Browsing by Author "Dey, Dipak K."

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    A Bayesian approach for mixed effects state-space models under skewness and heavy tails
    (2023) Hernandez-Velasco, Lina L.; Abanto-Valle, Carlos A.; Dey, Dipak K.; Castro, Luis M.
    Human immunodeficiency virus (HIV) dynamics have been the focus of epidemiological and biostatistical research during the past decades to understand the progression of acquired immunodeficiency syndrome (AIDS) in the population. Although there are several approaches for modeling HIV dynamics, one of the most popular is based on Gaussian mixed-effects models because of its simplicity from the implementation and interpretation viewpoints. However, in some situations, Gaussian mixed-effects models cannot (a) capture serial correlation existing in longitudinal data, (b) deal with missing observations properly, and (c) accommodate skewness and heavy tails frequently presented in patients' profiles. For those cases, mixed-effects state-space models (MESSM) become a powerful tool for modeling correlated observations, including HIV dynamics, because of their flexibility in modeling the unobserved states and the observations in a simple way. Consequently, our proposal considers an MESSM where the observations' error distribution is a skew-t. This new approach is more flexible and can accommodate data sets exhibiting skewness and heavy tails. Under the Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is implemented. To evaluate the properties of the proposed models, we carried out some exciting simulation studies, including missing data in the generated data sets. Finally, we illustrate our approach with an application in the AIDS Clinical Trial Group Study 315 (ACTG-315) clinical trial data set.
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    Bregman divergence to generalize Bayesian influence measures for data analysis
    (2021) De Oliveira, Melaine C.; Castro, Luis M.; Dey, Dipak K.; Sinha, Debajyoti
    For existing Bayesian cross-validated measure of influence of each observation on the posterior distribution, this paper considers a generalization using the Bregman Divergence (BD). We investigate various practically useful and desirable properties of these BD based measures to demonstrate the superiority of these measures compared to existing Bayesian measures of influence and Bayesian residual based diagnostics. We provide a practical and easily comprehensible method for calibrating these BD based measures. Also, we show how to compute our BD based measure via Markov chain Monte Carlo (MCMC) samples from a single posterior based on the full data. Using a Bayesian meta-analysis of clinical trials, we illustrate how our new measures of influence of observations have more useful practical roles for data analysis than popular Bayesian residual analysis tools. (c) 2020 Elsevier B.V. All rights reserved.
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    On Posterior Properties of the Two Parameter Gamma Family of Distributions
    (2021) Ramos, Pedro L.; Dey, Dipak K.; Louzada, Francisco; Ramos, Eduardo
    The gamma distribution has been extensively used in many areas of applications. In this paper, considering a Bayesian analysis we provide necessary and sufficient conditions to check whether or not improper priors Lead to proper posterior distributions. Further, we also discuss sufficient conditions to verify if the obtained posterior moments are finite. An interesting aspect of our findings are that one can check if the posterior is proper or improper and also if its posterior moments are finite by looking directly in the behavior of the proposed improper prior. To illustrate our proposed methodology these results are applied in different objective priors.
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    POWER LAWS DISTRIBUTIONS IN OBJECTIVE PRIORS
    (2023) Ramos, Pedro Luiz; Rodrigues, Francisco A.; Ramos, Eduardo; Dey, Dipak K.; Louzada, Francisco
    Using objective priors in Bayesian applications has become a common way of analyzing data without using subjective information. Formal rules are usually used to obtain these prior distributions, and the data provide the dominant infor-mation in the posterior distribution. However, these priors are typically improper, and may lead to an improper posterior. Here, for a general family of distribu-tions, we show that the objective priors obtained for the parameters either follow a power law distribution, or exhibit asymptotic power law behavior. As a result, we observe that the exponents of the model are between 0.5 and 1. Understand-ing this behavior allows us to use the exponent of the power law directly to verify whether such priors lead to proper or improper posteriors. The general family of distributions we consider includes essential models such as the exponential, gamma, Weibull, Nakagami-m, half-normal, Rayleigh, Erlang, and Maxwell Boltzmann dis-tributions, among others. In summary, we show that understanding the mechanisms that describe the shape of a prior provides essential information that can be used to understand the properties of posterior distributions.