normal-teskari-gamma Ehtimollar zichligi funktsiyasi
Parametrlar m {displaystyle mu,} Manzil (haqiqiy ) λ > 0 {displaystyle lambda> 0,} a > 0 {displaystyle alfa> 0,} β > 0 {displaystyle eta> 0,} Qo'llab-quvvatlash x ∈ ( − ∞ , ∞ ) , σ 2 ∈ ( 0 , ∞ ) {displaystyle xin (-infty, infty),!,; sigma ^ {2} in (0, infty)} PDF λ 2 π σ 2 β a Γ ( a ) ( 1 σ 2 ) a + 1 tugatish ( − 2 β + λ ( x − m ) 2 2 σ 2 ) {displaystyle {frac {sqrt {lambda}} {sqrt {2pi sigma ^ {2}}}} {frac {eta ^ {alfa}} {Gamma (alfa)}} chap ({frac {1} {sigma ^ {2) }}} ight) ^ {alfa +1} exp left (- {frac {2 eta + lambda (x-mu) ^ {2}} {2sigma ^ {2}}} ight)} Anglatadi E [ x ] = m {displaystyle operator nomi {E} [x] = mu}
E [ σ 2 ] = β a − 1 {displaystyle operator nomi {E} [sigma ^ {2}] = {frac {eta} {alfa -1}}} a > 1 {displaystyle alfa> 1} Rejim x = m (bir xil) , x = m (ko'p o'zgaruvchan) {displaystyle x = mu; {extrm {(univariate)}}, x = {oldsymbol {mu}}; {extrm {(multivariate)}}}
σ 2 = β a + 1 + 1 / 2 (bir xil) , σ 2 = β a + 1 + k / 2 (ko'p o'zgaruvchan) {displaystyle sigma ^ {2} = {frac {eta} {alfa + 1 + 1/2}}; {extrm {(univariate)}}, sigma ^ {2} = {frac {eta} {alfa + 1 + k / 2}}; {extrm {(ko'p o'zgaruvchan)}}} Varians Var [ x ] = β ( a − 1 ) λ {displaystyle operator nomi {Var} [x] = {frac {eta} {(alfa -1) lambda}}} a > 1 {displaystyle alfa> 1} Var [ σ 2 ] = β 2 ( a − 1 ) 2 ( a − 2 ) {displaystyle operator nomi {Var} [sigma ^ {2}] = {frac {eta ^ {2}} {(alfa -1) ^ {2} (alfa -2)}}} a > 2 {displaystyle alfa> 2}
Kov [ x , σ 2 ] = 0 {displaystyle operator nomi {Cov} [x, sigma ^ {2}] = 0} a > 1 {displaystyle alfa> 1}
Yilda ehtimollik nazariyasi va statistika , normal-teskari-gamma taqsimoti (yoki Gauss-teskari-gamma taqsimoti ) ko'p parametrli uzluksiz to'rt parametrli oiladir ehtimollik taqsimoti . Bu oldingi konjugat a normal taqsimot noma'lum bilan anglatadi va dispersiya .
Ta'rif
Aytaylik
x ∣ σ 2 , m , λ ∼ N ( m , σ 2 / λ ) {displaystyle xmid sigma ^ {2}, mu, lambda sim mathrm {N} (mu, sigma ^ {2} / lambda) ,!} bor normal taqsimot bilan anglatadi m {displaystyle mu} dispersiya σ 2 / λ {displaystyle sigma ^ {2} / lambda}
σ 2 ∣ a , β ∼ Γ − 1 ( a , β ) {displaystyle sigma ^ {2} mid alfa, eta sim Gamma ^ {- 1} (alfa, eta)!} bor teskari gamma taqsimoti . Keyin ( x , σ 2 ) {displaystyle (x, sigma ^ {2})}
( x , σ 2 ) ∼ N- Γ − 1 ( m , λ , a , β ) . {displaystyle (x, sigma ^ {2}) sim {ext {N -}} Gamma ^ {- 1} (mu, lambda, alfa va boshqalar)!} ( NIG {displaystyle {ext {NIG}}} N- Γ − 1 . {displaystyle {ext {N -}} Gamma ^ {- 1}.}
The normal-teskari-Wishart taqsimoti ko'p o'zgaruvchan tasodifiy o'zgaruvchilar bo'yicha aniqlangan normal-teskari-gamma taqsimotning umumlashtirilishi.
