Posterior Predictive Distribution

2015 ◽  
pp. 1-6
Author(s):  
Maria Maddalena Barbieri
Keyword(s):  
Predictive Distribution ◽  
Posterior Predictive Distribution

scholarly journals Modeling Chickenpox Dynamics with a Discrete Time Bayesian Stochastic Compartmental Model

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
A. Corberán-Vallet ◽  
F. J. Santonja ◽  
M. Jornet-Sanz ◽  
R.-J. Villanueva
Keyword(s):  
Discrete Time ◽  
Compartmental Model ◽  
Predictive Distribution ◽  
Model Parameters ◽  
Immunization Program ◽  
Posterior Predictive Distribution ◽  
Vaccination Strategies ◽  
Proposed Model ◽  
Routine Immunization Program ◽  
The Impact

We present a Bayesian stochastic susceptible-exposed-infectious-recovered model in discrete time to understand chickenpox transmission in the Valencian Community, Spain. During the last decades, different strategies have been introduced in the routine immunization program in order to reduce the impact of this disease, which remains a public health’s great concern. Under this scenario, a model capable of explaining closely the dynamics of chickenpox under the different vaccination strategies is of utter importance to assess their effectiveness. The proposed model takes into account both heterogeneous mixing of individuals in the population and the inherent stochasticity in the transmission of the disease. As shown in a comparative study, these assumptions are fundamental to describe properly the evolution of the disease. The Bayesian analysis of the model allows us to calculate the posterior distribution of the model parameters and the posterior predictive distribution of chickenpox incidence, which facilitates the computation of point forecasts and prediction intervals.

Consideration of Epistemic Uncertainty in Extreme Wave Height Estimation

2014 ◽  
Author(s):  
Ryota Wada ◽  
Takuji Waseda
Keyword(s):  
Gulf Of Mexico ◽  
Observational Data ◽  
Wave Height ◽  
Extreme Values ◽  
Epistemic Uncertainty ◽  
Extreme Value Distribution ◽  
Predictive Distribution ◽  
Extreme Value ◽  
Posterior Predictive Distribution ◽  
Value Estimation

Extreme value estimation of significant wave height is essential for designing robust and economically efficient ocean structures. But in most cases, the duration of observational wave data is not efficient to make a precise estimation of the extreme value for the desired period. When we focus on hurricane dominated oceans, the situation gets worse. The uncertainty of the extreme value estimation is the main topic of this paper. We use Likelihood-Weighted Method (LWM), a method that can quantify the uncertainty of extreme value estimation in terms of aleatory and epistemic uncertainty. We considered the extreme values of hurricane-dominated regions such as Japan and Gulf of Mexico. Though observational data is available for more than 30 years in Gulf of Mexico, the epistemic uncertainty for 100-year return period value is notably large. Extreme value estimation from 10-year duration of observational data, which is a typical case in Japan, gave a Coefficient of Variance of 43%. This may have impact on the design rules of ocean structures. Also, the consideration of epistemic uncertainty gives rational explanation for the past extreme events, which were considered as abnormal. Expected Extreme Value distribution (EEV), which is the posterior predictive distribution, defined better extreme values considering the epistemic uncertainty.

AN MLP TRAINING ALGORITHM TAKING INTO ACCOUNT KNOWN ERRORS ON INPUTS AND OUTPUTS

2002 ◽  
Vol 12 (05) ◽  
pp. 369-379
Author(s):  
J. SVENSSON
Keyword(s):  
Cost Function ◽  
A Priori ◽  
Hessian Matrix ◽  
Predictive Distribution ◽  
Training Algorithm ◽  
Posterior Predictive Distribution ◽  
Network Function ◽  
Error Bars ◽  
Mlp Network ◽  
Inputs And Outputs

A training algorithm is introduced that takes into account a priori known errors on both inputs and outputs in an MLP network. The new cost function introduced for this case is based on a linear approximation of the network function over the input distribution for a given input pattern. Update formulas, in the form of the gradient of the new cost function, is given for a MLP network, together with expressions for the Hessian matrix. This is later used to calculate error bars in a Bayesian framework. The error bars thus derived are discussed in relation to the more commonly used width of the target posterior predictive distribution. It will also be shown that the taking into account of known input uncertainties in the way suggested in this article will have a strong regularizing effect on the solution.

