
metropolis-hustings algoritam

statistiks, stattistikkal physics ennivayil, metropolis-hustings algoritam ennathu markov cheyin monde carlo (mcmc) reethiyaanu, athil ninnu nerittu sample edukkaan prayaasamulla oru probability vitharanathil ninnu kramarahithamaya saambilukalude oru shreni nedunnathinulla reethiyaanu. eee kramam vitharanam ekadeshamaayi kanakkakkaan upayogikkaam (udha.jii. oru histograam srishtikkaan) allengil oru integral kanakkakkaan (udha.jii. prathekshikkunna moolyam). metropolis-haastingukalum mattu emm. sii. emm. sii algorithangalum saadhaaranayaayi multy-dimensional vitharanagalil ninnu sample edukkaan upayogikkunnu, prathyekichu alavukalude ennam kooduthalaayirikkumbol. ekamaana vitharanagalkk saadhaaranayaayi mattu reethikalundu (udha.jii. vitharanathil ninnu nerittu swatantra saambilukal thirike nalkaan kazhiyunna adaptive rejection sample), iva mcmc reethikalil antharleenamaaya autocorylated saambilukalude prashnatthil ninnu mukthamaan.
charithram.
1953-ll "phaast computing mesheenukalude avastha kanakkukoottalukalude samavaakyam" enna thalakkettilulla oru prabandhatthinte aadya saha rachayithavaaya nicolas metropolisinte perilaanu algoritam ariyappedunnathu. rosenblath, marshell rosenblath, augusta echu. teller edward teller. varshangalolam eee algoritam "metropolis algoritam" ennaanu ariyappettirunnath. samamita nirddesha vitharanagalude kaaryatthil paper algoritam nirddeshichu, pakshe 1970 ll, dablyu.keey. hashings ithu kooduthal pothuvaaya kesilekk vyapippichu. metropolis-hustings algoritam" enna padatthinte aadya upayogam vyakthamallengilum, pothuvalkkaricha reethi oduvil rund perukalilum thiricharinju.
metropolis algoritam vikasippicheduthathint kredittumaayi bandhappett chila vivaadangal nilanilkkunnu. eee reethiyude computational vashangal parichithamaaya metropolis, staanislaav ulamumaayi oru muun lekhanathil "monde carlo" enna padam upayogikkukayum 1952 ll pareekshanangalil upayogicha bhraanthan aii computer roopakalppana cheyyukayum nirmmikkukayum cheytha saidhaanthika vibhaagathil groupine nayikkukayum cheythu. ennirunnaalum, 2003 nuu munbu algoritam vikasippichathinekkurich vishadamaaya vivaranam undaayirunnilla. maranathinu thottumunbu, marshell rosenblath 1953 leeii prasidheekaranatthinte ambathaam vaarshikathodanubandhichu lanlil nadanna 2003 leeii oru sammelanathil pankedutthu. eee konfaransil, "stattistikkal mekkaniksinulla monde carlo algorithathinte uthbhavam" enna thalakkettilulla oru avatharanathil rosenblath algorithatheyum athinte vikasanatheyum vivarichu. 50-aamm vaarshikasammelanam vivarikkunna 2005-leeii oru jenal lekhanathil govarnattis kooduthal charitraparamaaya vyakthatha nalkunnu. taanum bhaarya ariana chernnaanu eee jolie cheythathennum computer samayam nalkukayallaathe vikasanathil metropolis oru pankum vahichitillennum rosenblath vyakthamaakkunnu.
1953 leeii lekhanathile anchu rachayithaakkalum "pakalukalum raathrikalum" orumichu pravarthichuvennu thante ormmakkurippukalil prasthaavikkunna edward tellarinte oru vivaranathinu ithu virudhamaanu. ithinu vipareethamaayi, rosenblathinte vishadamaaya vivaranam "vishadamaaya kainamattiks pinthudarunnathinupakaram stattistikkal mechanics prayojanappedutthaanum samyojitha sharaasharikal edukkaanum" nirnaayakavum ennaal aadyakaala nirddeshavumaayi tellarine credit cheiyunnu. ithu, joonn voonn nyoomaanumaayi palappozhum charcha cheythathayi addeham parayunna oru vishayamaaya monde carlo sameepanathekkurichu chinthikkaan rosenblath paranju. augusta teller computer pravartthanam aarambhichathayi eriyaana rosenblath (2003 ll govarnattisinode) vivarichu, ennaal eriyaana thanne athu ettedukkukayum aadyam muthal coode ezhuthukayum cheythu. thante maranathinu thottumunbu rekhappeduthappetta oru vaakkalulla charithratthil, yadhaartha prashnam avatharippichathinum athu pariharichathinum computer prograaming cheythathinum rosenblath veendum tellare credit cheiyunnu.