Xarakteristikasi
Ehtimollar zichligi funktsiyasi f ( x , σ 2 ∣ m , λ , a , β ) = λ σ 2 π β a Γ ( a ) ( 1 σ 2 ) a + 1 tugatish ( − 2 β + λ ( x − m ) 2 2 σ 2 ) {displaystyle f (x, sigma ^ {2} mid mu, lambda, alfa, eta) = {frac {sqrt {lambda}} {sigma {sqrt {2pi}}}}, {frac {eta ^ {alfa}} { Gamma (alfa)}}, chap ({frac {1} {sigma ^ {2}}} ight) ^ {alfa +1} exp left (- {frac {2 eta + lambda (x-mu) ^ {2} } {2sigma ^ {2}}} kech)} Ko'p o'zgaruvchan shakl uchun qaerda x {displaystyle mathbf {x}} k × 1 {displaystyle k imes 1}
f ( x , σ 2 ∣ m , V − 1 , a , β ) = | V | − 1 / 2 ( 2 π ) − k / 2 β a Γ ( a ) ( 1 σ 2 ) a + 1 + k / 2 tugatish ( − 2 β + ( x − m ) ′ V − 1 ( x − m ) 2 σ 2 ) . {displaystyle f (mathbf {x}, sigma ^ {2} mid mu, mathbf {V} ^ {- 1}, alfa, eta) = | mathbf {V} | ^ {- 1/2} {(2pi) ^ {-k / 2}}, {frac {eta ^ {alfa}} {Gamma (alfa)}}, chap ({frac {1} {sigma ^ {2}}} ight) ^ {alfa + 1 + k / 2} exp chap (- {frac {2 eta + (mathbf {x} - {oldsymbol {mu}}) 'mathbf {V} ^ {- 1} (mathbf {x} - {oldsymbol {mu}}))} 2sigma ^ {2}}} kech).} qayerda | V | {displaystyle | mathbf {V} |} aniqlovchi ning k × k {displaystyle k imes k} matritsa V {displaystyle mathbf {V}} k = 1 {displaystyle k = 1} x , V , m {displaystyle mathbf {x}, mathbf {V}, {oldsymbol {mu}}} skalar .
Muqobil parametrlash Ruxsat berish ham mumkin γ = 1 / λ {displaystyle gamma = 1 / lambda}
f ( x , σ 2 ∣ m , γ , a , β ) = 1 σ 2 π γ β a Γ ( a ) ( 1 σ 2 ) a + 1 tugatish ( − 2 γ β + ( x − m ) 2 2 γ σ 2 ) {displaystyle f (x, sigma ^ {2} mid mu, gamma, alfa, eta) = {frac {1} {sigma {sqrt {2pi gamma}}}}, {frac {eta ^ {alpha}} {Gamma ( alfa)}}, chap ({frac {1} {sigma ^ {2}}} ight) ^ {alfa +1} exp left (- {frac {2gamma eta + (x-mu) ^ {2}} {2gamma sigma ^ {2}}} kech)} Ko'p o'zgaruvchan shaklda kovaryans matritsasini hisobga olish uchun tegishli o'zgarish bo'ladi V {displaystyle mathbf {V}} teskari V − 1 {displaystyle mathbf {V} ^ {- 1}}
Kümülatif taqsimlash funktsiyasi F ( x , σ 2 ∣ m , λ , a , β ) = e − β σ 2 ( β σ 2 ) a ( erf ( λ ( x − m ) 2 σ ) + 1 ) 2 σ 2 Γ ( a ) {displaystyle F (x, sigma ^ {2} mid mu, lambda, alfa, eta) = {frac {e ^ {- {frac {eta} {sigma ^ {2}}}} chap ({frac {eta} {) sigma ^ {2}}} ight) ^ {alfa} left (operator nomi {erf} left ({frac {{sqrt {lambda}} (x-mu)} {{sqrt {2}} sigma}} ight) + 1ight )} {2sigma ^ {2} Gamma (alfa)}}} Xususiyatlari
Marginal taqsimotlar Berilgan ( x , σ 2 ) ∼ N- Γ − 1 ( m , λ , a , β ) . {displaystyle (x, sigma ^ {2}) sim {ext {N -}} Gamma ^ {- 1} (mu, lambda, alfa va boshqalar)!} σ 2 {displaystyle sigma ^ {2}} teskari gamma taqsimoti :
σ 2 ∼ Γ − 1 ( a , β ) {displaystyle sigma ^ {2} sim Gamma ^ {- 1} (alfa, eta)!} esa a λ β ( x − m ) {displaystyle {sqrt {frac {alfa lambda} {eta}}} (x-mu)} t taqsimoti bilan 2 a {displaystyle 2alpha}
