Template:Normal distribution bayesian confidence bounds: Difference between revisions

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==Bayesian Confidence Bounds==
#REDIRECT [[The_Normal_Distribution#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.

Latest revision as of 04:31, 13 August 2012

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