avabodham.
density formula _ 3 nuu aanupaathikamaaya oru function formula _ 2 namukkariyaamenkil, formula _ 2-nte moolyangal kanakkakkaan kazhiyumenkil, metropolis-hustings algorithathinu probability density formula _ 1 upayogichu eathu probability vitharanathil ninnum saambilukal edukkaan kazhiyum. formula _ 2 saandrathaykku aanupaathikamaayirikkanam enna aavasyakatha, athinu krithyamaayi thulyamallaathathinaal, metropolis-husting algoritam prathyekichum upayogapradamaakkunnu, kaaranam aavashyamaaya normalisation ghadakam kanakkaakkunnath praayogikamaayi valare budhimuttaanu.
metropolis-hustings algoritam sample moolyangalude oru shreni srishtikkunnu, kooduthal kooduthal sample moolyangal uthppaadippikkappedum, moolyangalude vitharanam aavasyamulla vitharanathe kooduthal aduthu kanakkakkunnu. eee sample moolyangal aavarthichu nirmmikkunnu, adutha saambilinte vitharanam nilavile sample moolyathe maathram aasrayichirikkunnu, angane saambilukalude kramam oru markov shringalayaakkunnu. prathyekichum, oro aavarthanathilum, nilavile sample moolyathe adisthaanamaakki adutha sample moolyathinaayi algoritam oru sthaanaarthiye thiranjedukkunnu. thudarnnu, kurachu saadhyathayode, sthaanaarthi onnukil amgeekarikkappedunnu, eee saahacharyathil adutha aavarthanathil sthaanaarthi moolyam upayogikkunnu, allengil athu nirasikkappedunnu, eee saahacharyathil sthaanaarthi moolyam upekshikkukayum adutha aavarthanathil nilavile moolyam veendum upayogikkukayum cheiyunnu. aavasyamulla vitharanavumayi bandhappett karant, candidate sample moolyangalude function formula _ 2-nte moolyangal thaarathamyam cheythaanu sveekaaryathayude saadhyatha nirnnayikkunnath.
thithreekaranatthinaayi, metropolis-hustings algoritam enna prathyeka cases, nirddesha pravartthanam samamithiyulla metropolis-hustings algoritam, chuvade vivarichirikkunnu.
formula _ 2 aavasyamulla probability density function formula _ 1 (a) nuu aanupaathikamaaya oru functionayirikkatte.keey.eiii. oru lakshyam vitharanam).
sample sthalathekkurichu kramarahithamayi neengaan sramikkunnathiloodeyum chilappol neekkangal sweekarikkunnathiloodeyum chilappol sthaanathu thudarunnathiloodeyum eee algoritam munnott pookunnu. nilavile saambilumaayi bandhappett, nilavilulla pooyintinekkaal kooduthal saadhyathayulla oru poyintilekku neengaan nammal shramikkukayaanenkil, nilavile saambilumaayi bandhappett, density formula 1 aaya vitharanamanusarichu, puthiya nirddishta sample ethratholam saadhyamaanennu 29 enna sveekaaryatha anupatha formula soochippikkunnuvennu shradhikkuka.iii. oru formula _ 32) nuu anusrithamaaya formula _ 1-nte uyarnna saandrathayulla mekhalayile oru pooyint, njangal ellaayppozhum neekkam sweekarikkum. ennirunnaalum, saadhyathayulla oru poyintilekku neengaan shramikkukayaanenkil, chilappol nammal eee neekkam nirsikkum, saadhyathayil aapekshikamaya idivu koodumbol, puthiya pooyint nirasikkanulla saadhyatha kooduthalaanu. athinaal, nammal formula _ 1-nte uyarnna saandrathayulla pradeshangalil thaamasikkukayum (dhaaraalam saambilukal thirike nalkukayum cheyyum), athesamayam idaykkide kuranja saandrathayulla pradeshangal maathrame sandarshikkukayulloo. avabodhapoorvvam, athukondaanu eee algoritam pravartthikkukayum saandrata soothravaakyam _ 1 upayogichu aavasyamulla vitharanam pinthudarunna saambilukal nalkukayum cheiyunnathu.
