Template:Normal distribution bayesian confidence bounds - Revision history

Richard House: Redirected page to The Normal Distribution#Bayesian Confidence Bounds

2012-08-13T04:31:52Z

Redirected page to The Normal Distribution#Bayesian Confidence Bounds

← Older revision		Revision as of 04:31, 13 August 2012
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	~~===Bayesian Confidence Bounds===~~		#REDIRECT [[The_Normal_Distribution#Bayesian_Confidence_Bounds]]
	~~====Bounds on Parameters====~~
	~~From chapter for~~ [[~~Confidence Bounds~~]]~~, we know that the marginal posterior distribution of <math>\mu </math> can be written as:~~

	~~::<math>\begin{align}~~
	~~f(\mu \|Data)= & \int_{0}^{\infty }f(\mu ,\sigma \|Data)d\sigma \\~~
	~~= & \frac{\int_{0}^{\infty }L(Data\|\mu ,\sigma )\varphi (\mu )\varphi (\sigma )d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|\mu ,\sigma )\varphi (\mu )\varphi (\sigma )d\mu d\sigma }~~
	~~\end{align}</math>~~

	~~where:~~

	~~::<math>\varphi (\sigma )</math> = <math>\tfrac{1}{\sigma }</math> is the non-informative prior of <math>\sigma </math> .~~

	~~::<math>\varphi (\mu )</math> is a uniform distribution from - <math>\infty </math> to + <math>\infty </math> , the non-informative prior of <math>\mu .</math>~~

	~~Using the above prior distributions, <math>f(\mu \|Data)</math> can be rewritten as:~~


	~~::<math>f(\mu \|Data)=\frac{\int_{0}^{\infty }L(Data\|\mu ,\sigma )\tfrac{1}{\sigma }d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data\|\mu ,\sigma )\tfrac{1}{\sigma }d\mu d\sigma }</math>~~


	~~The one-sided upper bound of <math>\mu </math> is:~~


	~~::<math>CL=P(\mu \le {{\mu }_{U}})=\int_{-\infty }^{{{\mu }_{U}}}f(\mu \|Data)d\mu </math>~~


	~~The one-sided lower bound of <math>\mu </math> is:~~


	~~::<math>1-CL=P(\mu \le {{\mu }_{L}})=\int_{-\infty }^{{{\mu }_{L}}}f(\mu \|Data)d\mu </math>~~


	~~The two-sided bounds of <math>\mu </math> are:~~


	~~::<math>CL=P({{\mu }_{L}}\le \mu \le {{\mu }_{U}})=\int_{{{\mu }_{L}}}^{{{\mu }_{U}}}f(\mu \|Data)d\mu </math>~~


	~~The same method can be used to obtained the bounds of <math>\sigma </math>.~~

	~~====Bounds on Time (Type 1)====~~
	~~The reliable life for the normal distribution is:~~


	~~::<math>T=\mu +\sigma {{\Phi }^{-1}}(1-R)</math>~~


	~~The one-sided upper bound on time is:~~


	~~::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(T\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu +\sigma {{\Phi }^{-1}}(1-R)\le {{T}_{U}})</math>~~


	~~The above equation can be rewritten in terms of <math>\mu </math> as:~~


	~~::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le {{T}_{U}}-\sigma {{\Phi }^{-1}}(1-R))</math>~~


	~~From the posterior distribution of <math>\mu</math>:~~


	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{{{T}_{U}}-\sigma {{\Phi }^{-1}}(1-R)}L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }</math>


	~~The same method can be applied for one-sided lower bounds and two-sided bounds on time.~~

	~~====Bounds on Reliability (Type 2)====~~
	~~The one-sided upper bound on reliability is:~~


	~~::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le T-\sigma {{\Phi }^{-1}}(1-{{R}_{U}}))</math>~~


	~~From the posterior distribution of <math>\mu</math>:~~


	::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{T-\sigma {{\Phi }^{-1}}(1-{{R}_{U}})}L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }</math>


