Mohlala oa Lipotso tsa Puisano bakeng sa Sethala sa Metropolis
Moelelong oa li-simulation tsa Monte Carlo, sethala sa Metropolis ke algorithm ea bohlokoa mekheniking ea lipalo-palo le masimong a mang. Karolong ena, re tšohla ka ho khetheha mokhoa oa Metropolis-Hastings, algorithm e sebelisoang ho etsa sampole ho tsoa likabong tse rarahaneng tsa menyetla. Ka ho utloisisa mehato e ho algorithm ena, re ka etsa li-simulation tse nepahetseng le tse sebetsang hantle haholoanyane.
Selelekela ho Algorithm ea Metropolis
Algorithm ea Metropolis e hlahisitsoe ke Nicholas Metropolis le basebetsi-'moho le eena ka 1953. Mokhoa ona o sebelisoa ho etsa mohlala le ho etsisa boemo ba litsamaiso tsa 'mele, haholo-holo tse kenyeletsang likaroloana tse ngata tse kang likhase kapa metsi. Phetolelo ea sejoale-joale ea algorithm ena, Metropolis-Hastings, ke kakaretso e lumellang lisampole ho ntšoa ho tsoa kabong ea sepheo e sa tloaelehang.
Mehato ho Algorithm ea Metropolis
Ho utloisisa hore na algorithm ea Metropolis e sebetsa joang, ho bohlokoa ho tloaelana le mehato ena:
1. Ho qala: Qala ka ho kgetha tharollo ya pele ka mokhoa o sa reroang ho tloha sebakeng sa tharollo kapa kabo ya pele. Mohlala, re qala ka boemo ba mocheso kapa boemo ba dikarolwana.
2. Ho Hlahisa Mohato o Mocha: Hlahisa boemo bo bocha (tharollo e ncha) ka ho etsa phetoho e nyane boemong ba hona joale. Hangata sena se bitsoa mohato oa "tlhahiso". Phetoho ena hangata e nkiloe kabong e lekanang, joalo ka kabo ea Gaussian.
3. Ho Bala Karolelano ea Kamohelo: Bala karolelano ea kamohelo, e etsang qeto ea hore na re amohela kapa re hana mohato o reriloeng. Karolelano ena ke karolelano ea monyetla oa boemo bo bocha ho boemo ba hona joale. Mongolong oa lipalo, karolelano ena e fanoa ke:
\[
A = \min\left(1, \frac{P(\text{new})}{P(\text{current})}\right)
\]
moo \( P \) e leng kgonahalo ya boemo bo itseng.
4. Qeto e Sebelisang Karolelano ea Kamohelo: Bapisa karolelano ea kamohelo le boleng bo sa reroang bo nkiloeng kabong e tšoanang pakeng tsa 0 le 1. Haeba karolelano ea kamohelo e le kholo ho feta boleng bo sa reroang, amohela motsamao o mocha; ho seng joalo, o hane 'me u lule boemong ba hona joale.
5. Pheta-pheto: Pheta mehato ea 2 ho isa ho ea 4 bakeng sa palo e lakatsehang ea lipeto-pheto kapa ho fihlela sistimi e fihla tekanong.
Lipotso tsa Mehlala le Puisano
A re buisaneng ka mehlala e meng ea lipotso ho utloisisa sethala sa Metropolis hamolemo.
Mohlala oa Potso ea 1
Potso: O na le karoloana e boemong bo le bong \( x \) e angwang ke tshebetso ya eneji e ka bang teng \( U(x) = x^2 \). Sebedisa algorithm ya Metropolis ho etsisa kabo ya maemo a karoloana.
Puisano:
1. Ho qala: Qala ho tloha boemong \( x = 0 \).
2. Hlahisa Motsamao o Mocha: Hlahisa boemo bo bocha \( x' = x + \Delta x \), ka \( \Delta x \) e nkiloeng ho tsoa kabong ea Gaussian e nang le lefela le tloaelehileng.
3. Palo ea Karolelano ea Matla: Bala karolelano ea matla:
\[
\Delta U = U(x') – U(x) = x'^2 – x^2
\]
Kahoo, karolelano ea kamohelo ke:
\[
A = \metsotso\left(1, e^{-\Delta U}\right)
\]
4. Qeto: Haeba \( A \) e feta palo e sa reroang pakeng tsa 0 le 1, amohela \( x' \); ho seng jwalo, dula boemong \( x \).
5. Pheta-pheta: Pheta ts'ebetso ena ka, mohlala, mehato e 10,000.
Kabo ea boemo bo hlahang e tla latela kabo ea Gaussian e nang le lefela le tloaelehileng le phapang e lekanang le bokhoni, e leng se fellang ka kabo e bōpiloeng ke mosebetsi oa matla a ka bang teng.
Mohlala oa Potso ea 2
Potso: Sebelisa algorithm ea Metropolis ho lumellana le tlhaloso ea mosebetsi oa Bayesian. A re re re batla ho lumellana le leralla le bonolo ho sete ea data re sebelisa linear regression le MCMC.
Puisano:
1. Ho qala: Beha diparamitha tsa mohlala wa pele \( \beta = (m, c) \).
2. Ho Sisinya Mohato o Mocha: Sisinya diparamithara tse ntjha tsa kabo ya tlhahiso e tlwaelehileng ya multivariate. Mohlala, sebedisa kabo ya Gaussian bakeng sa di-variable \( m \) le \( c \).
3. Karolelano ea Kamohelo: Bala karolelano ea kamohelo ka:
\[
A = \min\left(1, \frac{L(m', c'| \text{data})P(m', c')}{L(m, c| \text{data})P(m, c)}\right)
\]
Moo \( L \) e leng monyetla, mme \( P \) e le pele ho paramethara.
4. Qeto: Bapisa karolelano le boleng bo sa reroang ba 0 ho isa ho 1 ho amohela kapa ho hana tlhahiso.
5. Phetoho: Tsamaisa ketsiso ka makhetlo a mangata ho fihlela ho fihlelleha ho kopana.
Ka mokhoa ona, re ka fumana kabo ea morao bakeng sa liparamente tsa regression, e leng se re fang mokhoa oa ho hakanya le ho toloka likamano ho data.
Qetello
Sethala sa Metropolis ho li-simulation tsa Monte Carlo se re lumella ho etsa sampole ho tsoa likabong tse rarahaneng tsa sepheo 'me se sebetsa e le motheo oa mokhoa oa Metropolis-Hastings. Ka ho sebelisa mokhoa ona mafapheng a fapaneng, re ka fihlela mohlala o nepahetseng haholoanyane le kutloisiso e qaqileng haholoanyane ea sistimi. Litšebelisong tse tlohang ho fisiks le baeloji ho isa saenseng ea khomphutha le lipalo-palo, algorithm ena e fana ka litharollo tse ntle le tse sebetsang mathateng a rarahaneng.
