Unbiased and consistent nested sampling via sequential Monte Carlo

We introduce a new class of sequential Monte Carlo methods called Nested Sampling via Sequential Monte Carlo (NS-SMC), which reframes the Nested Sampling method of Skilling (2006) in terms of sequential Monte Carlo techniques. This new framework allows one to obtain provably consistent estimates of marginal likelihood and posterior inferences when Markov chain Monte Carlo (MCMC) is used to produce new samples. An additional benefit is that marginal likelihood estimates are also unbiased. In contrast to NS, the analysis of NS-SMC does not require the (unrealistic) assumption that the simulated samples be independent. As the original NS algorithm is a special case of NS-SMC, this provides insights as to why NS seems to produce accurate estimates despite a typical violation of its assumptions. For applications of NS-SMC, we give advice on tuning MCMC kernels in an automated manner via a preliminary pilot run, and present a new method for appropriately choosing the number of MCMC repeats at each iteration. Finally, a numerical study is conducted where the performance of NS--SMC and temperature--annealed SMC is compared on several challenging and realistic problems. Matlab code for our experiments is made available at http://github.com/LeahPrice/SMC-NS .

[1]  Morteza Marzjarani,et al.  Simulation and the Monte Carlo Method (3rd ed.) , 2019, Technometrics.

[2]  Nick Whiteley,et al.  Negative association, ordering and convergence of resampling methods , 2017, The Annals of Statistics.

[3]  M. Hobson,et al.  Sampling Errors in Nested Sampling Parameter Estimation , 2017, Bayesian Analysis.

[4]  Robert John Nicholas Baldock,et al.  Classical Statistical Mechanics with Nested Sampling , 2017 .

[5]  Clément Walter,et al.  Point process-based Monte Carlo estimation , 2014, Stat. Comput..

[6]  James Ridgway,et al.  Computation of Gaussian orthant probabilities in high dimension , 2014, Statistics and Computing.

[7]  Johannes Buchner,et al.  A statistical test for Nested Sampling algorithms , 2014, Statistics and Computing.

[8]  Alexandros Beskos,et al.  On the convergence of adaptive sequential Monte Carlo methods , 2013, The Annals of Applied Probability.

[9]  James Ridgway,et al.  Leave Pima Indians alone: binary regression as a benchmark for Bayesian computation , 2015, 1506.08640.

[10]  P. Graff,et al.  Parameter estimation for compact binaries with ground-based gravitational-wave observations using the LALInference software library , 2014, 1409.7215.

[11]  B. Brewer.  Inference for Trans-dimensional Bayesian Models with Diffusive Nested Sampling , 2014, 1411.3921.

[12]  N. Chopin,et al.  Control functionals for Monte Carlo integration , 2014, 1410.2392.

[13]  Gareth W. Peters,et al.  Improving SMC sampler estimate by recycling all past simulated particles , 2014, 2014 IEEE Workshop on Statistical Signal Processing (SSP).

[14]  F. Cérou,et al.  Fluctuation Analysis of Adaptive Multilevel Splitting , 2014, 1408.6366.

[15]  Gábor Csányi,et al.  Nested sampling for materials: the case of hard spheres. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  P. Moral,et al.  Particle methods: An introduction with applications , 2014 .

[17]  Pierre Del Moral,et al.  Mean Field Simulation for Monte Carlo Integration , 2013 .

[18]  Antonietta Mira,et al.  Zero variance Markov chain Monte Carlo for Bayesian estimators , 2010, Stat. Comput..

[19]  Arnaud Doucet,et al.  An adaptive sequential Monte Carlo method for approximate Bayesian computation , 2011, Statistics and Computing.

[20]  J. Stoyanov.  The Oxford Handbook of Nonlinear Filtering , 2012 .

[21]  Pierre Del Moral,et al.  Sequential Monte Carlo for rare event estimation , 2012, Stat. Comput..

[22]  P. Moral,et al.  On adaptive resampling strategies for sequential Monte Carlo methods , 2012, 1203.0464.

[23]  David L Wild,et al.  Exploring the energy landscapes of protein folding simulations with Bayesian computation. , 2010, Biophysical journal.

[24]  Dirk P. Kroese,et al.  Efficient Monte Carlo simulation via the generalized splitting method , 2012, Stat. Comput..

[25]  N. Hengartner,et al.  Simulation and Estimation of Extreme Quantiles and Extreme Probabilities , 2011 .

[26]  Anthony N. Pettitt,et al.  Likelihood-free Bayesian estimation of multivariate quantile distributions , 2011, Comput. Stat. Data Anal..

[27]  Dirk P. Kroese,et al.  Handbook of Monte Carlo Methods , 2011 .

[28]  Arnaud Doucet,et al.  Inference for Lévy‐Driven Stochastic Volatility Models via Adaptive Sequential Monte Carlo , 2011 .

[29]  Brendon J. Brewer,et al.  Diffusive nested sampling , 2009, Stat. Comput..

[30]  Paul Fearnhead,et al.  An Adaptive Sequential Monte Carlo Sampler , 2010, 1005.1193.

[31]  C. Robert,et al.  Properties of nested sampling , 2008, 0801.3887.

[32]  Mark A. Girolami,et al.  Bayesian inference for differential equations , 2008, Theor. Comput. Sci..

[33]  A. Pettitt,et al.  Marginal likelihood estimation via power posteriors , 2008 .

[34]  Dirk P. Kroese,et al.  An Efficient Algorithm for Rare-event Probability Estimation, Combinatorial Optimization, and Counting , 2008 .

[35]  L. Koopmans,et al.  Bayesian strong gravitational-lens modelling on adaptive grids: objective detection of mass substructure in Galaxies , 2008, 0805.0201.

[36]  F. Feroz,et al.  Multimodal nested sampling: an efficient and robust alternative to Markov Chain Monte Carlo methods for astronomical data analyses , 2007, 0704.3704.

[37]  A. P. Dawid,et al.  Bayesian Statistics 8 , 2007 .

[38]  Iain Murray.  Advances in Markov chain Monte Carlo methods , 2007 .

[39]  J. Skilling.  Nested sampling for general Bayesian computation , 2006 .

[40]  Zoubin Ghahramani,et al.  Nested sampling for Potts models , 2005, NIPS.

[41]  P. Moral.  Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications , 2004 .

[42]  Michael A. West,et al.  BAYESIAN MODEL ASSESSMENT IN FACTOR ANALYSIS , 2004 .

[43]  P. Moral,et al.  Sequential Monte Carlo samplers , 2002, cond-mat/0212648.

[44]  Tim Hesterberg,et al.  Monte Carlo Strategies in Scientific Computing , 2002, Technometrics.

[45]  N. Chopin.  A sequential particle filter method for static models , 2002 .

[46]  Radford M. Neal.  Annealed importance sampling , 1998, Stat. Comput..

[47]  D. Dittmar.  Slice Sampling , 2000 .

[48]  Michael A. West,et al.  Bayesian Forecasting and Dynamic Models (2nd edn) , 1997, J. Oper. Res. Soc..

[49]  G. Kitagawa.  Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models , 1996 .

[50]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[51]  Reuven Y. Rubinstein,et al.  Simulation and the Monte Carlo method , 1981, Wiley series in probability and mathematical statistics.