	~~The same method can be used to calculate the one-sided lower bounds and the two-sided bounds on reliability.~~

Harry Guo: /* Bayesian Confidence Bounds */

2012-02-10T22:25:35Z

Bayesian Confidence Bounds

← Older revision		Revision as of 22:25, 10 February 2012
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	===Bayesian Confidence Bounds===		===Bayesian Confidence Bounds===
	====Bounds on Parameters====		====Bounds on Parameters====
	From ~~Chapter 5~~, we know that the marginal posterior distribution of <math>\mu </math> can be written as:		From chapter for [[Confidence Bounds]], we know that the marginal posterior distribution of <math>\mu </math> can be written as:

	::<math>\begin{align}		::<math>\begin{align}
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	\end{align}</math>		\end{align}</math>

	:where:		where:

	<math>\varphi (\sigma )</math> = <math>\tfrac{1}{\sigma }</math> is the non-informative prior of <math>\sigma </math> .		::<math>\varphi (\sigma )</math> = <math>\tfrac{1}{\sigma }</math> is the non-informative prior of <math>\sigma </math> .

	::<math>\varphi (\mu )</math> is a uniform distribution from - <math>\infty </math> to + <math>\infty </math> , the non-informative prior of <math>\mu .</math>		::<math>\varphi (\mu )</math> is a uniform distribution from - <math>\infty </math> to + <math>\infty </math> , the non-informative prior of <math>\mu .</math>
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	~~Eqn. (1SCBT)~~ can be rewritten in terms of <math>\mu </math> as:		The above equation can be rewritten in terms of <math>\mu </math> as:


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	From the posterior distribution of <math>\mu ~~\ \ :~~</math>		From the posterior distribution of <math>\mu</math>:


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	From the posterior distribution of <math>\mu ~~\ \ :~~</math>		From the posterior distribution of <math>\mu</math>:

Harry Guo: /* Bayesian Confidence Bounds */

2012-02-10T17:42:47Z

Bayesian Confidence Bounds

← Older revision		Revision as of 17:42, 10 February 2012
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	==Bayesian Confidence Bounds==		===Bayesian Confidence Bounds===
	===Bounds on Parameters===		====Bounds on Parameters====
	From Chapter 5, we know that the marginal posterior distribution of <math>\mu </math> can be written as:		From Chapter 5, we know that the marginal posterior distribution of <math>\mu </math> can be written as:

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	The same method can be used to obtained the bounds of <math>\sigma </math>.		The same method can be used to obtained the bounds of <math>\sigma </math>.

	===Bounds on Time (Type 1)===		====Bounds on Time (Type 1)====
	The reliable life for the normal distribution is:		The reliable life for the normal distribution is:

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	The same method can be applied for one-sided lower bounds and two-sided bounds on time.		The same method can be applied for one-sided lower bounds and two-sided bounds on time.

	===Bounds on Reliability (Type 2)===		====Bounds on Reliability (Type 2)====
	The one-sided upper bound on reliability is:		The one-sided upper bound on reliability is:

Harry Guo: moved Template:Nd bayesian confidence bounds to Template:Normal distribution bayesian confidence bounds

2012-02-10T17:35:29Z

moved Template:Nd bayesian confidence bounds to Template:Normal distribution bayesian confidence bounds

← Older revision	Revision as of 17:35, 10 February 2012
(No difference)

Nicolette Young: Created page with '==Bayesian Confidence Bounds== ===Bounds on Parameters=== From Chapter 5, we know that the marginal posterior distribution of $\mu$ can be written as: :: $\b…'$

2012-01-04T17:48:50Z

Created page with '==Bayesian Confidence Bounds== ===Bounds on Parameters=== From Chapter 5, we know that the marginal posterior distribution of <math>\mu </math> can be written as: ::<math>\b…'

