Last updated on 2020-02-04 by Jong Hee Park

Applied researchers interested in Bayesian statistics are increasingly attracted to R because of the ease of which one can code algorithms to sample from posterior distributions as well as the significant number of packages contributed to the Comprehensive R Archive Network (CRAN) that provide tools for Bayesian inference. This task view catalogs these tools. In this task view, we divide those packages into four groups based on the scope and focus of the packages. We first review R packages that provide Bayesian estimation tools for a wide range of models. We then discuss packages that address specific Bayesian models or specialized methods in Bayesian statistics. This is followed by a description of packages used for post-estimation analysis. Finally, we review packages that link R to other Bayesian sampling engines such as JAGS, OpenBUGS, WinBUGS, and Stan.

**Bayesian packages for general model fitting**

- The arm package contains R functions for Bayesian inference using lm, glm, mer and polr objects.
- BACCO is an R bundle for Bayesian analysis of random functions. BACCO contains three sub-packages: emulator, calibrator, and approximator, that perform Bayesian emulation and calibration of computer programs.
- bayesm provides R functions for Bayesian inference for various models widely used in marketing and micro-econometrics. The models include linear regression models, multinomial logit, multinomial probit, multivariate probit, multivariate mixture of normals (including clustering), density estimation using finite mixtures of normals as well as Dirichlet Process priors, hierarchical linear models, hierarchical multinomial logit, hierarchical negative binomial regression models, and linear instrumental variable models.
- LaplacesDemon seeks to provide a complete Bayesian environment, including numerous MCMC algorithms, Laplace Approximation with multiple optimization algorithms, scores of examples, dozens of additional probability distributions, numerous MCMC diagnostics, Bayes factors, posterior predictive checks, a variety of plots, elicitation, parameter and variable importance, and numerous additional utility functions.
- MCMCpack provides model-specific Markov chain Monte Carlo (MCMC) algorithms for wide range of models commonly used in the social and behavioral sciences. It contains R functions to fit a number of regression models (linear regression, logit, ordinal probit, probit, Poisson regression, etc.), measurement models (item response theory and factor models), changepoint models (linear regression, binary probit, ordinal probit, Poisson, panel), and models for ecological inference. It also contains a generic Metropolis sampler that can be used to fit arbitrary models.
- The mcmc package consists of an R function for a random-walk Metropolis algorithm for a continuous random vector.
- The nimble package provides a general MCMC system that allows customizable MCMC for models written in the BUGS/JAGS model language. Users can choose samplers and write new samplers. Models and samplers are automatically compiled via generated C++. The package also supports other methods such as particle filtering or whatever users write in its algorithm language.

