PDF нормального распределения
fμ,σ(x)=12π−−√σe−(x−μ)22σ2dx
но с точки зрения этоτ=1/σ2
gμ,τ(x)=τ−−√2π−−√e−τ(x−μ)22dx.
PDF-файл гамма-распределения
hα,β(τ)=1Γ(α)e−τβτ−1+αβ−αdτ.
Их произведение, слегка упрощенное простой алгеброй, поэтому
fμ,α,β(x,τ)=1βαΓ(α)2π−−√e−τ((x−μ)22+1β)τ−1/2+αdτdx.
Его внутренняя часть, очевидно, имеет вид , что делает ее кратной гамма-функции при интегрировании во всем диапазоне от τ = 0 до τ = ∞ . Следовательно, этот интеграл является немедленным (полученным, если знать, что интеграл гамма-распределения равен единице), давая предельное распределениеexp(−constant1×τ)×τconstant2dττ=0τ=∞
fμ,α,β(x)=β−−√Γ(α+12)2π−−√Γ(α)1(β2(x−μ)2+1)α+12.
Пытаясь соответствовать шаблону , отведенный для распределения показывает , есть ошибка в вопросе: PDF для распределения Стьюдента Студенческого фактически пропорционаленt
1k−−√s⎛⎝⎜⎜11+k−1(x−ls)2⎞⎠⎟⎟k+12
(the power of (x−l)/s is 2, not 1). Matching the terms indicates k=2α, l=μ, and s=1/αβ−−−√.
Notice that no Calculus was needed for this derivation: everything was a matter of looking up the formulas of the Normal and Gamma PDFs, carrying out some trivial algebraic manipulations involving products and powers, and matching patterns in algebraic expressions (in that order).
I don't know the steps of the calculation, but I do know the results from some book (cannot remember which one...). I usually keep it in mind directly... :-) The Studentt distribution with k degree freedom can be regarded as a Normal distribution with variance mixture Y , where Y follows inverse gamma distribution.
More precisely, X ~t(k) ,X =Y−−√ *Φ ,where Y ~IG(k/2,k/2) ,Φ is standard normal rv.
I hope this could help you in some sense.
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To simplify we assume mean0 . Using representation, we show the result for integer degrees of freedom.
1/τ−−−√X=Y
is equivalent to a Gaussian mixture with that prior: conditioned on τ , Y is Gaussian with precision τ , and the prior τ is as desired. Then it remains to show that 1/τ−−−√X is a t-distribution.
We can write
τ∼Γ(α,β)∼β2Γ(α,2)∼β2χ2(2α)
using a well-known result about gammas and Chi-squares (decompose a gamma as a sum of exponentials and combine the exponentials to normals to Chi squares)
This in turn implies that
Y∼X1(β/2)χ2(2α)−−−−−−−−−−√
=Xαβ−−−√χ22α/(2α)−−−−−−−√
which is a scaled t with k=2α and s=1/αβ−−−√ by variance of t. We can recenter our representation at μ and l would follow.
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