Template:Bayesian Confidence Bounds ED

Bayesian Confidence Bounds

Bounds on Parameters

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

[math]\displaystyle{ f(\lambda |Data)=\frac{L(Data|\lambda )\varphi (\lambda )}{\int_{0}^{\infty }L(Data|\lambda )\varphi (\lambda )d\lambda } }[/math]

where [math]\displaystyle{ \varphi (\lambda )=\tfrac{1}{\lambda } }[/math], is the non-informative prior of [math]\displaystyle{ \lambda }[/math].

With the above prior distribution, [math]\displaystyle{ f(\lambda |Data) }[/math] can be rewritten as:

[math]\displaystyle{ f(\lambda |Data)=\frac{L(Data|\lambda )\tfrac{1}{\lambda }}{\int_{0}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda } }[/math]

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

[math]\displaystyle{ CL=P(\lambda \le {{\lambda }_{U}})=\int_{0}^{{{\lambda }_{U}}}f(\lambda |Data)d\lambda }[/math]

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

[math]\displaystyle{ 1-CL=P(\lambda \le {{\lambda }_{L}})=\int_{0}^{{{\lambda }_{L}}}f(\lambda |Data)d\lambda }[/math]

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

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

Bounds on Time (Type 1)

The reliable life equation is:

[math]\displaystyle{ t=\frac{-\ln R}{\lambda } }[/math]

For the one-sided upper bound on time we have:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(t\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln R}{\lambda }\le {{T}_{U}}) }[/math]

The above equation can be rewritten in terms of [math]\displaystyle{ \lambda }[/math] as:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln R}{{{t}_{U}}}\le \lambda ) }[/math]

From the above posterior distribuiton equation, we have:

[math]\displaystyle{ CL=\frac{\int_{\tfrac{-\ln R}{{{t}_{U}}}}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }{\int_{0}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda } }[/math]

The above equation is solved w.r.t. [math]\displaystyle{ {{t}_{U}}. }[/math] The same method is applied for one-sided lower and two-sided bounds on time.

Bounds on Reliability (Type 2)

The one-sided upper bound on reliability is given by:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\exp (-\lambda t)\le {{R}_{U}}) }[/math]

The above equaation can be rewritten in terms of [math]\displaystyle{ \lambda }[/math] as:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln {{R}_{U}}}{t}\le \lambda ) }[/math]

From the equation for posterior distribution we have:

[math]\displaystyle{ CL=\frac{\int_{\tfrac{-\ln {{R}_{U}}}{t}}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }{\int_{0}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda } }[/math]

The above equation is solved w.r.t. [math]\displaystyle{ {{R}_{U}}. }[/math] The same method can be used to calculate one-sided lower and two sided bounds on reliability.