**Bayesian packages for specific models or methods**

- abc package implements several ABC algorithms for performing parameter estimation and model selection. Cross-validation tools are also available for measuring the accuracy of ABC estimates, and to calculate the misclassification probabilities of different models.
- abn is a package for modelling multivariate data using additive Bayesian networks. It provides routines to help determine optimal Bayesian network models for a given data set, where these models are used to identify statistical dependencies in messy, complex data.
- AdMit provides functions to perform the fitting of an adapative mixture of Student-t distributions to a target density through its kernel function. The mixture approximation can be used as the importance density in importance sampling or as the candidate density in the Metropolis-Hastings algorithm.
- The BaBooN package contains two variants of Bayesian Bootstrap Predictive Mean Matching to multiply impute missing data.
- bamlss provides an infrastructure for estimating probabilistic distributional regression models in a Bayesian framework. The distribution parameters may capture location, scale, shape, etc. and every parameter may depend on complex additive terms similar to a generalized additive model.
- The BART package provide flexible nonparametric modeling of covariates for continuous, binary, categorical and time-to-event outcomes.
- The bayesGARCH package provides a function which perform the Bayesian estimation of the GARCH(1,1) model with Student's t innovations.
- bayesImageS is an R package for Bayesian image analysis using the hidden Potts model.
- bayesmeta is an R package to perform meta-analyses within the common random-effects model framework.
- BayesSummaryStatLM provides two functions: one function that computes summary statistics of data and one function that carries out the MCMC posterior sampling for Bayesian linear regression models where summary statistics are used as input.
- BayesTree implements BART (Bayesian Additive Regression Trees) by Chipman, George, and McCulloch (2006).
- bayesQR supports Bayesian quantile regression using the asymmetric Laplace distribution, both continuous as well as binary dependent variables.
- BayesFactor provides a suite of functions for computing various Bayes factors for simple designs, including contingency tables, one- and two-sample designs, one-way designs, general ANOVA designs, and linear regression.
- BayesVarSel calculate Bayes factors in linear models and then to provide a formal Bayesian answer to testing and variable selection problems.
- BayHaz contains a suite of R functions for Bayesian estimation of smooth hazard rates via Compound Poisson Process (CPP) priors.
- BAYSTAR provides functions for Bayesian estimation of threshold autoregressive models.
- bbemkr implements Bayesian bandwidth estimation for Nadaraya-Watson type multivariate kernel regression with Gaussian error.
- BCE contains function to estimates taxonomic compositions from biomarker data using a Bayesian approach.
- BCBCSF provides functions to predict the discrete response based on selected high dimensional features, such as gene expression data.
- bcp implements a Bayesian analysis of changepoint problem using Barry and Hartigan product partition model.
- BDgraph provides statistical tools for Bayesian structure learning in undirected graphical models for multivariate continuous, discrete, and mixed data.
- BLR provides R functions to fit parametric regression models using different types of shrinkage methods.
- The BMA package has functions for Bayesian model averaging for linear models, generalized linear models, and survival models. The complementary package ensembleBMA uses the BMA package to create probabilistic forecasts of ensembles using a mixture of normal distributions.
- bmixture provides statistical tools for Bayesian estimation for the finite mixture of distributions, mainly mixture of Gamma, Normal and t-distributions.
- BMS is Bayesian Model Averaging library for linear models with a wide choice of (customizable) priors. Built-in priors include coefficient priors (fixed, flexible and hyper-g priors), and 5 kinds of model priors.
- Bmix is a bare-bones implementation of sampling algorithms for a variety of Bayesian stick-breaking (marginally DP) mixture models, including particle learning and Gibbs sampling for static DP mixtures, particle learning for dynamic BAR stick-breaking, and DP mixture regression.
- bnlearn is a package for Bayesian network structure learning (via constraint-based, score-based and hybrid algorithms), parameter learning (via ML and Bayesian estimators) and inference.
- BNSP is a package for Bayeisan non- and semi-parametric model fitting. It handles Dirichlet process mixtures and spike-slab for multivariate (and univariate) response analysis, with nonparametric models for the means, the variances and the correlation matrix.
- BoomSpikeSlab provides functions to do spike and slab regression via the stochastic search variable selection algorithm. It handles probit, logit, poisson, and student T data.
- bqtl can be used to fit quantitative trait loci (QTL) models. This package allows Bayesian estimation of multi-gene models via Laplace approximations and provides tools for interval mapping of genetic loci. The package also contains graphical tools for QTL analysis.
- bridgesampling provides R functions for estimating marginal likelihoods, Bayes factors, posterior model probabilities, and normalizing constants in general, via different versions of bridge sampling (Meng and Wong, 1996).
- bsamGP provides functions to perform Bayesian inference using a spectral analysis of Gaussian process priors. Gaussian processes are represented with a Fourier series based on cosine basis functions. Currently the package includes parametric linear models, partial linear additive models with/without shape restrictions, generalized linear additive models with/without shape restrictions, and density estimation model.
- bspec performs Bayesian inference on the (discrete) power spectrum of time series.
- bspmma is a package for Bayesian semiparametric models for meta-analysis.
- bsts is a package for time series regression using dynamic linear models using MCMC.
- BVS is a package for Bayesian variant selection and Bayesian model uncertainty techniques for genetic association studies.
- catnet is a package that handles discrete Bayesian network models and provides inference using the frequentist approach.
- coalescentMCMC provides a flexible framework for coalescent analyses in R.
- dclone provides low level functions for implementing maximum likelihood estimating procedures for complex models using data cloning and MCMC methods.
- deBInfer provides R functions for Bayesian parameter inference in differential equations using MCMC methods.
- dlm is a package for Bayesian (and likelihood) analysis of dynamic linear models. It includes the calculations of the Kalman filter and smoother, and the forward filtering backward sampling algorithm.
- EbayesThresh implements Bayesian estimation for thresholding methods. Although the original model is developed in the context of wavelets, this package is useful when researchers need to take advantage of possible sparsity in a parameter set.
- eco fits Bayesian ecological inference models in two by two tables using MCMC methods.
- ebdbNet can be used to infer the adjacency matrix of a network from time course data using an empirical Bayes estimation procedure based on Dynamic Bayesian Networks.
- eigenmodel estimates the parameters of a model for symmetric relational data (e.g., the above-diagonal part of a square matrix), using a model-based eigenvalue decomposition and regression using MCMC methods.
- EntropyMCMC is an R package for MCMC simulation and convergence evaluation using entropy and Kullback-Leibler divergence estimation.
- evdbayes provides tools for Bayesian analysis of extreme value models.
- exactLoglinTest provides functions for log-linear models that compute Monte Carlo estimates of conditional P-values for goodness of fit tests.
- factorQR is a package to fit Bayesian quantile regression models that assume a factor structure for at least part of the design matrix.
- FME provides functions to help in fitting models to data, to perform Monte Carlo, sensitivity and identifiability analysis. It is intended to work with models be written as a set of differential equations that are solved either by an integration routine from deSolve, or a steady-state solver from rootSolve.
- The
`gbayes()`

