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normal-teskari-istak

Notation	${ displaystyle ({ boldsymbol { mu}}, { boldsymbol { Sigma}}) sim mathrm {NIW} ({ boldsymbol { mu}} _ {0}, lambda, { boldsymbol { Psi}}, nu)}$
Parametrlar	${ displaystyle { boldsymbol { mu}} _ {0} in mathbb {R} ^ {D} ,}$ Manzil (ning vektori haqiqiy) ${ displaystyle lambda> 0 ,}$ (haqiqiy) ${ displaystyle { boldsymbol { Psi}} in mathbb {R} ^ {D times D}}$ teskari o'lchov matritsasi (pos. def.) ${ displaystyle nu> D-1 ,}$ (haqiqiy)
Qo'llab-quvvatlash	${ displaystyle { boldsymbol { mu}} in mathbb {R} ^ {D}; { boldsymbol { Sigma}} in mathbb {R} ^ {D times D}}$ kovaryans matritsasi (pos. def.)
PDF	${ displaystyle f ({ boldsymbol { mu}}, { boldsymbol { Sigma}} | { boldsymbol { mu}} _ {0}, lambda, { boldsymbol { Psi}}, nu ) = { mathcal {N}} ({ boldsymbol { mu}} | { boldsymbol { mu}} _ {0}, { tfrac {1} { lambda}} { boldsymbol { Sigma} }) { mathcal {W}} ^ {- 1} ({ boldsymbol { Sigma}} | { boldsymbol { Psi}}, nu)}$

Yilda ehtimollik nazariyasi va statistika, normal-teskari-Wishart taqsimoti (yoki Gauss-teskari-Vishart taqsimoti) ko'p o'zgaruvchan to'rt parametrli doimiy ehtimollik taqsimoti. Bu oldingi konjugat a ko'p o'zgaruvchan normal taqsimot noma'lum bilan anglatadi va kovaryans matritsasi (ning teskarisi aniqlik matritsasi).^[1]

Ta'rif

Aytaylik

${ displaystyle { boldsymbol { mu}} | { boldsymbol { mu}} _ {0}, lambda, { boldsymbol { Sigma}} sim { mathcal {N}} left ({ boldsymbol { mu}} { Big |} { boldsymbol { mu}} _ {0}, { frac {1} { lambda}} { boldsymbol { Sigma}} right)}$

bor ko'p o'zgaruvchan normal taqsimot bilan anglatadi ${ displaystyle { boldsymbol { mu}} _ {0}}$ va kovaryans matritsasi ${ displaystyle { tfrac {1} { lambda}} { boldsymbol { Sigma}}}$, qayerda

${ displaystyle { boldsymbol { Sigma}} | { boldsymbol { Psi}}, nu sim { mathcal {W}} ^ {- 1} ({ boldsymbol { Sigma}} | { boldsymbol { Psi}}, nu)}$

bor Wishart-ning teskari taqsimoti. Keyin ${ displaystyle ({ boldsymbol { mu}}, { boldsymbol { Sigma}})}$sifatida belgilangan normal-teskari-Wishart taqsimotiga ega

${ displaystyle ({ boldsymbol { mu}}, { boldsymbol { Sigma}}) sim mathrm {NIW} ({ boldsymbol { mu}} _ {0}, lambda, { boldsymbol { Psi}}, nu).}$

Xarakteristikasi

Ehtimollar zichligi funktsiyasi

${ displaystyle f ({ boldsymbol { mu}}, { boldsymbol { Sigma}} | { boldsymbol { mu}} _ {0}, lambda, { boldsymbol { Psi}}, nu ) = { mathcal {N}} chap ({ boldsymbol { mu}} { Big |} { boldsymbol { mu}} _ {0}, { frac {1} { lambda}} { boldsymbol { Sigma}} o'ng) { mathcal {W}} ^ {- 1} ({ boldsymbol { Sigma}} | { boldsymbol { Psi}}, nu)}$

PDF-ning to'liq versiyasi quyidagicha:^[2]

${ displaystyle f ({ boldsymbol { mu}}, { boldsymbol { Sigma}} | alpha, { boldsymbol { Psi}}, gamma, { boldsymbol { delta}}) = { frac { gamma ^ {D / 2} | { boldsymbol { Psi}} | ^ { alpha / 2} | { boldsymbol { Sigma}} | ^ {- { frac { alpha + D + 2 } {2}}}} {(2 pi) ^ {D / 2} 2 ^ { frac { alfa D} {2}} Gamma _ {D} ({ frac { alpha} {2} })}} { text {exp}} left {- { frac {1} {2}} (Tr ({ boldsymbol { Psi Sigma}} ^ {- 1}) + gamma ({ boldsymbol { mu}} - { boldsymbol { delta}}) ^ {T} { boldsymbol { Sigma}} ^ {- 1} ({ boldsymbol { mu}} - { boldsymbol { delta) }})) o'ng }}$

Bu yerda ${ displaystyle Gamma _ {D} [ cdot]}$ ko'p o'zgaruvchan gamma funktsiyasi va ${ displaystyle Tr ({ boldsymbol { Psi}})}$ berilgan matritsaning izidir.

Xususiyatlari

O'lchov

Marginal taqsimotlar

Qurilish bo'yicha marginal taqsimot ustida ${ displaystyle { boldsymbol { Sigma}}}$ bu Wishart-ning teskari taqsimoti, va shartli taqsimlash ustida ${ displaystyle { boldsymbol { mu}}}$ berilgan ${ displaystyle { boldsymbol { Sigma}}}$ a ko'p o'zgaruvchan normal taqsimot. The marginal taqsimot ustida ${ displaystyle { boldsymbol { mu}}}$ a ko'p o'zgaruvchan t-taqsimot.

