Béda révisi "Informasi Fisher"

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Dina [[statistik]], the '''informasi Fisher information''' ''I''(θ), thought of as the amount ofnyaeta [[information]] that an observable [[random variable]] carriesnu aboutbisa andiobservasi unobservablemawa kanyaho ngeunaan parameter nu teu ka observasi θ uponnu whichgumantung thekana [[probability distribution]] of ''X'' depends, is the [[variance]] of thengarupakeun [[score (statistics)|score]] [[variance]]. BecauseSabab theskor [[expectation]] ofnyaeta the score is zeronol, this maybisa bedituliskeun writtensalaku as
 
:<math>
</math>
 
wherenumana ''f'' is thengarupakeun [[probability density function]] ofvariabel random variable ''X''.
Informasi Fisher saterusna ngarupakeun ekspektasi kuadrat tina skor. Variabel random mawa informasi Fisher nu luhur nu ngakibatkeun nilai mutlak skor oge jadi luhur (inget yen skor ekspektasi nyaeta nol).
The Fisher information is thus the expectation of the square of the score. A random variable carrying high Fisher information implies that the absolute value of the score is frequently high (remember that the expectation of the score is zero).
 
ThisKonsep conceptieu isdipake namedkeur inngahargaan honorka ofahli thegenetis geneticistjeung and statisticianstatistikawan [[Ronald Fisher]].
 
Catetan yen informasi nu dihartikeun di luhur lain fungsi tina observasi sabagean, salaku variabel ''X'' geus ngabogaan ''average''. Konsep informasi gampang dipake keur ngabandingkeun dua metoda observasi dina proses random nu sarua.
Note that the information as defined above is not a function of a particular observation, as the random variable ''X'' has been averaged out. The concept of information is useful when comparing two methods of observation of some random process.
 
Informasi saperti nu geus dihartikeun di luhur bisa ditulis dina bentuk
Information as defined above may be written as
:<math>
I(\theta)=-E\left[
\right]
</math>
andsarta is thus the expection ofsaterusna log ofekspektasi thengarupakeun secondturunan derivativekadua ofti ''X'' withnu respectpakait tojeung ''&theta;''. InformationInformasi maysaterusna thusgeus bekatempo seenngarupakeun to be a measure of theukuran "sharpnesskaseukeutan" ofnu thengadukung supportkurva curvedeukeut near thekana [[maximum likelihood|maximum likelihood estimate]] of ''&theta;''. AKurva dukungan nu "bluntKodol" support curve (onenu withngabogaan anilai shallowminimum maximumdeet) wouldbakal havengabogaan lowturunan expectedekspektasi secondkadua derivativenu lemah, andsarta thussaterusna lowinformasi informationnu lemah; whilesabalikna bentuk nu aseukeut sharpbakal onengabogaan wouldnilai haveturunan akadua highnu expectedluhur secondsarta derivativesaterusna andnilai thusinformasi highnu informationluhur.
 
Informasi ngarupakeun tambahan, dina hal ieu informasi dicokot tina dua eksperimen [[independent]], ngarupakeun jumlah tina eta informasi:
Information is additive, in the sense that the information gathered by two [[independent]] experiments is the sum of the information of each of them:
 
:<math>
</math>
 
Hal ieu kusabab jumlah varian dua variabel random bebas ngarupakeun jumlah eta varian. Hal ieu nuturkeun yen informasi dina ukuran sampel random ''n'' nyaeta ''n'' kali dina ukuran hiji sampel(lamun eta observasi bebas).
This is because the variance of the sum of two independent random variables is the sum of their variances. It follows that the information in a random sample of size ''n'' is ''n'' times that in a sample of size one (if observations are independent).
 
TheInformasi informationieu provideddisaratkeun by aku [[sufficiency (statistics)|sufficient statistic]] isnyaeta samesarua asjeung that of the samplesampel ''X''. ThisIeu maygeus bekatempo seenku bymake usingkriteria faktorisasi Fisher's factorizationkeur criterionkacukupan for a sufficient statisticstatistis. IfLamun ''T(X)'' is sufficientcukup forkeur &theta;, thenmangka
 
:<math>
</math>
 
forkeur somesababaraha functionsfungsi ''g'' andjeung ''h'' (seetempo [[sufficient statistic]] forkeur akaterangan moreleuwih detailed explanationlengkep). The equality ofDina informationkanyataanna followspersamaan frominformasi thenuturkeun factbentuk that
 
:<math>
</math>
 
(whichnumana isieu thekasus case becausesabab ''h''(''X'') is independent ofngarupakeun &theta;) andbebas thesarta definitionharti forkeur informasi information givendiberekeun di aboveluhur. MoreLeuwih generallyumum, iflamun ''T=t(X)'' is angarupakeun [[statistic]], thenmangka
 
:<math>
I_T(\theta)\leq I_X(\theta)
</math>
nu sarua lamun jeung lamun ''T'' ngarupakeun kacukupan statistik.
with equality if and only if ''T'' is a sufficient statistic.
 
The [[Cram&eacute;r-Rao inequality]] statesnetepkeun thatyen the reciprocal of theinformasi Fisher informationbolak isbalik angarupakeun lowerwater boundhandap ondina thevarian variancekeur of anyunggal ''unbiased estimator of'' &theta;.
 
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