AP Statistics Curriculum 2007 Bayesian Other - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T16:40:20Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

IvoDinov: /* Probability and Statistics Ebook - Bayesian Inference for the Binomial and Poisson Distributions */

2009-10-26T20:46:27Z

‎Probability and Statistics Ebook - Bayesian Inference for the Binomial and Poisson Distributions

IvoDinov at 23:29, 22 October 2009

2009-10-22T23:29:50Z

JayZzz at 06:40, 2 June 2009

2009-06-02T06:40:27Z

JayZzz at 05:51, 28 May 2009

2009-05-28T05:51:35Z

JayZzz: New page: Bayesian Inference for the Binomial Distribution The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or fai...

2009-05-26T06:10:07Z

New page: Bayesian Inference for the Binomial Distribution The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or fai...

New page

Bayesian Inference for the Binomial Distribution

The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or failure with the trials being independent of one another and having the same probability of success. Suppose that there are n trials such that you have an observation of x successes from a binomial distribution of index n and parameter P

x ~ B(n,P)

subsequently, we can show that
p(x|P) = <math>{n \choose x}</math> <math> P^x </math> <math>(1 - P)^{n - x}</math> , (x = 0, 1, …, n)

p(x|P) is proportional to <math>P^x (1 - P)^{n - x}</math>

If the prior density has the form:
p(P) proportional to <math>P^{α - 1}</math>(1 – P)β – 1 , (P between 0 and 1)

then it follows the beta distribution
P ~ β(α,β)

From this we can appropriate the posterior which evidently has the form

p(P|x) is proportional to <math>P^{α + x – 1}</math>(1 – P)^{β + n – x – 1}

The posterior distribution of the Binomial is

(P|x) ~ β(α + x, β + n – x)

← Older revision		Revision as of 16:40, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Other}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Other}}

@@ Line 19: / Line 19: @@
 The posterior distribution of the [[EBook#Bernoulli_and_Binomial_Experiments |Binomial]] is
-: <math>(P|x) \sim \beta(\alpha + x, \beta + n – x)</math>.
+: <math> (P|x) \sim \beta(\alpha+x,\beta+n-x)</math>.
 ===Bayesian Inference for the Poisson Distribution===

@@ Line 1: / Line 1: @@
-'''Bayesian Inference for the Binomial Distribution
+==[[EBook | Probability and Statistics Ebook]] - Bayesian Inference for the Binomial and Poisson Distributions==
-The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or failure with the trials being independent of one another and having the same probability of success. Suppose that there are n trials such that you have an observation of x successes from a binomial distribution of index n and parameter P
+The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or failure with the trials being independent of one another and having the same probability of success. Suppose that there are n trials such that you have an observation of x successes from a [[EBook#Bernoulli_and_Binomial_Experiments |binomial distribution of index n and parameter P]]:
-x ~ B(n,P)
+We can show that
+: <math>p(x|P) = {n \choose x} P^x (1 - P)^{n - x}</math>, (x = 0, 1, …, n)
 If the prior density has the form:
-p(P) proportional to <math>P^{\alpha - 1}</math><math> (P-1)^{\beta - 1}</math> , (P between 0 and 1)
+: <math>p(P) \sim P^{\alpha - 1} (P-1)^{\beta - 1}</math>, (P between 0 and 1),
-P(x|<math>\lambda</math>) = (<math>\lambda^x</x!</math>)<math>e^{-\lambda}</math>
+From this we can appropriate the posterior which evidently has the form:
-Supose that you have n observations x=(x1, x2, …, xn) from such a distribution so that the likelihood is
+The posterior distribution of the [[EBook#Bernoulli_and_Binomial_Experiments |Binomial]] is
-L(<math>\lambda</math>|x) = <math>\lambda^T e^{(-n \lambda)}</math>, where T = <math>\sum{k_i}</math>
+A discrete random variable x is said to have a [[EBook#Poisson_Distribution |Poisson distribution]] of mean <math>\lambda</math> if it has the density:
-In Bayesian inference, the conjugate prior for the parameter <math>\lambda</math> of the Poisson distribution is the Gamma distribution.
+Suppose that you have n observations <math>x=(x_1, x_2, \cdots, x_n)</math> from such a distribution so that the likelihood is:
-<math>\lambda \sim</math> Gamma(<math>\alpha</math> , <math>\beta</math> )
+In Bayesian inference, the conjugate prior for the parameter <math>\lambda</math> of the [http://socr.ucla.edu/htmls/dist/Poisson_Distribution.html Poisson distribution] is the [http://socr.ucla.edu/htmls/dist/Gamma_Distribution.html Gamma distribution].
-The Poisson parameter <math>\lambda</math> is distributed accordingly to the parameterized Gamma density g in terms of a shape and inverse scale parameter <math>\alpha</math> and <math>\beta</math> respectively
+The Poisson parameter <math>\lambda</math> is distributed accordingly to the parametrized Gamma density ''g'' in terms of a shape and inverse scale parameter <math>\alpha</math> and <math>\beta</math> respectively:
-g(<math>\lambda</math>|<math>\alpha</math> , <math>\beta</math>) = <math>\displaystyle\frac{\beta^\alpha}{\Gamma(\alpha)}</math> <math>\lambda^{\alpha - 1} e^{-\beta \lambda}</math>
+Then, given the same sample of ''n'' measured values <math>k_i</math> from our likelihood and a prior of <math>\Gamma(\alpha, \beta)</math>, the posterior distribution becomes:
-For <math>\lambda</math> > 0
+: <math>\lambda \sim  \Gamma (\alpha + \displaystyle\sum_{i=1}^{\infty} k_i, \beta +n)</math>.
-Then, given the same sample of n measured values <math>k_i</math> from our likelihood and a prior of Gamma(<math>\alpha</math>, <math>\beta</math>), the posterior distribution becomes
+==See also==
-<math>\lambda \sim</math> Gamma (<math>\alpha + \displaystyle\sum_{i=1}^{\infty} k_i</math> , <math>\beta</math> + n)
+<hr>
-The posterior mean E[<math>\lambda</math>] approaches the maximum likelihood estimate in the limit as <math>\alpha</math> and <math>\beta</math> approach 0.
+{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Other}}