function in Hmisc derives the posterior (and optionally) the predictive distribution when both the prior and the likelihood are Gaussian, and when the statistic of interest comes from a two-sample problem. - ggmcmc is a tool for assessing and diagnosing convergence of Markov Chain Monte Carlo simulations, as well as for graphically display results from full MCMC analysis.
- gRain is a package for probability propagation in graphical independence networks, also known as Bayesian networks or probabilistic expert systems.
- The HI package has functions to implement a geometric approach to transdimensional MCMC methods and random direction multivariate Adaptive Rejection Metropolis Sampling.
- The hbsae package provides functions to compute small area estimates based on a basic area or unit-level model. The model is fit using restricted maximum likelihood, or in a hierarchical Bayesian way.
- iterLap performs an iterative Laplace approximation to build a global approximation of the posterior (using mixture distributions) and then uses importance sampling for simulation based inference.
- The function
`krige.bayes()`

in the geoR package performs Bayesian analysis of geostatistical data allowing specification of different levels of uncertainty in the model parameters. Ssee the Spatial view for more information. - The lmm package contains R functions to fit linear mixed models using MCMC methods.
- matchingMarkets implements a structural model based on a Gibbs sampler to correct for the bias from endogenous matching (e.g. group formation or two-sided matching).
- MCMCglmm is package for fitting Generalised Linear Mixed Models using MCMC methods.
- The
`mcmcsamp()`

function in lme4 allows MCMC sampling for the linear mixed model and generalized linear mixed model. - The mlogitBMA Provides a modified function
`bic.glm()`

of the BMA package that can be applied to multinomial logit (MNL) data. - The MNP package fits multinomial probit models using MCMC methods.
- mombf performs model selection based on non-local priors, including MOM, eMOM and iMOM priors..
- NetworkChange is an R package for change point analysis in longitudinal network data. It implements a hidden Markovmultilinear tensor regression model. Model diagnostic tools using marginal likelihoods and WAIC are provided.
- openEBGM calculates Empirical Bayes Geometric Mean (EBGM) and quantile scores from the posterior distribution using the Gamma-Poisson Shrinker (GPS) model to find unusually large cell counts in large, sparse contingency tables.
- pacbpred perform estimation and prediction in high-dimensional additive models, using a sparse PAC-Bayesian point of view and a MCMC algorithm.
- predmixcor provides functions to predict the binary response based on high dimensional binary features modeled with Bayesian mixture models.
- prevalence provides functions for the estimation of true prevalence from apparent prevalence in a Bayesian framework. MCMC sampling is performed via JAGS/rjags.
- profdpm facilitates profile inference (inference at the posterior mode) for a class of product partition models.
- The pscl package provides R functions to fit item-response theory models using MCMC methods and to compute highest density regions for the Beta distribution and the inverse gamma distribution.
- The PAWL package implements parallel adaptive Metropolis-Hastings and sequential Monte Carlo samplers for sampling from multimodal target distributions.
- PReMiuM is a package for profile regression, which is a Dirichlet process Bayesian clustering where the response is linked non-parametrically to the covariate profile.
- revdbayes provides functions for the Bayesian analysis of extreme value models using direct random sampling from extreme value posterior distributions.
- The
`hitro.new()`

