Student t als Mischung aus Gauß

23

Unter Verwendung der studentischen t-Verteilung mit $k > 0$ Freiheitsgraden haben der Ortsparameter $l$ und der Skalenparameter $s$ eine Dichte

\frac{Γ (\frac{k + 1}{2})}{Γ (\frac{k}{2} \sqrt{k π s^{2}})} {1 + k^{- 1} (\frac{x - l}{s})}^{- (k + 1) / 2},

$\frac{\Gamma \left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\sqrt{k \pi s^2}\right)} \left\{ 1 + k^{-1}\left( \frac{x-l}{s}\right)\right\}^{-(k+1)/2},$

wie man zeigt, dass die Student- Verteilung als eine Mischung von Gauß-Verteilungen geschrieben werden kann, indem man , , und die Verbindungsdichte , um die Randdichte ? Was sind die Parameter des resultierenden $t$ $X\sim N(\mu,\sigma^2)$ $\tau = 1/\sigma^2\sim\Gamma(\alpha,\beta)$ $f(x,\tau|\mu)$ $f(x|\mu)$ $t$ -Verteilung als Funktionen von ? $\mu,\alpha,\beta$

Durch die Integration der gemeinsamen bedingten Dichte in die Gamma-Verteilung habe ich mich in der Analysis verirrt.

distributions mixture

— Salih Ucan
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31

Das PDF einer Normalverteilung ist

f_{μ, σ} (x) = \frac{1}{\sqrt{2 π} σ} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}} d x

$f_{\mu, \sigma}(x) = \frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{(x-\mu )^2}{2 \sigma ^2}}dx$

aber in Bezug auf ist es $\tau = 1/\sigma^2$

g_{μ, τ} (x) = \frac{\sqrt{τ}}{\sqrt{2 π}} e^{- \frac{τ (x - μ)^{2}}{2}} d x .

$g_{\mu, \tau}(x) = \frac{\sqrt{\tau}}{\sqrt{2 \pi}} e^{-\frac{\tau(x-\mu )^2}{2 }}dx.$

Das PDF einer Gamma-Distribution ist

h_{α, β} (τ) = \frac{1}{Γ (α)} e^{- \frac{τ}{β}} τ^{- 1 + α} β^{- α} d τ .

$h_{\alpha, \beta}(\tau) = \frac{1}{\Gamma(\alpha)}e^{-\frac{\tau}{\beta }} \tau^{-1+\alpha } \beta ^{-\alpha }d\tau.$

Ihr Produkt, leicht vereinfacht mit einfacher Algebra, ist daher

f_{μ, α, β} (x, τ) = \frac{1}{β^{α} Γ (α) \sqrt{2 π}} e^{- τ (\frac{(x - μ)^{2}}{2} + \frac{1}{β})} τ^{- 1 / 2 + α} d τ d x .

$f_{\mu, \alpha, \beta}(x,\tau) =\frac{1}{\beta^\alpha\Gamma(\alpha)\sqrt{2 \pi}} e^{-\tau\left(\frac{(x-\mu )^2}{2 } + \frac{1}{\beta}\right)} \tau^{-1/2+\alpha}d\tau dx.$

Its inner part evidently has the form $\exp(-\text{constant}_1 \times \tau) \times \tau^{\text{constant}_2}d\tau$ , making it a multiple of a Gamma function when integrated over the full range $\tau=0$ to $\tau=\infty$ . That integral therefore is immediate (obtained by knowing the integral of a Gamma distribution is unity), giving the marginal distribution

f_{μ, α, β} (x) = \frac{\sqrt{β} Γ (α + \frac{1}{2})}{\sqrt{2 π} Γ (α)} \frac{1}{{(\frac{β}{2} (x - μ)^{2} + 1)}^{α + \frac{1}{2}}} .

$f_{\mu, \alpha, \beta}(x) = \frac{\sqrt{\beta } \Gamma \left(\alpha +\frac{1}{2}\right) }{\sqrt{2\pi } \Gamma (\alpha )}\frac{1}{\left(\frac{\beta}{2} (x-\mu )^2+1\right)^{\alpha +\frac{1}{2}}}.$

Trying to match the pattern provided for the $t$ distribution shows there is an error in the question: the PDF for the Student t distribution actually is proportional to

\frac{1}{\sqrt{k} s} {(\frac{1}{1 + k^{- 1} {(\frac{x - l}{s})}^{2}})}^{\frac{k + 1}{2}}

$\frac{1}{\sqrt{k} s }\left(\frac{1}{1+k^{-1}\left(\frac{x-l}{s}\right)^2}\right)^{\frac{k+1}{2}}$

(the power of $(x-l)/s$ is $2$ , not $1$ ). Matching the terms indicates $k = 2 \alpha$ , $l=\mu$ , and $s = 1/\sqrt{\alpha\beta}$ .

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).

— whuber
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10

Inspired by this answer I made an animation of the t distribution as a mixture of normal distributions. It is available here: sumsar.net/blog/2013/12/t-as-a-mixture-of-normals

— Rasmus Bååth

1

@whuber: Technically, for that kind of matching there is always an implicit use of calculus in your recognition that you can integrate out the gamma density using its known integral form. (This is the statistician's equivalent of hiding the broccoli by mixing it in with the meat and potatoes.) A clever way of hiding the calculus!

— Reinstate Monica

1

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 Student $t$ distribution with $k$ degree freedom can be regarded as a Normal distribution with variance mixture $Y$ , where $Y$ follows inverse gamma distribution. More precisely, $X$ ~ $t(k)$ , $X$ = $\sqrt Y$ * $\Phi$ ,where $Y$ ~ $IG(k/2,k/2)$ , $\Phi$ is standard normal rv. I hope this could help you in some sense.

— Jingjings
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0

To simplify we assume mean $0$ . Using representation, we show the result for integer degrees of freedom.

\sqrt{1 / τ} X = Y

$\sqrt{1/\tau} X =Y$ is equivalent to a Gaussian mixture with that prior: conditioned on

τ

$\tau$ ,

Y

$Y$ is Gaussian with precision

τ

$\tau$ , and the prior

τ

$\tau$ is as desired. Then it remains to show that

\sqrt{1 / τ} X

$\sqrt{1/\tau} X$ is a t-distribution. We can write

τ \sim Γ (α, β) \sim \frac{β}{2} Γ (α, 2) \sim \frac{β}{2} χ^{2} (2 α)

$\tau \sim \Gamma(\alpha, \beta) \sim \frac{\beta}{2} \Gamma(\alpha, 2) \sim \frac{\beta}{2} \chi^2(2\alpha)$ 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 \sim X \frac{1}{\sqrt{(β / 2) χ^{2} (2 α)}}

$Y \sim X \frac{1}{\sqrt{(\beta/2) \chi^2(2\alpha)} }$

= \frac{X \sqrt{α β}}{\sqrt{χ_{2 α}^{2} / (2 α)}}

$= \frac{ X \sqrt{\alpha \beta} }{\sqrt{ \chi^2_{2\alpha}/(2\alpha)}}$ which is a scaled t with

k = 2 α

$k=2\alpha$ and

s = 1 / \sqrt{α β}

$s=1/\sqrt{\alpha \beta}$ by variance of t. We can recenter our representation at

μ

$\mu$ and

l

$l$ would follow.

— Romil Sirohi
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