← Older revision		Revision as of 06:40, 2 June 2009
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	(P\|x) ~ β(α + x, β + n – x)		(P\|x) ~ β(α + x, β + n – x)
		+
		+
		+
		+	'''Bayesian Inference for the Poisson Distribution'''
		+
		+	A discrete random variable x is said to have a Poisson distribution of mean <math>\lambda</math> if it has the density
		+
		+
		+	P(x\|<math>\lambda</math>) = (<math>\lambda^x</x!</math>)<math>e^{-\lambda}</math>
		+
		+	Supose that you have n observations x=(x1, x2, …, xn) from such a distribution so that the likelihood is
		+
		+
		+	L(<math>\lambda</math>\|x) = <math>\lambda^T e^{(-n \lambda)}</math>, where T = <math>\sum{k_i}</math>
		+
		+	In Bayesian inference, the conjugate prior for the parameter <math>\lambda</math> of the Poisson distribution is the Gamma distribution.
		+
		+	<math>\lambda \sim</math> Gamma(<math>\alpha</math> , <math>\beta</math> )
		+
		+
		+	The Poisson parameter <math>\lambda</math> is distributed accordingly to the parameterized Gamma density g in terms of a shape and inverse scale parameter <math>\alpha</math> and <math>\beta</math> respectively
		+
		+
		+	g(<math>\lambda</math>\|<math>\alpha</math> , <math>\beta</math>) = <math>\displaystyle\frac{\beta^\alpha}{\Gamma(\alpha)}</math> <math>\lambda^{\alpha - 1} e^{-\beta \lambda}</math>
		+	For <math>\lambda</math> > 0
		+
		+
		+	Then, given the same sample of n measured values <math>k_i</math> from our likelihood and a prior of Gamma(<math>\alpha</math>, <math>\beta</math>), the posterior distribution becomes
		+
		+
		+	<math>\lambda \sim</math> Gamma (<math>\alpha + \displaystyle\sum_{i=1}^{\infty} k_i</math> , <math>\beta</math> + n)
		+
		+	The posterior mean E[<math>\lambda</math>] approaches the maximum likelihood estimate in the limit as <math>\alpha</math> and <math>\beta</math> approach 0.

@@ Line 1: / Line 1: @@
-Bayesian Inference for the Binomial Distribution
+'''Bayesian Inference for the Binomial Distribution
 The parameters of interest in this section is the probability P of success in a number of trials which can result in either success or failure with the trials being independent of one another and having the same probability of success. Suppose that there are n trials such that you have an observation of x successes from a binomial distribution of index n and parameter P
@@ Line 9: / Line 10: @@
 p(x|P) is proportional to <math>P^x (1 - P)^{n - x}</math>
 If the prior density has the form:
-p(P) proportional to <math>P^{α - 1}</math>(1 – P)β – 1 , (P between 0 and 1)
+p(P) proportional to <math>P^{\alpha - 1}</math><math> (P-1)^{\beta - 1}</math> , (P between 0 and 1)
 then it follows the beta distribution
 P ~ β(α,β)
 From this we can appropriate the posterior which evidently has the form
-p(P|x) is proportional to <math>P^{α + x – 1}</math>(1 – P)^{β + n – x – 1}
+p(P|x) is proportional to <math>P^{\alpha + x - 1}</math><math>(1-P)^{\beta + n - x - 1}</math>
 The posterior distribution of the Binomial is
 (P|x) ~ β(α + x, β + n – x)