function in Runuran provides an MCMC sampler based on the Hit-and-Run algorithm in combination with the Ratio-of-Uniforms method. - RSGHB can be used to estimate models using a hierarchical Bayesian framework and provides flexibility in allowing the user to specify the likelihood function directly instead of assuming predetermined model structures.
- rstiefel simulates random orthonormal matrices from linear and quadratic exponential family distributions on the Stiefel manifold using the Gibbs sampling method. The most general type of distribution covered is the matrix-variate Bingham-von Mises-Fisher distribution.
- RxCEcolInf fits the R x C inference model described in Greiner and Quinn (2009).
- SamplerCompare provides a framework for running sets of MCMC samplers on sets of distributions with a variety of tuning parameters, along with plotting functions to visualize the results of those simulations.
- SampleSizeMeans contains a set of R functions for calculating sample size requirements using three different Bayesian criteria in the context of designing an experiment to estimate a normal mean or the difference between two normal means.
- SampleSizeProportions contains a set of R functions for calculating sample size requirements using three different Bayesian criteria in the context of designing an experiment to estimate the difference between two binomial proportions.
- sbgcop estimates parameters of a Gaussian copula, treating the univariate marginal distributions as nuisance parameters as described in Hoff(2007). It also provides a semiparametric imputation procedure for missing multivariate data.
- SimpleTable provides a series of methods to conduct Bayesian inference and sensitivity analysis for causal effects from 2 x 2 and 2 x 2 x K tables.
- sna, an R package for social network analysis, contains functions to generate posterior samples from Butt's Bayesian network accuracy model using Gibbs sampling.
- spBayes provides R functions that fit Gaussian spatial process models for univariate as well as multivariate point-referenced data using MCMC methods.
- spikeslab provides functions for prediction and variable selection using spike and slab regression.
- spikeSlabGAM implements Bayesian variable selection, model choice, and regularized estimation in (geo-)additive mixed models for Gaussian, binomial, and Poisson responses.
- spTimer fits, spatially predict and temporally forecast large amounts of space-time data using Bayesian Gaussian Process Models, Bayesian Auto-Regressive (AR) Models, and Bayesian Gaussian Predictive Processes based AR Models.
- ssgraph is for Bayesian inference in undirected graphical models using spike-and-slab priors for multivariate continuous, discrete, and mixed data.
- ssMousetrack estimates previously compiled state-space modeling for mouse-tracking experiments using the rstan package, which provides the R interface to the Stan C++ library for Bayesian estimation.
- stochvol provides efficient algorithms for fully Bayesian estimation of stochastic volatility (SV) models.
- The tgp package implements Bayesian treed Gaussian process models: a spatial modeling and regression package providing fully Bayesian MCMC posterior inference for models ranging from the simple linear model, to nonstationary treed Gaussian process, and others in between.
- vbmp is a package for variational Bayesian multinomial probit regression with Gaussian process priors. It estimates class membership posterior probability employing variational and sparse approximation to the full posterior. This software also incorporates feature weighting by means of Automatic Relevance Determination.
- The
`vcov.gam()`

function the mgcv package can extract a Bayesian posterior covariance matrix of the parameters from a fitted`gam`

object. - zic provides functions for an MCMC analysis of zero-inflated count models including stochastic search variable selection.

**Post-estimation tools**

- BayesPostEst allows to generate and plot postestimation quantities after estimating Bayesian regression models. Functionality includes predicted probabilities and first differences as well as model checks. The functions can be used with MCMC output generated by any Bayesian estimation tool including JAGS, BUGS, MCMCpack, and Stan.
- BayesValidate implements a software validation method for Bayesian softwares.
- MCMCvis performs key functions (visualizes, manipulates, and summarizes) for MCMC analysis. Functions support simple and straightforward subsetting of model parameters within the calls, and produce presentable and 'publication-ready' output. MCMC output may be derived from Bayesian model output fit with JAGS, Stan, or other MCMC samplers.
- The boa package provides functions for diagnostics, summarization, and visualization of MCMC sequences. It imports draws from BUGS format, or from plain matrices. boa provides the Gelman and Rubin, Geweke, Heidelberger and Welch, and Raftery and Lewis diagnostics, the Brooks and Gelman multivariate shrink factors.
- The coda (Convergence Diagnosis and Output Analysis) package is a suite of functions that
can be used to summarize, plot, and and diagnose convergence from MCMC samples.
coda also defines an
`mcmc`

