Filtering via approximate Bayesian computation

Approximate Bayesian computation (ABC) has become a popular technique to facilitate Bayesian inference from complex models. In this article we present an ABC approximation designed to perform biased filtering for a Hidden Markov Model when the likelihood function is intractable. We use a sequential Monte Carlo (SMC) algorithm to both fit and sample from our ABC approximation of the target probability density. This approach is shown to, empirically, be more accurate w.r.t. the original filter than competing methods. The theoretical bias of our method is investigated; it is shown that the bias goes to zero at the expense of increased computational effort. Our approach is illustrated on a constrained sequential lasso for portfolio allocation to 15 constituents of the FTSE 100 share index.

[1]  R. C. Merton,et al.  On Estimating the Expected Return on the Market: An Exploratory Investigation , 1980 .

[2]  W. Sharpe,et al.  Mean-Variance Analysis in Portfolio Choice and Capital Markets , 1987 .

[3]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[4]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[5]  Yoram Singer,et al.  On‐Line Portfolio Selection Using Multiplicative Updates , 1998, ICML.

[6]  Jeffrey K. Uhlmann,et al.  New extension of the Kalman filter to nonlinear systems , 1997, Defense, Security, and Sensing.

[7]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[8]  M. Feldman,et al.  Population growth of human Y chromosomes: a study of Y chromosome microsatellites. , 1999, Molecular biology and evolution.

[9]  T. Kozubowski,et al.  Multivariate geometric stable distributions in financial applications , 1999 .

[10]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[11]  Duan Li,et al.  Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation , 2000 .

[12]  Neil J. Gordon,et al.  Editors: Sequential Monte Carlo Methods in Practice , 2001 .

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

[14]  Jun S. Liu,et al.  Monte Carlo strategies in scientific computing , 2001 .

[15]  Paul Marjoram,et al.  Markov chain Monte Carlo without likelihoods , 2003, Proceedings of the National Academy of Sciences of the United States of America.

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

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

[18]  G. Pagès,et al.  Optimal quantization methods for nonlinear filtering with discrete-time observations , 2005 .

[19]  Eric Moulines,et al.  Inference in hidden Markov models , 2010, Springer series in statistics.

[20]  A. Doucet,et al.  Efficient Block Sampling Strategies for Sequential Monte Carlo Methods , 2006 .

[21]  Jean-Michel Marin,et al.  DATA PROCESSING : COMPARISON OF BAYESIAN REGULARIZED PARTICLE FILTERS by Roberto Casarin , 2007 .

[22]  F. Fabozzi.  Robust Portfolio Optimization and Management , 2007 .

[23]  Michael A. West,et al.  Dynamic matrix-variate graphical models , 2007 .

[24]  A. Doucet,et al.  A Tutorial on Particle Filtering and Smoothing: Fifteen years later , 2008 .

[25]  Václav Smídl,et al.  Variational Bayesian Filtering , 2008, IEEE Transactions on Signal Processing.

[26]  A. Lo.  Hedge Funds: An Analytic Perspective , 2008 .

[27]  P. Moral,et al.  On Adaptive Resampling Procedures for Sequential Monte Carlo Methods , 2008 .

[28]  M. Pascual,et al.  Inapparent infections and cholera dynamics , 2008, Nature.

[29]  David Madigan,et al.  Algorithms for Sparse Linear Classifiers in the Massive Data Setting , 2008 .

[30]  J. Marin,et al.  Adaptivity for ABC algorithms: the ABC-PMC scheme , 2008 .

[31]  Christoforos Anagnostopoulos,et al.  Online optimization for variable selection in data streams , 2008, ECAI.

[32]  Raman Uppal,et al.  A Generalized Approach to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms , 2009, Manag. Sci..

[33]  C. Robert,et al.  Adaptive approximate Bayesian computation , 2008, 0805.2256.

[34]  F. Campillo,et al.  Convolution Particle Filter for Parameter Estimation in General State-Space Models , 2009, IEEE Transactions on Aerospace and Electronic Systems.

[35]  Victor DeMiguel,et al.  Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? , 2009 .

[36]  A. Doucet,et al.  Smoothing algorithms for state–space models , 2010 .

[37]  D. Hunter.  Dynamic mean-variance portfolio analysis under model risk , 2009 .

[38]  Dimitrios I. Fotiadis,et al.  Bayesian Methods for fMRI Time-Series Analysis Using a Nonstationary Model for the Noise , 2010, IEEE Transactions on Information Technology in Biomedicine.

[39]  Arnaud Doucet,et al.  On the Utility of Graphics Cards to Perform Massively Parallel Simulation of Advanced Monte Carlo Methods , 2009, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

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

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