×

Variational approximation for importance sampling. (English) Zbl 1505.62391

Summary: We propose an importance sampling algorithm with proposal distribution obtained from variational approximation. This method combines the strength of both importance sampling and variational method. On one hand, this method avoids the bias from variational method. On the other hand, variational approximation provides a way to design the proposal distribution for the importance sampling algorithm. Theoretical justification of the proposed method is provided. Numerical results show that using variational approximation as the proposal can improve the performance of importance sampling and sequential importance sampling.

MSC:

62-08 Computational methods for problems pertaining to statistics
62F15 Bayesian inference
65C05 Monte Carlo methods

Software:

PRMLT
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ali, SM; Silvey, SD, A general class of coefficients of divergence of one distribution from another, J R Stat Soc Ser B, 28, 131-142 (1966) · Zbl 0203.19902
[2] Armagan, A.; Dunson, D., Sparse variational analysis of linear mixed models for large data sets, Stat Probab Lett, 81, 1056-1062 (2011) · Zbl 1219.62045 · doi:10.1016/j.spl.2011.02.029
[3] Beal, MJ; Ghahramani, Z., The variational Bayesian EM algorithm for incomplete data: with application to scoring graphical model structures, Bayesian Stat, 7, 453-464 (2003)
[4] Bishop, CM, Pattern recognition and machine learning (2006), Berlin: Springer, Berlin · Zbl 1107.68072
[5] Blei, DM; Jordan, MI, Variational inference for Dirichlet process mixtures, Bayesian Anal, 1, 121-143 (2006) · Zbl 1331.62259 · doi:10.1214/06-BA104
[6] Blei, DM; Kucukelbir, A.; Mcauliffe, JD, Variational inference: a review for statisticians, J Am Stat Assoc, 112, 859-877 (2017) · doi:10.1080/01621459.2017.1285773
[7] Blei, DM; Ng, AY; Jordan, MI, Latent dirichlet allocation, J Mach Learn Res, 3, 993-1022 (2003) · Zbl 1112.68379
[8] Bugallo, MF; Elvira, V.; Martino, L.; Luengo, D.; Miguez, J.; Djuric, PM, Adaptive importance sampling: the past, the present, and the future, IEEE Signal Process Mag, 34, 60-79 (2017) · doi:10.1109/MSP.2017.2699226
[9] Cappé, O.; Douc, R.; Guillin, A.; Marin, J-M; Robert, CP, Adaptive importance sampling in general mixture classes, Stat Comput, 18, 447-459 (2008) · doi:10.1007/s11222-008-9059-x
[10] Cappé, O.; Guillin, A.; Marin, J-M; Robert, CP, Population Monte Carlo, J Comput Graph Stat, 13, 907-929 (2004) · doi:10.1198/106186004X12803
[11] Dempster, AP; Laird, NM; Rubin, DB, Maximum likelihood from incomplete data via the EM algorithm, J R Stat Soc Ser B, 39, 1-38 (1977) · Zbl 0364.62022
[12] Depraetere, N.; Vandebroek, M., A comparison of variational approximations for fast inference in mixed logit models, Comput Stat, 32, 93-125 (2017) · Zbl 1417.62043 · doi:10.1007/s00180-015-0638-y
[13] Dieng, AB; Tran, D.; Ranganath, R.; Paisley, J.; Blei, D., Variational inference via \(\chi\) upper bound minimization, Adv Neural Inf Process Syst, 30, 2732-2741 (2017)
[14] Doucet, A.; Godsill, S.; Andrieu, C., On sequential Monte Carlo sampling methods for Bayesian filtering, Stat Comput, 10, 197-208 (2000) · doi:10.1023/A:1008935410038
[15] Dowling M, Nassar J, Djurić PM, Bugallo MF (2018) Improved adaptive importance sampling based on variational inference. In: Proceedings of the 26th European signal processing conference (EUSIPCO), IEEE, pp 1632-1636