scholarly journals Comparing Gaussian Graphical Models with the Posterior Predictive Distribution and Bayesian Model Selection

2019 ◽  
Author(s):  
Donald Ray Williams ◽  
Philippe Rast ◽  
Luis Pericchi ◽  
Joris Mulder
Keyword(s):  
Model Selection ◽  
Hypothesis Testing ◽  
Graphical Models ◽  
Bayesian Model ◽  
Bayesian Model Selection ◽  
Predictive Distribution ◽  
Network Structures ◽  
Gaussian Graphical Models ◽  
Posterior Predictive Distribution ◽  
Leibler Divergence

Gaussian graphical models are commonly used to characterize conditional independence structures (i.e., networks) of psychological constructs. Recently attention has shifted from estimating single networks to those from various sub-populations. The focus is primarily to detect differences or demonstrate replicability. We introduce two novel Bayesian methods for comparing networks that explicitly address these aims. The first is based on the posterior predictive distribution, with Kullback-Leibler divergence as the discrepancy measure, that tests differences between two multivariate normal distributions. The second approach makes use of Bayesian model selection, with the Bayes factor, and allows for gaining evidence for invariant network structures. This overcomes limitations of current approaches in the literature that use classical hypothesis testing, where it is only possible to determine whether groups are significantly different from each other. With simulation we show the posterior predictive method is approximately calibrated under the null hypothesis ($\alpha = 0.05$) and has more power to detect differences than alternative approaches. We then examine the necessary sample sizes for detecting invariant network structures with Bayesian hypothesis testing, in addition to how this is influenced by the choice of prior distribution. The methods are applied to post-traumatic stress disorder symptoms that were measured in four groups. We end by summarizing our major contribution, that is proposing two novel methods for comparing GGMs, which extends beyond the social-behavioral sciences. The methods have been implemented in the R package BGGM.

scholarly journals A Bayesian-Deep Learning Model for Estimating COVID-19 Evolution in Spain

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2921
Author(s):  
Stefano Cabras
Keyword(s):  
Deep Learning ◽  
Predictive Distribution ◽  
Expert Elicitation ◽  
Gamma Model ◽  
Expected Number ◽  
Posterior Predictive Distribution ◽  
Future Evolution ◽  
Deep Learning Model ◽  
Modelling Approach

This work proposes a semi-parametric approach to estimate the evolution of COVID-19 (SARS-CoV-2) in Spain. Considering the sequences of 14-day cumulative incidence of all Spanish regions, it combines modern Deep Learning (DL) techniques for analyzing sequences with the usual Bayesian Poisson-Gamma model for counts. The DL model provides a suitable description of the observed time series of counts, but it cannot give a reliable uncertainty quantification. The role of expert elicitation of the expected number of counts and its reliability is DL predictions’ role in the proposed modelling approach. Finally, the posterior predictive distribution of counts is obtained in a standard Bayesian analysis using the well known Poisson-Gamma model. The model allows to predict the future evolution of the sequences on all regions or estimates the consequences of eventual scenarios.

The Survivor Problem: Simple Linear Regression with MCMC

2019 ◽  
pp. 269-307
Author(s):  
Therese M. Donovan ◽  
Ruth M. Mickey
Keyword(s):  
Linear Regression ◽  
Gibbs Sampling ◽  
Formal Education ◽  
Predictive Distribution ◽  
Simple Linear Regression ◽  
Posterior Predictive Distribution ◽  
Reality Show ◽  
The Relationship ◽  
Joint Posterior Distribution

While one of the most common uses of Bayes’ Theorem is in the statistical analysis of a dataset (i.e., statistical modeling), this chapter examines another application of Gibbs sampling: parameter estimation for simple linear regression. In the “Survivor Problem,” the chapter considers the relationship between how many days a contestant lasts in a reality-show competition as a function of how many years of formal education they have. This chapter is a bit more complicated than the previous chapter because it involves estimation of the joint posterior distribution of three parameters. As in earlier chapters, the estimation process is described in detail on a step-by-step basis. Finally, the posterior predictive distribution is estimated and discussed. By the end of the chapter, the reader will have a firm understanding of the following concepts: linear equation, sums of squares, posterior predictive distribution, and linear regression with Markov Chain Monte Carlo and Gibbs sampling.

