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{{Probability distribution |
  name      =Normal-Wishart|
  type      =density|
  pdf_image  =|
  cdf_image  =|
  notation =<math> (\boldsymbol\mu,\boldsymbol\Lambda) \sim \mathrm{NW}(\boldsymbol\mu_0,\lambda,\mathbf{W},\nu)</math>|
  parameters =<math>\boldsymbol\mu_0\in\mathbb{R}^D\,</math> [[location parameter|location]] (vector of [[real number|real]])<br /><math>\lambda > 0\,</math> (real)<br /><math>\mathbf{W} \in\mathbb{R}^{D\times D}</math> scale matrix ([[positive-definite matrix|pos. def.]])<br /><math>\nu > D-1\,</math> (real)|
  support    =<math>\boldsymbol\mu\in\mathbb{R}^D ; \boldsymbol\Lambda \in\mathbb{R}^{D\times D}</math> [[covariance matrix]] ([[positive-definite matrix|pos. def.]])|
  pdf        =<math>f(\boldsymbol\mu,\boldsymbol\Lambda|\boldsymbol\mu_0,\lambda,\mathbf{W},\nu) = \mathcal{N}(\boldsymbol\mu|\boldsymbol\mu_0,(\lambda\boldsymbol\Lambda)^{-1})\ \mathcal{W}(\boldsymbol\Lambda|\mathbf{W},\nu)</math>|
  cdf        =|
  mean      =|
  median    =|
  mode      =|
  variance  =|
  skewness  =|
  kurtosis  =|
  entropy    =|
  mgf        =|
  char      =|
}}
In [[probability theory]] and [[statistics]], the '''normal-Wishart distribution''' (or '''Gaussian-Wishart distribution''') is a multivariate four-parameter family of continuous [[probability distribution]]s. It is the [[conjugate prior]] of a [[multivariate normal distribution]] with unknown [[mean]] and [[precision matrix]] (the inverse of the [[covariance matrix]]).<ref name="bishop">Bishop, Christopher M. (2006). ''Pattern Recognition and Machine Learning.'' Springer Science+Business Media. Page 690.</ref>


==Definition==
Suppose


:<math>  \boldsymbol\mu|\boldsymbol\mu_0,\lambda,\boldsymbol\Lambda \sim \mathcal{N}(\boldsymbol\mu|\boldsymbol\mu_0,(\lambda\boldsymbol\Lambda)^{-1}) </math>
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has a [[multivariate normal distribution]] with [[mean]] <math>\boldsymbol\mu_0</math> and [[covariance matrix]] <math>(\lambda\boldsymbol\Lambda)^{-1}</math>, where
 
:<math>\boldsymbol\Lambda|\mathbf{W},\nu \sim \mathcal{W}(\boldsymbol\Lambda|\mathbf{W},\nu)</math>
has a [[Wishart distribution]]. Then <math>(\boldsymbol\mu,\boldsymbol\Lambda) </math>
has a normal-Wishart distribution, denoted as
:<math> (\boldsymbol\mu,\boldsymbol\Lambda) \sim \mathrm{NW}(\boldsymbol\mu_0,\lambda,\mathbf{W},\nu) .
</math>
 
==Characterization==
 
===Probability density function===
 
: <math>f(\boldsymbol\mu,\boldsymbol\Lambda|\boldsymbol\mu_0,\lambda,\mathbf{W},\nu) = \mathcal{N}(\boldsymbol\mu|\boldsymbol\mu_0,(\lambda\boldsymbol\Lambda)^{-1})\ \mathcal{W}(\boldsymbol\Lambda|\mathbf{W},\nu)</math>
 
==Properties==
 
===Scaling===
 
===Marginal distributions===
By construction, the [[marginal distribution]] over <math>\boldsymbol\Lambda</math> is a [[Wishart distribution]], and the [[conditional distribution]] over <math>\boldsymbol\mu</math> given <math>\boldsymbol\Lambda</math> is a [[multivariate normal distribution]]. The [[marginal distribution]] over <math>\boldsymbol\mu</math> is a [[multivariate t-distribution]].
 