[16] Hofman, JM; Wiggins, CH, Bayesian approach to network modularity, Phys Rev Lett, 100, 258701 (2008) · doi:10.1103/PhysRevLett.100.258701
[17] Hughes MC, Sudderth E (2013) Memoized online variational inference for Dirichlet process mixture models. Adv Neural Inf Process Syst 1133-1141
[18] Jordan, MI; Ghahramani, Z.; Jaakkola, TS; Saul, LK, An introduction to variational methods for graphical models, Mach Learn, 37, 183-233 (1999) · Zbl 0945.68164 · doi:10.1023/A:1007665907178
[19] Kong A (1992) A note on importance sampling using standardized weights. University of Chicago, Dept. of Statistics, Tech. Rep 348
[20] Kong, A.; Liu, JS; Wong, WH, Sequential imputations and Bayesian missing data problems, J Am Stat Assoc, 89, 278-288 (1994) · Zbl 0800.62166 · doi:10.1080/01621459.1994.10476469
[21] Kullback, S.; Leibler, RA, On information and sufficiency, Ann Math Stat, 22, 79-86 (1951) · Zbl 0042.38403 · doi:10.1214/aoms/1177729694
[22] Liu, JS; Chen, R., Sequential Monte Carlo methods for dynamic systems, J Am Stat Assoc, 93, 1032-1044 (1998) · Zbl 1064.65500 · doi:10.1080/01621459.1998.10473765
[23] Martino, L.; Elvira, V.; Louzada, F., Effective sample size for importance sampling based on discrepancy measures, Signal Process, 131, 386-401 (2017) · doi:10.1016/j.sigpro.2016.08.025
[24] Naesseth C, Linderman S, Ranganath R, Blei D (2018) Variational sequential Monte Carlo. In: Proceedings of the twenty-first international conference on artificial intelligence and statistics, proceedings of machine learning research, pp 968-977
[25] Neal, RM, Annealed importance sampling, Stat Comput, 11, 125-139 (2001) · doi:10.1023/A:1008923215028
[26] Owen AB (2013) Monte Carlo theory, methods and examples. http://statweb.stanford.edu/ owen/mc/
[27] O’Hagan, A.; White, A., Improved model-based clustering performance using Bayesian initialization averaging, Comput Stat, 34, 201-231 (2019) · Zbl 1417.62177 · doi:10.1007/s00180-018-0855-2
[28] Robbins, H.; Monro, S., A stochastic approximation method, Ann Math Stat, 22, 400-407 (1951) · Zbl 0054.05901 · doi:10.1214/aoms/1177729586
[29] Sanguinetti, G.; Lawrence, ND; Rattray, M., Probabilistic inference of transcription factor concentrations and gene-specific regulatory activities, Bioinformatics, 22, 2775-2781 (2006) · doi:10.1093/bioinformatics/btl473
[30] Sason, I.; Verdú, S., \(f\)-divergence inequalities, IEEE Trans Inf Theory, 62, 5973-6006 (2016) · Zbl 1359.94363 · doi:10.1109/TIT.2016.2603151
[31] Wang P, Blunsom P (2013) Collapsed variational Bayesian inference for hidden Markov models. AISTATS 599-607
[32] Xing EP, Jordan MI, Russell S (2002) A generalized mean field algorithm for variational inference in exponential families. In: Proceedings of the nineteenth conference on uncertainty in artificial intelligence, Morgan Kaufmann Publishers Inc., pp 583-591
[33] You, C.; Ormerod, JT; Mueller, S., On variational Bayes estimation and variational information criteria for linear regression models, Aust N Z J Stat, 56, 73-87 (2014) · Zbl 1334.62123 · doi:10.1111/anzs.12063
[34] Zreik, R.; Latouche, P.; Bouveyron, C., The dynamic random subgraph model for the clustering of evolving networks, Comput Stat, 32, 501-533 (2017) · Zbl 1417.62182 · doi:10.1007/s00180-016-0655-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.