गाऊसी के मिश्रण के रूप में छात्र टी

Question

आज़ादी के डिग्री, स्थान पैरामीटर l और स्केल पैरामीटर s घनत्व वाले छात्र t- वितरण का उपयोग करना$k > 0$$l$$s$

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

कैसे पता चलता है कि छात्र -distribution अनुमति से गाऊसी वितरण का एक मिश्रण के रूप में लिखा जा सकता है एक्स ~ एन ( μ , σ 2 ) , τ = 1 / σ 2 ~ Γ ( α , β ) , और संयुक्त घनत्व को एकीकृत च ( x , inal | μ ) सीमांत घनत्व प्राप्त करने के लिए f ( x | μ ) ? परिणामस्वरूप टी के पैरामीटर क्या हैं$t$$X\sim N(\mu,\sigma^2)$$\tau = 1/\sigma^2\sim\Gamma(\alpha,\beta)$$f(x,\tau|\mu)$$f(x|\mu)$$t$ कार्य के रूप -distribution ?$\mu,\alpha,\beta$

मैं गामा वितरण के साथ संयुक्त सशर्त घनत्व को एकीकृत करके पथरी में खो गया।

Answer 1

एक सामान्य वितरण की पीडीएफ है

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

लेकिन के मामले में यह है$\tau = 1/\sigma^2$

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

एक गामा वितरण की पीडीएफ है

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

उनका उत्पाद, आसान बीजगणित के साथ थोड़ा सरल है, इसलिए

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

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

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

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

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

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

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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).

Comments

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

@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

Answer 2

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.

Answer 3

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

$$\sqrt{1/\tau} X =Y$$ is equivalent to a Gaussian mixture with that prior: conditioned on $\tau$, $Y$ is Gaussian with precision $\tau$, and the prior $\tau$ is as desired. Then it remains to show that $\sqrt{1/\tau} X$ is a t-distribution. We can write $$\tau \sim \Gamma(\alpha, \beta) \sim \frac{\beta}{2} \Gamma(\alpha, 2) \sim \frac{\beta}{2} \chi^2(2\alpha)$$ 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 $$Y \sim X \frac{1}{\sqrt{(\beta/2) \chi^2(2\alpha)} }$$ $$= \frac{ X \sqrt{\alpha \beta} }{\sqrt{ \chi^2_{2\alpha}/(2\alpha)}}$$ which is a scaled t with $k=2\alpha$ and $s=1/\sqrt{\alpha \beta}$ by variance of t. We can recenter our representation at $\mu$ and $l$ would follow.