Parametrlarning orqa taqsimlanishi

Faraz qilaylik, namuna olish zichligi ko'p o'zgaruvchan normal taqsimotdir

${ displaystyle { boldsymbol {y_ {i}}} | { boldsymbol { mu}}, { boldsymbol { Sigma}} sim { mathcal {N}} _ {p} ({ boldsymbol {) mu}}, { boldsymbol { Sigma}})}$

qayerda ${ displaystyle { boldsymbol {y}}}$ bu ${ displaystyle n times p}$ matritsa va ${ displaystyle { boldsymbol {y_ {i}}}}$ (uzunlik ${ displaystyle p}$) qator ${ displaystyle i}$ matritsaning

Namuna taqsimotining o'rtacha va kovaryans matritsasi noma'lum bo'lganligi sababli, biz o'rtacha va kovaryans parametrlari bo'yicha oldin Normal-Teskari-Vishartni joylashtiramiz.

${ displaystyle ({ boldsymbol { mu}}, { boldsymbol { Sigma}}) sim mathrm {NIW} ({ boldsymbol { mu}} _ {0}, lambda, { boldsymbol { Psi}}, nu).}$

Olingan o'rtacha va kovaryans matritsasi uchun orqa tomon taqsimoti ham Normal-Teskari-Vishart bo'ladi

${ displaystyle ({ boldsymbol { mu}}, { boldsymbol { Sigma}} | y) sim mathrm {NIW} ({ boldsymbol { mu}} _ {n}, lambda _ {n }, { boldsymbol { Psi}} _ {n}, nu _ {n}),}$

qayerda

${ displaystyle { boldsymbol { mu}} _ {n} = { frac { lambda { boldsymbol { mu}} _ {0} + n { bar { boldsymbol {y}}}} { lambda + n}}}$

${ displaystyle lambda _ {n} = lambda + n}$

${ displaystyle nu _ {n} = nu + n}$

${ displaystyle { boldsymbol { Psi}} _ {n} = { boldsymbol { Psi + S}} + { frac { lambda n} { lambda + n}} ({ boldsymbol {{ bar) {y}} - mu _ {0}}}) ({ boldsymbol {{ bar {y}} - mu _ {0}}}) ^ {T} ~~~ mathrm {bilan,} ~ ~ { boldsymbol {S}} = sum _ {i = 1} ^ {n} ({ boldsymbol {y_ {i} - { bar {y}}}}) ({ boldsymbol {y_ {i} - { bar {y}}}}) ^ {T}}$.

Qo'shimchaning orqa qismidan namuna olish uchun ${ displaystyle ({ boldsymbol { mu}}, { boldsymbol { Sigma}})}$, ulardan oddiygina namunalar olinadi ${ displaystyle { boldsymbol { Sigma}} | { boldsymbol {y}} sim { mathcal {W}} ^ {- 1} ({ boldsymbol { Psi}} _ {n}, nu _ {n})}$, keyin chizish ${ displaystyle { boldsymbol { mu}} | { boldsymbol { Sigma, y}} sim { mathcal {N}} _ {p} ({ boldsymbol { mu}} _ {n}, { boldsymbol { Sigma}} / lambda _ {n})}$. Yangi kuzatishning orqa prognozidan chizish uchun chizilgan ${ displaystyle { boldsymbol { tilde {y}}} | { boldsymbol { mu, Sigma, y}} sim { mathcal {N}} _ {p} ({ boldsymbol { mu}} , { boldsymbol { Sigma}})}$ , ning allaqachon chizilgan qiymatlarini hisobga olgan holda ${ displaystyle { boldsymbol { mu}}}$ va ${ displaystyle { boldsymbol { Sigma}}}$.^[3]

Normal-teskari-Vishart tasodifiy o'zgarishini yaratish

Tasodifiy o'zgarishni yaratish to'g'ridan-to'g'ri:

1. Namuna ${ displaystyle { boldsymbol { Sigma}}}$ dan Wishart-ning teskari taqsimoti parametrlari bilan ${ displaystyle { boldsymbol { Psi}}}$ va ${ displaystyle nu}$
2. Namuna ${ displaystyle { boldsymbol { mu}}}$ dan ko'p o'zgaruvchan normal taqsimot o'rtacha bilan ${ displaystyle { boldsymbol { mu}} _ {0}}$ va dispersiya ${ displaystyle { boldsymbol { tfrac {1} { lambda}}} { boldsymbol { Sigma}}}$

Tegishli tarqatishlar

• The normal-Wishart taqsimoti aslida farq bilan emas, balki aniqlik bilan parametrlangan bir xil taqsimotdir. Agar ${ displaystyle ({ boldsymbol { mu}}, { boldsymbol { Sigma}}) sim mathrm {NIW} ({ boldsymbol { mu}} _ {0}, lambda, { boldsymbol { Psi}}, nu)}$ keyin ${ displaystyle ({ boldsymbol { mu}}, { boldsymbol { Sigma}} ^ {- 1}) sim mathrm {NW} ({ boldsymbol { mu}} _ {0}, lambda , { boldsymbol { Psi}} ^ {- 1}, nu)}$ .
• The normal-teskari-gamma taqsimoti bir o'lchovli ekvivalentdir.
• The ko'p o'zgaruvchan normal taqsimot va Wishart-ning teskari taqsimoti bu taqsimot amalga oshiriladigan tarkibiy taqsimotlardir.

Izohlar

Adabiyotlar

• Bishop, Kristofer M. (2006). Naqshni tanib olish va mashinada o'rganish. Springer Science + Business Media.
• Murphy, Kevin P. (2007). "Gauss taqsimotini kon'yugate Bayes tahlillari." [2]