oru vitharanathil ninnu nerittu swatantra saambilukal srishtikkunna adaptive rejection sample polulla oru algorithavumaayi thaarathamyappeduthumbol, metropolis-hastingukalkkum mattu mcmc algorithangalkkum niravadhi poraaymakalundu
maruvashathu, ettavum lalithamaaya nirasikkal sample reethikal "alavukalude saapam" anubavikkunnu, avide nirasikkanulla saadhyatha alavukalude ennathinte pravartthanamenna nilayil ganyamaayi vardhikkunnu. metropolis-hastingukalkkum mattu emm. sii. emm. sii reethikalkkum eee prashnam athrayilla, athinaal sample cheyyenda vitharanathinte alavukalude ennam kooduthalaayirikkumbol palappozhum labhyamaya eaka parihaaramaanith. thalphalamaayi, hierarchical bayeshyan modalukalil ninnum mattu uyarnna alavilulla stattistikkal modalukalil ninnum saambilukal nirmmikkunnathinulla thiranjedukkunna reethikalaanu mcmc reethikal.
multiveriyettu vitharanagalil, mukalil vivarichathupole classic metropolis-hustings algorithathil oru puthiya multy dimensional sample pooyint thiranjedukkunnathu ulppedunnu. alavukalude ennam kooduthalaayirikkumbol, upayogikkaan anuyojyamaaya jumping vitharanam kandethunnathu budhimuttaayirikkum, kaaranam vyathyastha vyakthigatha alavukal valare vyathyasthamaaya reethikalil pravartthikkunnu, koodaathe amithamaya mandagathiyilulla mixing ozhivakkaan ellah alavukalkkum jumping veethi (mukalil kaanuka) oresamayam "sariyaaya" aayirikkanam. atharam saahacharyangalil palappozhum mikacha reethiyil pravartthikkunna oru badal sameepanamaaya gibbs sampling, ellah alavukalkkum oresamayam oru sample thiranjedukkunnathinupara oro alavinum mattullavaril ninnu vevvere oru puthiya sample thiranjedukkunnathu ulppedunnu. athuvazhi, uyarnna alavilulla sthalathu ninnu sample edukkunna prashnam cheria alavilulla saambilukalude prashnangalude oru shekharamaayi kurayum. mikka saadhaarana sreniparamaaya modalukalileyum pole, oro variable mattu veriyabilukalude oru cheria ennam maathram vyavastha cheiyunna vyakthigatha kramarahithamaya veriyabilukalude oru koottam multiveriyettu vitharanam ulkkollumbol ithu prathyekichum baadhakamaan. oro variable mattullavayude ettavum puthiya moolyangalil vyavastha cheythukondu vyakthigatha variableukal oru samayam onnil onnu sample cheiyunnu. multiveriyettu vitharanathinte krithyamaaya roopathe aashrayich eee vyakthigatha saambilukal thiranjedukkaan vividha algorithangal upayogikkaam adaptive rejection sample reethikal, adaptive rejection metropolis sample algoritam, lalithamaaya oru dimensional metropolis-haasting steppu, allengil slies sample ennivayaanu chila saadhyathakal.
aupachaarikamaaya uthbhavam.
metropolis-hustings algorithathinte uddeshyam aavasyamulla vitharana soothravaakyam _ 1 anusarichu avasthakalude oru shekharam srishtikkuka ennathaanu. ithu nirvahikkunnathin, algoritam oru markov prakriya upayogikkunnu, ithu lakshanangalillaatha oru advitheeya nischala vitharana soothravaakyathil athunnu.
