Template:Bayesian Confidence Bounds ED

Bounds on Parameters
From Chapter 5, we know that the posterior distribution of $$\lambda $$ can be written as:


 * $$f(\lambda |Data)=\frac{L(Data|\lambda )\varphi (\lambda )}{\int_{0}^{\infty }L(Data|\lambda )\varphi (\lambda )d\lambda }$$

where $$\varphi (\lambda )=\tfrac{1}{\lambda }$$, is the non-informative prior of $$\lambda $$.

With the above prior distribution, $$f(\lambda |Data)$$ can be rewritten as:


 * $$f(\lambda |Data)=\frac{L(Data|\lambda )\tfrac{1}{\lambda }}{\int_{0}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }$$

The one-sided upper bound of $$\lambda $$ is:


 * $$CL=P(\lambda \le {{\lambda }_{U}})=\int_{0}^f(\lambda |Data)d\lambda $$

The one-sided lower bound of $$\lambda $$ is:


 * $$1-CL=P(\lambda \le {{\lambda }_{L}})=\int_{0}^f(\lambda |Data)d\lambda $$

The two-sided bounds of $$\lambda $$ are:


 * $$CL=P({{\lambda }_{L}}\le \lambda \le {{\lambda }_{U}})=\int_^f(\lambda |Data)d\lambda $$

Bounds on Time (Type 1)
The reliable life equation is:


 * $$t=\frac{-\ln R}{\lambda }$$

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


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(t\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln R}{\lambda }\le {{T}_{U}})$$

The above equation can be rewritten in terms of $$\lambda $$ as:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln R}\le \lambda )$$

From the above posterior distribuiton equation, we have:


 * $$CL=\frac{\int_{\tfrac{-\ln R}}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }{\int_{0}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }$$

The above equation is solved w.r.t. $${{t}_{U}}.$$ 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:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\exp (-\lambda t)\le {{R}_{U}})$$

The above equaation can be rewritten in terms of $$\lambda $$ as:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\frac{-\ln {{R}_{U}}}{t}\le \lambda )$$

From the equation for posterior distribution we have:


 * $$CL=\frac{\int_{\tfrac{-\ln {{R}_{U}}}{t}}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }{\int_{0}^{\infty }L(Data|\lambda )\tfrac{1}{\lambda }d\lambda }$$

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