object and related methods which are used by other packages. It can easily import MCMC output from WinBUGS, OpenBUGS, and JAGS, or from plain matrices. coda contains the Gelman and Rubin, Geweke, Heidelberger and Welch, and Raftery and Lewis diagnostics. - plotMCMC extends coda by adding convenience functions to make it easier to create multipanel plots. The graphical parameters have sensible defaults and are easy to modify via top-level arguments.
- ramps implements Bayesian geostatistical analysis of Gaussian processes using a reparameterized and marginalized posterior sampling algorithm.

**Packages for learning Bayesian statistics**

- AtelieR is a GTK interface for teaching basic concepts in statistical inference, and doing elementary bayesian statistics (inference on proportions, multinomial counts, means and variances).
- The BaM package is an R package associated with Jeff Gill's book, "Bayesian Methods: A Social and Behavioral Sciences Approach, Second Edition" (CRC Press, 2007).
- BayesDA provides R functions and datasets for "Bayesian Data Analysis, Second Edition" (CRC Press, 2003) by Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin.
- The Bolstad package contains a set of R functions and data sets for the book Introduction to Bayesian Statistics, by Bolstad, W.M. (2007).
- The LearnBayes package contains a collection of functions helpful in learning the basic tenets of Bayesian statistical inference. It contains functions for summarizing basic one and two parameter posterior distributions and predictive distributions and MCMC algorithms for summarizing posterior distributions defined by the user. It also contains functions for regression models, hierarchical models, Bayesian tests, and illustrations of Gibbs sampling.

**Packages that link R to other sampling engines**

- bayesmix is an R package to fit Bayesian mixture models using JAGS.
- BayesX provides functionality for exploring and visualizing estimation results obtained with the software package BayesX.
- Boom provides a C++ library for Bayesian modeling, with an emphasis on Markov chain Monte Carlo.
**BRugs**provides an R interface to OpenBUGS. It works under Windows and Linux.**BRugs**used to be available from CRAN, now it is located at the CRANextras repository.- brms implements Bayesian multilevel models in R using Stan. A wide range of distributions and link functions are supported, allowing users to fit linear, robust linear, binomial, Pois- son, survival, response times, ordinal, quantile, zero-inflated, hurdle, and even non-linear models all in a multilevel context.
- There are two packages that can be used to interface R with WinBUGS. R2WinBUGS provides a set of functions to call WinBUGS on a Windows system and a Linux system.
- There are three packages that provide R interface with Just Another Gibbs Sampler (JAGS): rjags, R2jags, and runjags.
- All of these BUGS engines use graphical models for model specification. As such, the gR task view may be of interest.
- rstan provides R functions to parse, compile, test, estimate, and analyze Stan models by accessing the header-only Stan library provided by the `StanHeaders' package. The Stan project develops a probabilistic programming language that implements full Bayesian statistical inference via MCMC and (optionally penalized) maximum likelihood estimation via optimization.
- pcFactorStan provides convenience functions and pre-programmed Stan models related to the paired comparison factor model. Its purpose is to make fitting paired comparison data using Stan easy.
- R2BayesX provides an R interface to estimate structured additive regression (STAR) models with 'BayesX'.

The Bayesian Inference Task View is written by Jong Hee Park (Seoul National University, South Korea), Andrew D. Martin (University of Michigan, Ann Arbor, MI, USA), and Kevin M. Quinn (UC Berkeley, Berkeley, CA, USA). Please email the task view maintainer with suggestions.

- Bayesian Statistics and Marketing (bayesm)
- BayesX
- Just Another Gibbs Sampler (JAGS)
- MCMCpack
- The BUGS Project (WinBUGS)
- OpenBUGS
- BRugs in CRANextras
- NIMBLE
- Stan
- Task view: gR
- Bioconductor package: vbmp
- BOA

3 months ago by Gilles Kratzer

Modelling Multivariate Data with Additive Bayesian Networks

2 years ago by Yu-Sung Su

Data Analysis Using Regression and Multilevel/Hierarchical Models

6 years ago by Yvonnick Noel

A GTK GUI for teaching basic concepts in statistical inference, and doing elementary bayesian tests.