oru markov prakriya athinte parivartthana saadhyathakalude formula _ 39 kondu nirvachikkunnu, eathenkilum stettu formula _ 12 ll ninnu mattethenkilum stettu formula _ 20 lake maranullah saadhyatha. inipparayunna rund vyavasthakal paalikkumbol ithinu oru advitheeya nischala vitharana formula _ 37 undu
metropolis-hustings algoritam melpparanja rund vyavasthakal niravettunna oru markov prakriya (parivartthana saadhyathakal nirmmikkunnathiloode) roopakalppana cheiyunnathu ulkkollunnu, athaayath athinte nischala vitharana soothravaakyam _ 37 formula _ 1 aayi thiranjedukkunnu. algorithathinte derivation vishadamaaya balance vyavasthayil ninnaan aarambhikkunnath
ithu veendum ezhuthappettirikkunnathu
parivartthanatthe rund upa ghattangalayi verthirikkuka ennathaanu sameepanam; nirddeshavum sveekaaryatha-nirasikalum. nirddesha vitharana soothravaakyam _ 56 ennathu oru stettu formula _ 20 nirddeshikkunnathinulla vyavasthaapitha sambhaavyathayaan, koodaathe sveekaaryatha vitharana soothravaakyam _ 59 ennathu nirddishta stettu formula _ 20 sweekarikkaanulla saadhyathayaanu. parivartthana saadhyatha avayude ulppannamaayi ezhuthaam
munbathe samavaakyathil eee bandham cherkkumbol, namukk undu
deriveshante adutha ghattam mukalil paranja vyavastha niravettunna oru sveekaaryatha anupatham thiranjedukkuka ennathaanu. oru pothu choys metropolis choys aannu
eee metropolis sveekaaryatha anupatha formula 64-nuu, formula 65 allengil formula 66, eathu vidhathilum, vyavastha thripthippedunnu.
metropolis-hustings algoritam inipparayunna reethiyil ezhuthaam
nirddishta vyavasthakal palikkappettaal, save cheytha avasthakalude prayogaparamaaya vitharanam formula 78 formula 1 neey sameepikkum. formula 1 phalapradamaayi kanakkakkaan aavashyamaaya aavarthanangalude ennam (formula 80) formula 1 umm nirddesha vitharanavum thammilulla bandhavum esttimettinte aavasyamulla krithyathayum ulppedeyulla gadakangalude ennathe aasrayichirikkunnu. vyathirikthamaya avastha edangalile vitharanathinu, athu markov prakriyayude svayam bandhita samayathinte kramathilaayirikkanam.
oru pothuprashnam, eathu vitharana soothravaakyam _ 56 aanu oraal upayogikkendathu allengil sariyaaya vilayiruthalinu aavashyamaaya aavarthanangalude ennam vyakthamallenn shradhikkendathu pradhaanamaan; iva randum reethiyude swatantra paaraamettarukalaanu, avaa kaiyilulla prathyeka prashnavumaayi krameekarikkanam.
samkhyaa samyojanathil upayogikkuka.
metropolis-hustings algoritam oru integral kanakkukoottuka ennathaanu. prathyekamaayi, oru space formula 84 umm formula 86, formula 87 nuu mukalilulla probability distribution formula 1 umm pariganikkuka. metropolis-hustingukalkku eee roopathinte avibhaajyaghadakam kanakkakkaan kazhiyum
ivide formula _ 89 ennathu palishayude (alakkaavunna) pravarthanamaanu. 
udaaharanathinu, oru staattistik formula _ 90 umm athinte probability distribution formula _ 91 umm pariganikkuka, ithu oru naamamaathramaaya vitharanamaanu. formula _ 91 nte wall formula _ 93-nuu formula _ 91 kanakkaakkuka ennathaanu lakshyamenn karuthuka. aupachaarikamaayi, formula _ 91 ingane ezhuthaam
athinaal, indikkettar function formula 98-nte prathekshikkunna moolyam kanakkaakkunnathiloode formula 91 kanakkakkaan kazhiyum, ithu formula 99 aayirikkumbol 1 umm poojyavumaanu.
formula _ 93 formula _ 91-nte vaalilaayathinaal, formula _ 91-nte vaalilulla formula _ 90 upayogichu oru stettu formula _ 12 varaykkaanulla saadhyatha formula _ 91-nuu aanupaathikamaanu, ithu nirvachanam anusarichu cheruthaanu. metropolis-hustings algoritam ivide sample (apoorva) avasthakalkku kooduthal saadhyatha nalkaanum angane vaalukalil formula _ 91 kanakkakkaan upayogikkunna saambilukalude ennam vardhippikkaanum upayogikkaam. ithu cheyyaan kazhiyum e.jii. aaa avasthakalkku anukoolamaayi oru sample vitharana soothravaakyam _ 37 upayogichu (udha.jii. formula _ 109 ulla formula _ 108).
ghattam ghattamaayulla nirddeshangal.
