Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene...
Transcript of Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene...
![Page 1: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/1.jpg)
Applica'onsofChernoffBound
BarnaSaha
![Page 2: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/2.jpg)
Es'ma'ngSampleSize
• Letpbetheunknownprobabilitythatagenemutates.
• En'redatasetsize=N• Samplesize=n• Inthesampleofthemhavebeenmutated• Es'matedprobabilityofmuta'on Isthisareliable
es,mate?
![Page 3: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/3.jpg)
Whenisareliablees'mate?
• Mustsa'sfy
• Or, Confidenceparameter
Errortolerance
![Page 4: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/4.jpg)
Es'ma'ngSampleSize
• Defineindicatorrandomvariableswhichis1ifthei-thsampledelementhasthedesiredproperty(muta'on/i-phone8query..)and0otherwise.
![Page 5: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/5.jpg)
Es'ma'ngSampleSize
• Wehave
![Page 6: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/6.jpg)
Es'ma'ngSampleSize
• • Wewant
![Page 7: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/7.jpg)
Es'ma'ngSampleSize
• • Wewant
![Page 8: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/8.jpg)
Es'ma'ngSampleSize
• • Wewant
![Page 9: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/9.jpg)
Repea'ngReservoirSampling
![Page 10: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/10.jpg)
Repea'ngReservoirSampling
• Numberofitems=100• Accordingtotheproofofthereservoirsamplingeachitemhas1/100chanceofbeingstoredinthereservoir
• Considerthe1stitemanddefineanindicatorrandomvariablewhichis1ifthethe1stitemisstoredattheendofthealgorithmonitsi-thrun.
• Werunthealgorithm“m”'mes.
![Page 11: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/11.jpg)
Repea'ngReservoirSampling
• Define
• Wewantthefrequencyofallitemstobewithinwithhighprobability.
![Page 12: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/12.jpg)
Repea'ngReservoirSampling
• Wewantthefrequencyofallitemstobewithinwithhighprobability.
• Forthe1stitem
![Page 13: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/13.jpg)
Repea'ngReservoirSampling
• Wewantthefrequencyofallitemstobewithinwithhighprobability.
• Forthe1stitem
• Forthe2nditem
• Forthe100thitem
![Page 14: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/14.jpg)
Repea'ngReservoirSampling
• Wewantthefrequencyofallitemstobewithinwithhighprobability.
• Probthereexistsatleastoneitemoutof100suchthatitsfrequencyisnotintherangeis
![Page 15: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/15.jpg)
Chernoff+UnionBound
• RandomLoadBalancing– SupposeacontentdeliverynetworklikeYouTubereceivesamillioncontentrequestsperminute.Eachrequestneedstobeservedfromoneofthe1000servers.Howshouldonedistributetheloadsothatnoserverisoverloaded.
![Page 16: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/16.jpg)
Chernoff+UnionBound
• RandomLoadBalancing– SupposeacontentdeliverynetworklikeYouTubereceivesamillioncontentrequestsperminute.Eachrequestneedstobeservedfromoneofthe1000servers.Howshouldonedistributetheloadsothatnoserverisoverloaded.
– Assigneachrequesttoarandomserver.
![Page 17: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/17.jpg)
RandomLoadBalancing
• Lettherebe“n”requestsand“k”servers• Considerserver“i”• Defineanindicatorrandomvariablewhichwillbe1ifrequestjisassignedtoserveriand0otherwise.
• Loadonmachinei:
• [ApplytheChernoffBound]
![Page 18: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/18.jpg)
RandomLoadBalancing
• ApplyingtheChernoffBoundweget
![Page 19: Applicaons of Chernoff BoundEs’mang Sample Size • Let p be the unknown probability that a gene mutates. • En’re dataset size=N • Sample size=n • In the sample of them](https://reader034.fdocuments.in/reader034/viewer/2022042306/5ed1b7c276a00015387f8edc/html5/thumbnails/19.jpg)
RandomLoadBalancing
• ApplyUnionBound
• Prob(thereexistsatleastoneserverwhichisoverloaded)