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| ===Bayesian Confidence Bounds===
| | #REDIRECT [[The_Normal_Distribution#Bayesian_Confidence_Bounds]] |
| ====Bounds on Parameters====
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| From Chapter 5, we know that the marginal posterior distribution of <math>\mu </math> can be written as:
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| ::<math>\begin{align}
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| f(\mu |Data)= & \int_{0}^{\infty }f(\mu ,\sigma |Data)d\sigma \\
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| = & \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 }
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| \end{align}</math>
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| :where:
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| <math>\varphi (\sigma )</math> = <math>\tfrac{1}{\sigma }</math> is the non-informative prior of <math>\sigma </math> .
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| ::<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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| Using the above prior distributions, <math>f(\mu |Data)</math> can be rewritten as:
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| ::<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>
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| The one-sided upper bound of <math>\mu </math> is:
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| ::<math>CL=P(\mu \le {{\mu }_{U}})=\int_{-\infty }^{{{\mu }_{U}}}f(\mu |Data)d\mu </math>
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| The one-sided lower bound of <math>\mu </math> is:
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| ::<math>1-CL=P(\mu \le {{\mu }_{L}})=\int_{-\infty }^{{{\mu }_{L}}}f(\mu |Data)d\mu </math>
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| The two-sided bounds of <math>\mu </math> are:
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| ::<math>CL=P({{\mu }_{L}}\le \mu \le {{\mu }_{U}})=\int_{{{\mu }_{L}}}^{{{\mu }_{U}}}f(\mu |Data)d\mu </math>
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| The same method can be used to obtained the bounds of <math>\sigma </math>.
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| ====Bounds on Time (Type 1)====
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| The reliable life for the normal distribution is:
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| ::<math>T=\mu +\sigma {{\Phi }^{-1}}(1-R)</math>
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| The one-sided upper bound on time is:
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| ::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(T\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu +\sigma {{\Phi }^{-1}}(1-R)\le {{T}_{U}})</math>
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| Eqn. (1SCBT) can be rewritten in terms of <math>\mu </math> as:
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| ::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le {{T}_{U}}-\sigma {{\Phi }^{-1}}(1-R))</math>
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| From the posterior distribution of <math>\mu \ \ :</math>
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| ::<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>
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| The same method can be applied for one-sided lower bounds and two-sided bounds on time.
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| ====Bounds on Reliability (Type 2)====
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| The one-sided upper bound on reliability is:
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| ::<math>CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le T-\sigma {{\Phi }^{-1}}(1-{{R}_{U}}))</math>
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| From the posterior distribution of <math>\mu \ \ :</math>
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| ::<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>
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| The same method can be used to calculate the one-sided lower bounds and the two-sided bounds on reliability.
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