Joining and splitting models with Markov melding.

Analysing multiple evidence sources is often feasible only via a modular approach, with separate submodels specified for smaller components of the available evidence. Here we introduce a generic framework that enables fully Bayesian analysis in this setting. We propose a generic method for forming a suitable joint model when joining submodels, and a convenient computational algorithm for fitting this joint model in stages, rather than as a single, monolithic model. The approach also enables splitting of large joint models into smaller submodels, allowing inference for the original joint model to be conducted via our multi-stage algorithm. We motivate and demonstrate our approach through two examples: joining components of an evidence synthesis of A/H1N1 influenza, and splitting a large ecology model.

[1]  Paul J. Birrell,et al.  Synthesising evidence to estimate pandemic (2009) A/H1N1 influenza severity in 2009-2011 , 2014, 1408.7025.

[2]  P Besbeas,et al.  Integrating Mark–Recapture–Recovery and Census Data to Estimate Animal Abundance and Demographic Parameters , 2002, Biometrics.

[3]  David J Spiegelhalter,et al.  Bias modelling in evidence synthesis , 2009, Journal of the Royal Statistical Society. Series A,.

[4]  David M. Eddy,et al.  Meta-analysis by the confidence profile method , 1992 .

[5]  Geoffrey E. Hinton.  Training Products of Experts by Minimizing Contrastive Divergence , 2002, Neural Computation.

[6]  S. Brooks,et al.  A Bayesian approach to combining animal abundance and demographic data , 2004 .

[7]  Mark Jit,et al.  Calibration of Complex Models through Bayesian Evidence Synthesis , 2015, Medical decision making : an international journal of the Society for Medical Decision Making.

[8]  C. Sempi,et al.  Copula Theory: An Introduction , 2010 .

[9]  David J. Lunn,et al.  Fully Bayesian hierarchical modelling in two stages, with application to meta-analysis , 2013, Journal of the Royal Statistical Society. Series C, Applied statistics.

[10]  Christian Genest,et al.  Combining Probability Distributions: A Critique and an Annotated Bibliography , 1986 .

[11]  D M Eddy,et al.  Synthesis of environmental evidence: nitrogen dioxide epidemiology studies. , 1992, Journal of the Air & Waste Management Association.

[12]  David J. Spiegelhalter,et al.  Conflict Diagnostics in Directed Acyclic Graphs, with Applications in Bayesian Evidence Synthesis , 2013, 1310.0628.

[13]  David J. Lunn,et al.  The BUGS Book: A Practical Introduction to Bayesian Analysis , 2013 .

[14]  Christopher F. Parmeter,et al.  Applied Nonparametric Econometrics , 2015 .

[15]  David B. Dunson,et al.  Robust and Scalable Bayes via a Median of Subset Posterior Measures , 2014, J. Mach. Learn. Res..

[16]  A. Dawid,et al.  Hyper Markov Laws in the Statistical Analysis of Decomposable Graphical Models , 1993 .

[17]  Kaplan,et al.  ‘Combining Probability Distributions from Experts in Risk Analysis’ , 2000, Risk analysis : an official publication of the Society for Risk Analysis.

[18]  David B. Dunson,et al.  Robust Bayesian Inference via Coarsening , 2015, Journal of the American Statistical Association.

[19]  Alex J. Sutton,et al.  Multiparameter evidence synthesis in epidemiology and medical decision‐making: current approaches , 2006 .

[20]  Manuele Leonelli,et al.  Bayesian decision support in complex modular systems : an algebraic and graphical approach , 2015 .

[21]  W. Gilks,et al.  Following a moving target—Monte Carlo inference for dynamic Bayesian models , 2001 .

[22]  Chong Wang,et al.  Asymptotically Exact, Embarrassingly Parallel MCMC , 2013, UAI.

[23]  A. Tversky,et al.  On the Reconciliation of Probability Assessments , 1979 .

[24]  Artem Lebedev,et al.  Revealing the True Incidence of Pandemic A(H1N1)pdm09 Influenza in Finland during the First Two Seasons — An Analysis Based on a Dynamic Transmission Model , 2016, PLoS Comput. Biol..

[25]  David Draper,et al.  Assessment and Propagation of Model Uncertainty , 2011 .

[26]  Li-Jung Liang,et al.  A Hierarchical Semiparametric Regression Model for Combining HIV‐1 Phylogenetic Analyses Using Iterative Reweighting Algorithms , 2007, Biometrics.

[27]  S. Richardson,et al.  Bayesian graphical models for regression on multiple data sets with different variables , 2008, Biostatistics.

[28]  James O. Berger,et al.  Modularization in Bayesian analysis, with emphasis on analysis of computer models , 2009 .

[29]  Takafumi Kanamori,et al.  Density Ratio Estimation in Machine Learning , 2012 .

[30]  Arnaud Doucet,et al.  On Markov chain Monte Carlo methods for tall data , 2015, J. Mach. Learn. Res..

[31]  Nicky J Welton,et al.  Evidence Synthesis for Decision Making in Healthcare , 2012 .

[32]  Michael Branson,et al.  A note on the power prior , 2009, Statistics in medicine.

[33]  M. Rosińska,et al.  Conflict diagnostics for evidence synthesis in a multiple testing framework , 2017, 1702.07304.

[34]  A E Ades,et al.  Mixed treatment comparison with multiple outcomes reported inconsistently across trials: Evaluation of antivirals for treatment of influenza A and B , 2008, Statistics in medicine.

[35]  Bent Natvig,et al.  Extensions of a Conflict Measure of Inconsistencies in Bayesian Hierarchical Models , 2009 .

[36]  Roman Hovorka,et al.  Rapid model exploration for complex hierarchical data: application to pharmacokinetics of insulin aspart , 2015, Statistics in medicine.

[37]  Pj Green.  Introducing Highly Structured Stochastic Systems , 2003 .

[38]  R. Wilkinson.  Approximate Bayesian computation (ABC) gives exact results under the assumption of model error , 2008, Statistical applications in genetics and molecular biology.

[39]  D. Commenges,et al.  Evidence synthesis through a degradation model applied to myocardial infarction , 2012, Lifetime Data Analysis.

[40]  A. Raftery,et al.  Inference for Deterministic Simulation Models: The Bayesian Melding Approach , 2000 .

[41]  Andrew Gelman,et al.  Sampling for Bayesian Computation with Large Datasets , 2005 .

[42]  Xiangyu Wang,et al.  Parallel MCMC via Weierstrass Sampler , 2013, ArXiv.

[43]  James S. Clark,et al.  Estimating seed and pollen movement in a monoecious plant: a hierarchical Bayesian approach integrating genetic and ecological data , 2011, Molecular ecology.

[44]  Isabelle Albert,et al.  A Bayesian Evidence Synthesis for Estimating Campylobacteriosis Prevalence , 2011, Risk analysis : an official publication of the Society for Risk Analysis.

[45]  Michael A. West,et al.  Combined Parameter and State Estimation in Simulation-Based Filtering , 2001, Sequential Monte Carlo Methods in Practice.

[46]  Timothy J. Robinson,et al.  Sequential Monte Carlo Methods in Practice , 2003 .