New page

==Bayesian Confidence Bounds==
===Bounds on Parameters===
From Chapter 5, we know that the marginal posterior distribution of <math>\mu </math> can be written as:

::<math>\begin{align}
f(\mu |Data)= & \int_{0}^{\infty }f(\mu ,\sigma |Data)d\sigma \\
= & \frac{\int_{0}^{\infty }L(Data|\mu ,\sigma )\varphi (\mu )\varphi (\sigma )d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|\mu ,\sigma )\varphi (\mu )\varphi (\sigma )d\mu d\sigma }
\end{align}</math>

:where:

<math>\varphi (\sigma )</math> = <math>\tfrac{1}{\sigma }</math> is the non-informative prior of <math>\sigma </math> .

::<math>\varphi (\mu )</math> is a uniform distribution from - <math>\infty </math> to + <math>\infty </math> , the non-informative prior of <math>\mu .</math>

Using the above prior distributions, <math>f(\mu |Data)</math> can be rewritten as:

::<math>f(\mu |Data)=\frac{\int_{0}^{\infty }L(Data|\mu ,\sigma )\tfrac{1}{\sigma }d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|\mu ,\sigma )\tfrac{1}{\sigma }d\mu d\sigma }</math>

The one-sided upper bound of <math>\mu </math> is:

::<math>CL=P(\mu \le {{\mu }_{U}})=\int_{-\infty }^{{{\mu }_{U}}}f(\mu |Data)d\mu </math>

The one-sided lower bound of <math>\mu </math> is:

::<math>1-CL=P(\mu \le {{\mu }_{L}})=\int_{-\infty }^{{{\mu }_{L}}}f(\mu |Data)d\mu </math>

The two-sided bounds of <math>\mu </math> are:

::<math>CL=P({{\mu }_{L}}\le \mu \le {{\mu }_{U}})=\int_{{{\mu }_{L}}}^{{{\mu }_{U}}}f(\mu |Data)d\mu </math>

The same method can be used to obtained the bounds of <math>\sigma </math>.

===Bounds on Time (Type 1)===
The reliable life for the normal distribution is:

::<math>T=\mu +\sigma {{\Phi }^{-1}}(1-R)</math>

The one-sided upper bound on time is:

::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(T\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu +\sigma {{\Phi }^{-1}}(1-R)\le {{T}_{U}})</math>

Eqn. (1SCBT) can be rewritten in terms of <math>\mu </math> as:

::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le {{T}_{U}}-\sigma {{\Phi }^{-1}}(1-R))</math>

From the posterior distribution of <math>\mu \ \ :</math>

::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{{{T}_{U}}-\sigma {{\Phi }^{-1}}(1-R)}L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }</math>

The same method can be applied for one-sided lower bounds and two-sided bounds on time.

===Bounds on Reliability (Type 2)===
The one-sided upper bound on reliability is:

::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le T-\sigma {{\Phi }^{-1}}(1-{{R}_{U}}))</math>

From the posterior distribution of <math>\mu \ \ :</math>

::<math>CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{T-\sigma {{\Phi }^{-1}}(1-{{R}_{U}})}L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }</math>

The same method can be used to calculate the one-sided lower bounds and the two-sided bounds on reliability.

Template:Normal distribution bayesian confidence bounds - Revision history

Richard House: Redirected page to The Normal Distribution#Bayesian Confidence Bounds

Harry Guo: /* Bayesian Confidence Bounds */

Harry Guo: /* Bayesian Confidence Bounds */

Harry Guo: moved Template:Nd bayesian confidence bounds to Template:Normal distribution bayesian confidence bounds

Nicolette Young: Created page with '==Bayesian Confidence Bounds== ===Bounds on Parameters=== From Chapter 5, we know that the marginal posterior distribution of \mu can be written as: ::\b…'

Nicolette Young: Created page with '==Bayesian Confidence Bounds== ===Bounds on Parameters=== From Chapter 5, we know that the marginal posterior distribution of $\mu$ can be written as: :: $\b…'$