metropolis-haasting algoritam pinthudarunnathinu ettavum puthiya moolya sample formula _ 9 aanennu karuthuka, aduthathaayi nammal probability density formula _ 70 ulla oru puthiya nirddesha stettu formula _ 20 varaykkukayum oru moolyam kanakkaakkukayum cheiyunnu.
evide
ithu saadhyathayaanu (udha.jii., bayeshyan posteeriyar) nirddishta sample formula _ 20 umm munbathe sample formula _ 9 umm thammilulla anupatham, oppam
rund dishakalilulla nirddesha saandrathayude anupaathamaan (formula _ 9 muthal formula _ 20 vareyum nere marichum).
nirddesha saandrata samamithiyaanengil ithu 1 nuu thulyamaanu.
thudarnnu puthiya stettu formula _ 120 inipparayunna niyamangalkkanusrithamaayi thiranjedukkunnu.
markov cheyin oru ekapaksheeyamaaya prarambha moolya soothravaakyathil ninnaan aarambhikkunnath, eee prarambha avastha "marannupokunnathuvare" niravadhi aavarthanangalkkayi algoritam pravarthippikkunnu. upekshikkappedunna eee saambilukal "bayne-inn" ennariyappedunnu. formula _ 12-nte amgeekritha moolyangalude bakkiyulla koottam vitharana formula _ 1-ll ninnulla oru saambiline prathinidheekarikkunnu.
nirddesha saandrata taargett distribution formula _ 1-nte aakruthiyumaayi poruthappedunnundel algoritam mikacha reethiyil pravartthikkunnu, athil ninnu nerittulla sample edukkaan budhimuttaanu, athaayath formula _ 128.
oru gausiyan nirddesha saandrata formula _ 14 upayogikkukayaanenkil, bayne-inn kaalayalavil variance paaraameettar formula _ 130 tyoon cheyyendathundu.
avasana formula _ 131 saambilukalude oru jaalakathil sweekarikkunna nirddishta saambilukalude bhagamaaya "sveekaaryatha nirakk" kanakkakkiyaanu ithu saadhaaranayaayi cheiyunnathu.
aavasyamulla sveekaaryatha nirakk taargett vitharanathe aasrayichirikkunnu, ennirunnaalum oru ekamaana gausiyan vitharanathinu anuyojyamaaya sveekaaryatha nirakk ekadesham 50 shathamaanamaanennu saidhaanthikamaayi kaanichittundu, ithu oru formula _ 131-dimensional gausiyan taargett vitharanathinu ekadesham 23 shathamaanamaayi kurayunnu. bernstain-voonn maise sidhaantham upayogichu sthaapikkaan kazhiyunnathupole, palappozhum oru multiveriyettu normal distribution pinthudarunnathinaal mathiyaya saadhaarana bayeshyan posteeriyarukalil ninnu sample edukkumbol eee maargganirddeshangal nannayi pravartthikkum.
soothravaakyam _ 130 valare cheruthaanenkil, shringhala "pathukke kalakkum" (i.iii.sveekaaryatha nirakk uyarnnathayirikkum, ennaal thudarchayaaya saambilukal pathukke sthalathinu chuttum neengum, koodaathe shringhala pathukke formula _ 1 lake maathram othucherum). maruvashathu,
formula _ 130 valare valuthaanenkil, sveekaaryatha nirakk valare kuravaayirikkum, kaaranam nirddeshangal valare kuranja probability saandrathayulla pradeshangalil erangaan saadhyathayund, athinaal formula _ 136 valare cheruthayirikkum, veendum shringhala valare pathukke othucherum. muun khandikayil paraamarshichirikkunna saidhaanthika kanakkukalkk anusrithamaayi ellah saambilukaludeyum 30 sathamaanam kramatthil algorithangal sweekarikkunnathinu oraal saadhaaranayaayi nirddesha vitharanam tyoon cheiyunnu.
bayeshyan anumaanam.
oru stattistikkal modalinte pinbhaagathe vitharanathil ninnu saambilukal varaykkaan mcmc upayogikkaam.
sveekaaryathayude saadhyatha nalkunnathu
soothravaakyam _ 137
formula 138 ennathu saadhyathayaanu, formula 139 ennathu munkoor probability densityum formula 140 ennathu (vyavasthaapitha) nirddesha saadhyathayumaanu.

