Template:Normal distribution bayesian confidence bounds: Difference between revisions

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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>:





Revision as of 22:25, 10 February 2012

Bayesian Confidence Bounds

Bounds on Parameters

From chapter for Confidence Bounds, we know that the marginal posterior distribution of [math]\displaystyle{ \mu }[/math] can be written as:

[math]\displaystyle{ \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]\displaystyle{ \varphi (\sigma ) }[/math] = [math]\displaystyle{ \tfrac{1}{\sigma } }[/math] is the non-informative prior of [math]\displaystyle{ \sigma }[/math] .
[math]\displaystyle{ \varphi (\mu ) }[/math] is a uniform distribution from - [math]\displaystyle{ \infty }[/math] to + [math]\displaystyle{ \infty }[/math] , the non-informative prior of [math]\displaystyle{ \mu . }[/math]

Using the above prior distributions, [math]\displaystyle{ f(\mu |Data) }[/math] can be rewritten as:


[math]\displaystyle{ 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]\displaystyle{ \mu }[/math] is:


[math]\displaystyle{ CL=P(\mu \le {{\mu }_{U}})=\int_{-\infty }^{{{\mu }_{U}}}f(\mu |Data)d\mu }[/math]


The one-sided lower bound of [math]\displaystyle{ \mu }[/math] is:


[math]\displaystyle{ 1-CL=P(\mu \le {{\mu }_{L}})=\int_{-\infty }^{{{\mu }_{L}}}f(\mu |Data)d\mu }[/math]


The two-sided bounds of [math]\displaystyle{ \mu }[/math] are:


[math]\displaystyle{ 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]\displaystyle{ \sigma }[/math].

Bounds on Time (Type 1)

The reliable life for the normal distribution is:


[math]\displaystyle{ T=\mu +\sigma {{\Phi }^{-1}}(1-R) }[/math]


The one-sided upper bound on time is:


[math]\displaystyle{ 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]\displaystyle{ \mu }[/math] as:


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


From the posterior distribution of [math]\displaystyle{ \mu }[/math]:


[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \mu }[/math]:


[math]\displaystyle{ 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.