distribution in order to reflect more accurately the mass uncertainties associated with RV studies.
Ming-hui Chen - One of the best experts on this subject based on the ideXlab platform.
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propriety of the Posterior Distribution and existence of the mle for regression models with covariates missing at random
Journal of the American Statistical Association, 2004Co-Authors: Ming-hui Chen, Qi-man Shao, Joseph G IbrahimAbstract:Characterizing model identifiability in the presence of missing covariate data is a very important issue in missing data problems. In this article, we characterize the propriety of the Posterior Distribution of the regression coefficients for some general classes of regression models, including the class of generalized linear models (GLM's) and parametric survival models with right-censored data. Toward this goal, we derive some very general and easy-to-check conditions for the matrix of covariates. We also derive sufficient conditions for the existence of the maximum likelihood estimates and establish novel results for checking propriety of the Posterior when the sample size is large. Several theorems are given to establish propriety of the Posterior and the existence of the maximum likelihood estimator. The conditions reduce to solving a system of linear equations, which can be carried out using software such as MAPLE, IMSL, or SAS. We assume that the missing covariates are missing at random and assume ...
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Propriety of Posterior Distribution for dichotomous quantal response models
Proceedings of the American Mathematical Society, 2000Co-Authors: Ming-hui Chen, Qi-man ShaoAbstract:In this article, we investigate the property of Posterior Distribution for dichotomous quantal response models using a uniform prior Distribution on the regression parameters. Sufficient and necessary conditions for the propriety of the Posterior Distribution with a general link function are established. In addition, the sufficient conditions for the existence of the Posterior moments and the Posterior moment generating function are also obtained. Finally, the relationship between the propriety of Posterior Distribution and the existence of the maximum likelihood estimate is examined.