== Posterior distribution of the parameters ==
 
{{Empty section|date=March 2013}}
 
== Generating normal-Wishart random variates ==
Generation of random variates is straightforward:
# Sample <math>\boldsymbol\Lambda</math> from a [[Wishart distribution]] with parameters <math>\mathbf{W}</math> and <math>\nu</math>
# Sample <math>\boldsymbol\mu</math> from a [[multivariate normal distribution]] with mean <math>\boldsymbol\mu_0</math> and variance <math>(\lambda\boldsymbol\Lambda)^{-1}</math>
 
== Related distributions ==
* The [[normal-inverse Wishart distribution]] is essentially the same distribution parameterized by variance rather than precision.
* The [[normal-gamma distribution]] is the one-dimensional equivalent.
* The [[multivariate normal distribution]] and [[Wishart distribution]] are the component distributions out of which this distribution is made.
 
==Notes==
{{reflist}}
 
== References ==
* Bishop, Christopher M. (2006). ''Pattern Recognition and Machine Learning.'' Springer Science+Business Media.
 
{{ProbDistributions|multivariate}}
 
{{DEFAULTSORT:Normal-Wishart Distribution}}
[[Category:Multivariate continuous distributions]]
[[Category:Conjugate prior distributions]]
[[Category:Normal distribution]]
[[Category:Probability distributions]]

Latest revision as of 00:19, 7 January 2015


Amazon is easily the most the biggest marketplaces online that sells to people all around the globe. Since the reason such substantial website, it reaches an shoppers each and every day. Virtually anything can be found in this site, from cups and plates to books and televisions. A lot of people actually sell products through Amazon because built able to reach so many potential customers through out. This is not a public auction site like Ebay, so all items get their price listed and shipping will be extra if it's not seen on Amazon's Prime Membership. Using tips help to make it money selling on Amazon can help anyone flourish.

In addition to being cool, Macs are at ease. While PCs are open to each different types of viruses, an Apple buyer can assured that they will not have to deal with frequent system wrecks. This is just one more amazing selling machine point for Apple. Even though I guarantee Windows 7 does the job, people no longer trust this method. Therefore, companies such as HP are struggling to outlive.

Don't make your mistake in thinking you can start making money without ANY investment. Neglect the idea of greenbacks for nothing. Here is the bare minimum you should expect to buy. If you aren't willing to spend this much money, truly go find another venture. $8.95 per year for a web page.$6.00 per month for hosting web page.$50 to $100 for a good web editor like DreamWeaver.$400 for a pretty good computer (check Dell's Refurbished site most effective deals).and an investment in products if ensure be selling physical products online.

Branding is a conceptual issue. But what relating to the more practical aspects of packaging? What type of materials should companies use? Just how can products be packaged and shipped safely without damage? If you're ready to see more info on the amazing selling machine course look into the web-site. As an expert in packaging, it's very much you give such recommendations. Many businesses to help package and ship unique products, they will don't know how to package the products properly. As the result, products get damaged during cross over. This leads to loss of revenues.

The reason they are cheap is really because the local book stores near colleges buy books from people. The ones cannot get regarding end up selling on amazon. With fierce competition on amazon it drastically drops rates.

Selling individually on Amazon will probably cost you more in fees than selling individually on eBay, but your product will have greater visibility on Amazon . com. On eBay, your listing is proscribed to ten days. On Amazon, items can be listed indefinitely, as long as you continue to pay your monthly fee.

Amazon gives sellers a "shipping credit", based on an item's league. This does not really cover complete shipping amount, but usually does. End up getting is fixed by The amazon website. You cannot solicit more from the customer, very well as if the shipping credit does not cover your shipping costs, you must ship them. Since the shipping credit is fixed, it can be figured in when setting your selling price.

Remember, have your free content produced from high quality, informative, and entertaining content, and perform three things above. That's about that to do in order to produce a back-end selling machine! Just keep with your mind, when reaching your free item, overlook about your back-end gross sales.or else you'll be losing out on prime selling space!

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