5 years ago by Florian Meinfelder

Bayesian Bootstrap Predictive Mean Matching - Multiple and Single Imputation for Discrete Data

3 months ago by Nikolaus Umlauf

Bayesian Additive Models for Location, Scale, and Shape (and Beyond)

8 years ago by Kjetil Halvorsen

Functions and Datasets for the book "Bayesian Data Analysis"

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Computation of Bayes Factors for Common Designs

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Bayesian Estimation of the GARCH(1,1) Model with Student-t Innovations

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Bayesian Methods for Image Segmentation using a Potts Model

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Generate Postestimation Quantities for Bayesian MCMC Estimation

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MCMC Sampling of Bayesian Linear Models via Summary Statistics

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Bayes Factors, Model Choice and Variable Selection in Linear Models

6 years ago by Edward M.H. Lin

On Bayesian analysis of Threshold autoregressive model (BAYSTAR)

6 years ago by Han Lin Shang

Bayesian bandwidth estimation for multivariate kernel regression with Gaussian error

4 years ago by Longhai Li

Bias-Corrected Bayesian Classification with Selected Features

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Bayesian composition estimator: estimating sample (taxonomic) composition from biomarker data

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Bayesian Structure Learning in Graphical Models using Birth-Death MCMC

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Bayesian Estimation for Finite Mixture of Distributions

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Bayesian Network Structure Learning, Parameter Learning and Inference

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Bridge Sampling for Marginal Likelihoods and Bayes Factors

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Bayesian Spectral Analysis Models using Gaussian Process Priors

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Bayesian Variant Selection: Bayesian Model Uncertainty Techniques for Genetic Association Studies

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Data Cloning and MCMC Tools for Maximum Likelihood Methods

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Bayesian Inference for Differential Equations

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Bayesian and Likelihood Analysis of Dynamic Linear Models

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Empirical Bayes Thresholding and Related Methods

9 months ago by Peter Hoff

Semiparametric Factor and Regression Models for Symmetric Relational Data

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Probabilistic Forecasting using Ensembles and Bayesian Model Averaging

a year ago by Didier Chauveau

MCMC Simulation and Convergence Evaluation using Entropy and Kullback-Leibler Divergence Estimation

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A Flexible Modelling Environment for Inverse Modelling, Sensitivity, Identifiability and Monte Carlo Analysis

7 months ago by Xavier Fernández i Marín

Tools for Analyzing MCMC Simulations from Bayesian Inference

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Simulation from distributions supported by nested hyperplanes

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Approximate Probability Densities by Iterated Laplace Approximations

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Tools to Visualize, Manipulate, and Summarize MCMC Output

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Mixed GAM Computation Vehicle with Automatic Smoothness Estimation

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Bayesian Model Averaging for Multinomial Logit Models

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Bayesian Model Selection and Averaging for Non-Local and Local Priors

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MCMC, Particle Filtering, and Programmable Hierarchical Modeling

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EBGM Disproportionality Scores for Adverse Event Data Mining

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PAC-Bayesian Estimation and Prediction in Sparse Additive Models.

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Stan Models for the Paired Comparison Factor Model

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Classification rule based on Bayesian mixture models with feature selection bias corrected

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Dirichlet Process Bayesian Clustering, Profile Regression

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Estimate Structured Additive Regression Models with 'BayesX'

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Ratio-of-Uniforms Sampling for Bayesian Extreme Value Analysis

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'R x C Ecological Inference With Optional Incorporation of Survey Information'

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Functions for Hierarchical Bayesian Estimation: A Flexible Approach

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Random Orthonormal Matrix Generation and Optimization on the Stiefel Manifold

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A Framework for Comparing the Performance of MCMC Samplers

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Calculating sample size requirements when estimating the difference between two binomial proportions

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Semiparametric Bayesian Gaussian Copula Estimation and Imputation

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Bayesian Inference and Sensitivity Analysis for Causal Effects from 2 x 2 and 2 x 2 x K Tables in the Presence of Unmeasured Confounding

7 years ago by Udaya B. Kogalur

Prediction and variable selection using spike and slab regression

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Bayesian Variable Selection and Model Choice for Generalized Additive Mixed Models

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Bayesian Graphical Estimation using Spike-and-Slab Priors

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Efficient Bayesian Inference for Stochastic Volatility (SV) Models