Ko'p o'zgaruvchan holda, ning marginal taqsimoti x {displaystyle mathbf {x}} ko'p o'zgaruvchan t tarqatish :
x ∼ t 2 a ( m , β a V − 1 ) {displaystyle mathbf {x} sim t_ {2alpha} ({oldsymbol {mu}}, {frac {eta} {alfa}} mathbf {V} ^ {- 1})!} Xulosa Miqyosi Eksponent oilasi Axborot entropiyasi Kullback - Leybler divergensiyasi Ehtimollarni maksimal darajada baholash
Ushbu bo'lim bo'sh. Siz yordam berishingiz mumkin unga qo'shilish . (2010 yil iyul )
Parametrlarning orqa taqsimlanishi
Maqolalarni ko'ring normal-gamma taqsimoti va oldingi konjugat .
Parametrlarni talqin qilish
Maqolalarni ko'ring normal-gamma taqsimoti va oldingi konjugat .
Normal-teskari-gamma tasodifiy o'zgarishlarni yaratish
Tasodifiy o'zgarishni yaratish to'g'ridan-to'g'ri:
Namuna σ 2 {displaystyle sigma ^ {2}} a {displaystyle alfa} β {displaystyle eta} Namuna x {displaystyle x} m {displaystyle mu} σ 2 / λ {displaystyle sigma ^ {2} / lambda} Tegishli tarqatishlar
The normal-gamma taqsimoti tomonidan parametrlangan bir xil taqsimot aniqlik dan ko'ra dispersiya Ko'p o'lchovli o'rtacha va umuman noma'lum ijobiy aniq kovaryans matritsasini yaratishga imkon beradigan ushbu taqsimotning umumlashtirilishi σ 2 V {displaystyle sigma ^ {2} mathbf {V}} σ 2 {displaystyle sigma ^ {2}} normal-teskari-Wishart taqsimoti Shuningdek qarang
Adabiyotlar
Denison, Devid G. T.; Xolms, Kristofer S.; Mallik, Bani K .; Smit, Adrian F. M. (2002) Lineer bo'lmagan tasniflash va regressiya uchun Bayes usullari , Vili. ISBN 0471490369 Koch, Karl-Rudolf (2007) Bayesiya statistikasiga kirish (2-nashr), Springer. ISBN 354072723X Diskret o'zgaruvchan Diskret o'zgaruvchan Doimiy o'zgaruvchan Doimiy o'zgaruvchan Doimiy o'zgaruvchan Doimiy o'zgaruvchan Aralashtirilgan uzluksiz diskret bir o'zgaruvchidir Ko'p o'zgaruvchan (qo'shma) Yo'naltirilgan Degeneratsiya va yakka Oilalar
![{displaystyle operator nomi {E} [x] = mu}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d60f5921cca1c75d673eb70db395bf3a88f9170f)
![{displaystyle operator nomi {E} [sigma ^ {2}] = {frac {eta} {alfa -1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b74baba053fd81d56d62de618558ac7af62ade55)
![{displaystyle operator nomi {Var} [x] = {frac {eta} {(alfa -1) lambda}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c11eddb529a936912263edfb0c46ce2a42adfbd5)
![{displaystyle operator nomi {Var} [sigma ^ {2}] = {frac {eta ^ {2}} {(alfa -1) ^ {2} (alfa -2)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d089f3b7da4ce1f13940b4731eb531932850d0e)
![{displaystyle operator nomi {Cov} [x, sigma ^ {2}] = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df7006f5738ee174c6c35e1694f1c4ac3b2c9c42)
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