Editing Importance sampling (section)

===Multiple and adaptive importance sampling ===
When different proposal distributions, <math>g_i(x)</math> , <math>i=1,\ldots,n,</math> are jointly used for drawing the samples <math>x_1, \ldots, x_n, </math> different proper weighting functions can be employed (e.g., see <ref>{{Cite book|publisher = ACM|date = 1995-01-01|location = New York, NY, USA|isbn = 978-0-89791-701-8|pages = [https://archive.org/details/computergraphics00sigg/page/419 419–428]|doi = 10.1145/218380.218498|first1 = Eric|last1 = Veach|first2 = Leonidas J.|last2 = Guibas| title=Proceedings of the 22nd annual conference on Computer graphics and interactive techniques - SIGGRAPH '95 | chapter=Optimally combining sampling techniques for Monte Carlo rendering |citeseerx = 10.1.1.127.8105| s2cid=207194026 |chapter-url = https://archive.org/details/computergraphics00sigg/page/419}}</ref><ref>{{Cite journal|title = Safe and Effective Importance Sampling|journal = Journal of the American Statistical Association|date = 2000-03-01|issn = 0162-1459|pages = 135–143|volume = 95|issue = 449|doi = 10.1080/01621459.2000.10473909|first1 = Art|last1 = Owen|first2 = Yi Zhou|last2 = Associate|citeseerx = 10.1.1.36.4536| s2cid=119761472 }}</ref><ref>{{Cite journal|title = Efficient Multiple Importance Sampling Estimators|journal = IEEE Signal Processing Letters|date = 2015-10-01|issn = 1070-9908|pages = 1757–1761|volume = 22|issue = 10|doi = 10.1109/LSP.2015.2432078|first1 = V.|last1 = Elvira|first2 = L.|last2 = Martino|first3 = D.|last3 = Luengo|first4 = M.F.|last4 = Bugallo|arxiv = 1505.05391|bibcode = 2015ISPL...22.1757E| s2cid=14504598 }}</ref><ref>{{Cite journal|last1=Elvira|first1=Víctor|last2=Martino|first2=Luca|last3=Luengo|first3=David|last4=Bugallo|first4=Mónica F.|title=Improving population Monte Carlo: Alternative weighting and resampling schemes|journal=Signal Processing|volume=131|pages=77–91|doi=10.1016/j.sigpro.2016.07.012|arxiv=1607.02758|year=2017|s2cid=205171823 }}</ref>). In an adaptive setting, the proposal distributions, <math>g_{i,t}(x)</math> , <math>i=1,\ldots,n,</math> and <math>t=1,\ldots,T,</math> are updated each iteration <math>t</math> of the adaptive importance sampling algorithm. Hence, since a population of proposal densities is used, several suitable combinations of sampling and weighting schemes can be employed.<ref>{{Cite journal|title = Population Monte Carlo|journal = Journal of Computational and Graphical Statistics|date = 2004-12-01|issn = 1061-8600|pages = 907–929|volume = 13|issue = 4|doi = 10.1198/106186004X12803|first1 = O.|last1 = Cappé|first2 = A.|last2 = Guillin|first3 = J. M.|last3 = Marin|first4 = C. P.|last4 = Robert| s2cid=119690181 }}</ref><ref>{{Cite journal|last1=Martino|first1=L.|last2=Elvira|first2=V.|last3=Luengo|first3=D.|last4=Corander|first4=J.|date=2017-05-01|title=Layered adaptive importance sampling|journal=Statistics and Computing|language=en|volume=27|issue=3|pages=599–623|doi=10.1007/s11222-016-9642-5|issn=0960-3174|arxiv=1505.04732|s2cid=2508031 }}</ref><ref>{{Cite journal|title = Adaptive importance sampling in general mixture classes|journal = Statistics and Computing|date = 2008-04-25|issn = 0960-3174|pages = 447–459|volume = 18|issue = 4|doi = 10.1007/s11222-008-9059-x|first1 = Olivier|last1 = Cappé|first2 = Randal|last2 = Douc|first3 = Arnaud|last3 = Guillin|first4 = Jean-Michel|last4 = Marin|first5 = Christian P.|last5 = Robert|arxiv = 0710.4242| s2cid=483916 }}</ref><ref>{{Cite journal|title = Adaptive Multiple Importance Sampling|journal = Scandinavian Journal of Statistics|date = 2012-12-01|issn = 1467-9469|pages = 798–812|volume = 39|issue = 4|doi = 10.1111/j.1467-9469.2011.00756.x|first1 = Jean-Marie|last1 = Cornuet|first2 = Jean-Michel|last2 = Marin|first3 = Antonietta|last3 = Mira|author3-link=Antonietta Mira|first4 = Christian P.|last4 = Robert|arxiv = 0907.1254| s2cid=17191248 }}</ref><ref>{{Cite journal|title = An Adaptive Population Importance Sampler: Learning From Uncertainty|journal = IEEE Transactions on Signal Processing|date = 2015-08-01|issn = 1053-587X|pages = 4422–4437|volume = 63|issue = 16|doi = 10.1109/TSP.2015.2440215|first1 = L.|last1 = Martino|first2 = V.|last2 = Elvira|first3 = D.|last3 = Luengo|first4 = J.|last4 = Corander|bibcode = 2015ITSP...63.4422M|citeseerx = 10.1.1.464.9395| s2cid=17017431 }}</ref><ref>{{Cite journal|title = Adaptive importance sampling in signal processing|journal = Digital Signal Processing|date = 2015-12-01|pages = 36–49|volume = 47|series = Special Issue in Honour of William J. (Bill) Fitzgerald|doi = 10.1016/j.dsp.2015.05.014|first1 = Mónica F.|last1 = Bugallo|first2 = Luca|last2 = Martino|first3 = Jukka|last3 = Corander|doi-access = free}}</ref><ref>{{Cite journal|last1=Bugallo|first1=M. F.|last2=Elvira|first2=V.|last3=Martino|first3=L.|last4=Luengo|first4=D.|last5=Miguez|first5=J.|last6=Djuric|first6=P. M.|date=July 2017|title=Adaptive Importance Sampling: The past, the present, and the future|journal=IEEE Signal Processing Magazine|volume=34|issue=4|pages=60–79|doi=10.1109/msp.2017.2699226|issn=1053-5888|bibcode=2017ISPM...34...60B|s2cid=5619054 }}</ref>