Estimation in Parallel Randomized Experiments

1981 ◽  
Vol 6 (4) ◽  
pp. 377-401 ◽  
Author(s):  
Donald B. Rubin
Keyword(s):  
Empirical Bayes ◽  
Predictive Distribution ◽  
Model Specification ◽  
Randomized Experiments ◽  
Posterior Predictive Distribution ◽  
Bayesian Techniques ◽  
Simulation Techniques ◽  
Simple Simulation ◽  
The One ◽  
New Treatments

Many studies comparing new treatments to standard treatments consist of parallel randomized experiments. In the example considered here, randomized experiments were conducted in eight schools to determine the effectiveness of special coaching programs for the SAT. The purpose here is to illustrate Bayesian and empirical Bayesian techniques that can be used to help summarize the evidence in such data about differences among treatments, thereby obtaining improved estimates of the treatment effect in each experiment, including the one having the largest observed effect. Three main tools are illustrated: 1) graphical techniques for displaying sensitivity within an empirical Bayes framework, 2) simple simulation techniques for generating Bayesian posterior distributions of individual effects and the largest effect, and 3) methods for monitoring the adequacy of the Bayesian model specification by simulating the posterior predictive distribution in hypothetical replications of the same treatments in the same eight schools.

scholarly journals Differential Privacy Applications to Bayesian and Linear Mixed Model Estimation

2013 ◽  
Vol 5 (1) ◽  
Author(s):  
John M. Abowd ◽  
Matthew J. Schneider ◽  
Lars Vilhuber
Keyword(s):  
Random Effects ◽  
Small Area ◽  
Mixed Model ◽  
Linear Mixed Model ◽  
Differential Privacy ◽  
Predictive Distribution ◽  
Random Errors ◽  
Time Dimension ◽  
Posterior Predictive Distribution ◽  
Computationally Intensive

We consider a particular maximum likelihood estimator (MLE) and a computationally intensive Bayesian method for differentially private estimation of the linear mixed-effects model (LMM) with normal random errors. The LMM is important because it is used in small-area estimation and detailed industry tabulations that present significant challenges for confidentiality protection of the underlying data. The differentially private MLE performs well compared to the regular MLE, and deteriorates as the protection increases for a problem in which the small-area variation is at the county level. More dimensions of random effects are needed to adequately represent the time dimension of the data, and for these cases the differentially private MLE cannot be computed. The direct Bayesian approach for the same model uses an informative, reasonably diffuse prior to compute the posterior predictive distribution for the random effects. The empirical differential privacy of this approach is estimated by direct computation of the relevant odds ratios after deleting influential observations according to various criteria.

scholarly journals Identifying model violations under the multispecies coalescent model using P2C2M.SNAPP

PeerJ ◽  
2020 ◽  
Vol 8 ◽  
pp. e8271
Author(s):  
Drew J. Duckett ◽  
Tara A. Pelletier ◽  
Bryan C. Carstens
Keyword(s):  
Posterior Distribution ◽  
Incomplete Lineage Sorting ◽  
R Package ◽  
Predictive Distribution ◽  
Species Trees ◽  
Lineage Sorting ◽  
Posterior Predictive Distribution ◽  
Coalescent Model ◽  
Multispecies Coalescent ◽  
Phylogenetic Estimation

Phylogenetic estimation under the multispecies coalescent model (MSCM) assumes all incongruence among loci is caused by incomplete lineage sorting. Therefore, applying the MSCM to datasets that contain incongruence that is caused by other processes, such as gene flow, can lead to biased phylogeny estimates. To identify possible bias when using the MSCM, we present P2C2M.SNAPP. P2C2M.SNAPP is an R package that identifies model violations using posterior predictive simulation. P2C2M.SNAPP uses the posterior distribution of species trees output by the software package SNAPP to simulate posterior predictive datasets under the MSCM, and then uses summary statistics to compare either the empirical data or the posterior distribution to the posterior predictive distribution to identify model violations. In simulation testing, P2C2M.SNAPP correctly classified up to 83% of datasets (depending on the summary statistic used) as to whether or not they violated the MSCM model. P2C2M.SNAPP represents a user-friendly way for researchers to perform posterior predictive model checks when using the popular SNAPP phylogenetic estimation program. It is freely available as an R package, along with additional program details and tutorials.

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