New insights into approximate bayesian computation

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 51, Issue 1, 2015, Pages 376-403

  Biau, Gérard ^a   Cérou, Frédéric ^b   Guyader, Arnaud ^c  

^a SORBONNE UNIVERSITÉ   (France)

^b INRIA   (France)

^c UNIVERSITÉ RENNES 2   (France)

Author keywords

Approximate Bayesian Computation; Conditional density estimation; Mathematical statistics; Nearest neighbor methods; Nonparametric estimation

Indexed keywords

EID: 84920937418     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1214/13-AIHP590     Document Type: Article

References (55)

1. I. S. Abramson. On bandwidth variation in kernel estimates - A square root law. Ann. Statist. 10 (1982) 1217-1223. MR0673656
2. D. M. Bashtannyk and R. J. Hyndman. Bandwidth selection for kernel conditional density estimation. Comput. Statist. Data Anal. 36 (2001) 279-298. MR1836204
3. M. Beaumont, J.-M. Cornuet, J.-M. Marin and C. P. Robert. Adaptive approximate Bayesian computation. Biometrika 96 (2009) 983-990. MR2767283
4. M. A. Beaumont, W. Zhang and D. J. Balding. Approximate Bayesian computation in population genetics. Genetics 162 (2002) 2025-2035.
5. G. Biau, F. Cérou and A. Guyader. On the rate of convergence of the bagged nearest neighbor estimate. J. Mach. Learn. Res. 11 (2010) 687-712. MR2600626
6. M. Blum. Approximate Bayesian computation: A nonparametric perspective. J. Amer. Statist. Assoc. 105 (2010) 1178-1187. MR2752613
7. L. Breiman, W. Meisel and E. Purcell. Variable kernel estimates of multivariate densities. Technometrics 19 (1977) 135-144.
8. F. Cérou and A. Guyader. Nearest neighbor classification in infinite dimension. ESAIM Probab. Stat. 10 (2006) 340-355. MR2247925
9. T. M. Cover. Estimation by the nearest neighbor rule. IEEE Trans. Inform. Theory 14 (1968) 50-55.
10. M. de Guzmán. Differentiation of Integrals in Rn. Lecture Notes in Mathematics 481. Springer, Berlin, 1975. MR0457661
11. L. Devroye. Necessary and sufficient conditions for the pointwise convergence of nearest neighbor regression function estimates. Z. Wahrsch. Verw. Gebiete 61 (1982) 467-481. MR0682574
12. L. Devroye and A. Krzyzak. New multivariate product density estimates. J. Multivariate Anal. 82 (2002) 88-110. MR1918616
13. L. Devroye, L. Györfi and G. Lugosi. A Probabilistic Theory of Pattern Recognition. Springer, New York, 1996. MR1383093
14. J. Fan and T. H. Yim. A crossvalidation method for estimating conditional densities. Biometrika 91 (2004) 819-834. MR2126035
15. O. P. Faugeras. A quantile-copula approach to conditional density estimation. J. Multivariate Anal. 100 (2009) 2083-2099. MR2543088
16. P. Fearnhead and D. Prangle. Constructing summary statistics for approximate Bayesian computation: Semi-automatic approximate Bayesian computation. J. Roy. Statist. Soc. Ser. B 74 (2012) 419-474. MR2925370
17. E. Fix and J. L. Hodges. Discriminatory analysis - Nonparametric discrimination: Consistency properties. Project 21-49-004, Report Number 4, USAF School of Aviation Medicine, Randolph Field, TX, 1951.
18. Y. X. Fu and W. H. Li. Estimating the age of the common ancestor of a sample of DNA sequences. Mol. Biol. Evol. 14 (1997) 195-199.
19. L. Györfi and M. Kohler. Nonparametric estimation of conditional distributions. IEEE Trans. Inform. Theory 53 (2007) 1872-1879. MR2317148
20. P. Hall and J. S. Marron. Variable window width kernel estimates of probability densities. Probab. Theory Related Fields 80 (1988) 37-49. MR0970470
21. P. Hall, J. Racine and Q. Li. Cross-validation and the estimation of conditional probability densities. J. Amer. Statist. Assoc. 99 (2004) 1015-1026. MR2109491
22. B. H. Hansen. Nonparametric conditional density estimation. Technical report, Univ. Wisconsin, 2004.
23. G. H. Hardy, J. E. Littlewood and G. Pólya. Inequalities. Cambridge Univ. Press, Cambridge, 1988. MR0944909
24. W. K. Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 (1970) 97-109.
25. R. J. Hyndman, D. M. Bashtannyk and G. K. Grunwald. Estimating and visualizing conditional densities. J. Comput. Graph. Statist. 5 (1996) 315-336. MR1422114
26. B. Jessen, J. Marcinkiewicz and A. Zygmund. Note on the differentiability of multiple integrals. Fund. Math. 25 (1935) 217-234.
27. M. C. Jones. Variable kernel density estimates and variable kernel density estimates. Aust. J. Stat. 32 (1990) 361-371. MR1098587
28. P. Joyce and P. Marjoran. Approximately sufficient statistics and Bayesian computation. Stat. Appl. Genet. Mol. Biol. 7 (2008) Art. ID 26. MR2438407
29. E. Kaufmann and R.-D. Reiss. On conditional distributions of nearest neighbors. J. Multivariate Anal. 42 (1992) 67-76. MR1177518
30. D. O. Loftsgaarden and C. P. Quesenberry. A nonparametric estimate of a multivariate density function. Ann. Math. Statist. 36 (1965) 1049-1051. MR0176567
31. Y. P. Mack and M. Rosenblatt. Multivariate k-nearest neighbor density estimates. J. Multivariate Anal. 9 (1979) 1-15. MR0530638
32. J. M. Marin and C. P. Robert. Bayesian Core: A Practical Approach to Computational Bayesian Statistics. Springer, New York, 2007. MR2289769
33. J. M. Marin, N. Pillai, C. P. Robert and J. Rousseau. Relevant statistics for Bayesian model choice. J. R. Stat. Soc. Ser B. To appear, 2014.
34. J. M. Marin, P. Pudlo, C. P. Robert and R. Ryder. Approximate Bayesian computational methods. Stat. Comput. 22 (2012) 1167-1180. MR2992292
35. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller. Equations of state calculations by fast computing machines. J. Chem. Phys. 21 (1953) 1087-1091.
36. D. S. Moore and J. W. Yackel. Consistency properties of nearest neighbor density function estimators. Ann. Statist. 5 (1977) 143-154. MR0426275
37. D. S. Moore and J. W. Yackel. Large sample properties of nearest neighbor density function estimators. In Statistical Decision Theory and Related Topics II: Proceedings of a Symposium Held at Purdue University, May 17-19, 1976, S. S. Gupta and D. S. Moore (Eds) 269-279. Academic Press, New York, 1977. MR0431497
38. E. A. Nadaraya. On estimating regression. Theory Probab. Appl. 9 (1964) 141-142.
39. E. A. Nadaraya. On nonparametric estimates of density functions and regression curves. Theory Probab. Appl. 10 (1965) 186-190.
40. E. Parzen. On the estimation of a probability density function and the mode. Ann. Math. Statist. 33 (1962) 1065-1076. MR0143282
41. J. K. Pritchard, M. T. Seielstad, A. Perez-Lezaun and M. W. Feldman. Population growth of human Y chromosomes: A study of Y chromosome microsatellites. Mol. Biol. Evol. 16 (1999) 1791-1798.
42. B. D. Ripley. Stochastic Simulation. Wiley, New York, 1982. MR2299137
43. C. P. Robert and G. Casella. Monte Carlo Statistical Methods, 2nd edition. Springer, New York, 2004. MR2080278
44. C. P. Robert, J.-M. Cornuet, J.-M. Marin and N. S. Pillai. Lack of confidence in approximate Bayesian computation model choice. Proc. Natl. Acad. Sci. USA 108 (2011) 15112-15117.
45. M. Rosenblatt. Conditional probability density and regression estimates. In Multivariate Analysis II, P. R. Krishnaiah (Ed.) 25-31. Academic Press, New York, 1969. MR0254987
46. R. M. Royall. A class of non-parametric estimates of a smooth regression function. Ph.D. Thesis, Stanford Univ., 1966. MR2615964
47. D. Rubin. Bayesianly justifiable and relevant frequency calculations for the applied statistician. Ann. Statist. 12 (1984) 1151-1172. MR0760681
48. S. A. Sisson, Y. Fan and M. M. Tanaka. Sequential Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. USA 104 (2007) 1760-1765. MR2301870
49. E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton, 1970. MR0290095
50. C. J. Stone. Consistent nonparametric regression. Ann. Statist. 5 (1977) 595-645. MR0443204
51. S. Tavaré, D. Balding, R. Griffith and P. Donnelly. Inferring coalescence times from DNA sequence data. Genetics 145 (1997) 505-518.
52. G. S. Watson. Smooth regression analysis. Sankhya A 26 (1964) 359-372. MR0185765
53. R. L. Wheeden and A. Zygmund. Measure and Integral. An Introduction to Real Analysis. Marcel Dekker, New York, 1977. MR0492146
54. R. D. Wilkinson. Approximate Bayesian computation (ABC) gives exact results under the assumption of model error. Stat. Appl. Genet. Mol. Biol. 12 (2008) 129-141. MR3071024
55. A. Zygmund. Trigonometric Series, Vol. II. Cambridge Univ. Press, Cambridge, 1959. MR0107776