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Bayesian Confidence Bounds

Bounds on Parameters

Bayesian Bounds use non-informative prior distributions for both parameters. From Chapter 6, we know that if the prior distribution of η and β are independent, the posterior joint distribution of η and β can be written as:

[math]\displaystyle{ f(\eta ,\beta |Data)= \dfrac{L(Data|\eta ,\beta )\varphi (\eta )\varphi (\beta )}{\int_{0}^{\infty }\int_{0}^{\infty }L(Data|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\eta d\beta } }[/math]

The marginal distribution of η is:

[math]\displaystyle{ f(\eta |Data) =\int_{0}^{\infty }f(\eta ,\beta |Data)d\beta = \dfrac{\int_{0}^{\infty }L(Data|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\beta }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(Data|\eta ,\beta )\varphi (\eta )\varphi (\beta )d\eta d\beta } }[/math]

where: [math]\displaystyle{ \varphi (\beta )=\frac{1}{\beta } }[/math] is the non-informative prior of β. [math]\displaystyle{ \varphi (\eta )=\frac{1}{\eta } }[/math] is the non-informative prior of η. Using these non-informative prior distributions, [math]\displaystyle{ f(\eta|Data) }[/math] can be rewritten as:

[math]\displaystyle{ f(\eta |Data)=\dfrac{\int_{0}^{\infty }L(Data|\eta ,\beta )\frac{1}{\beta } \frac{1}{\eta }d\beta }{\int_{0}^{\infty }\int_{0}^{\infty }L(Data|\eta ,\beta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } }[/math]

The one-sided upper bounds of η is:

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

The one-sided lower bounds of η is:

[math]\displaystyle{ 1-CL=P(\eta \leq \eta _{L})=\int_{0}^{\eta _{L}}f(\eta |Data)d\eta }[/math] The two-sided bounds of η is:

[math]\displaystyle{ CL=P(\eta _{L}\leq \eta \leq \eta _{U})=\int_{\eta _{L}}^{\eta _{U}}f(\eta |Data)d\eta }[/math] Same method is used to obtain the bounds of β.

Bounds on Reliability

[math]\displaystyle{ CL=\Pr (R\leq R_{U})=\Pr (\eta \leq T\exp (-\frac{\ln (-\ln R_{U})}{\beta })) }[/math] From the posterior distribution of η, we have:

[math]\displaystyle{ CL=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } EQNREF BayesCLR }[/math]

Eqn. (EQNREF BayesCLR ) is solved numerically for R_U. The same method can be used to calculate the one sided lower bounds and two-sided bounds on reliability.

Bounds on Time

From Chapter 5, we know that:

[math]\displaystyle{ CL=\Pr (T\leq T_{U})=\Pr (\eta \leq T_{U}\exp (-\frac{\ln (-\ln R)}{\beta })) }[/math] From the posterior distribution of η, we have:

[math]\displaystyle{ CL=\dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{ \ln (-\ln R)}{\beta })}L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\frac{1}{\beta }\frac{1}{\eta }d\eta d\beta } EQNREF BayesCLT }[/math]

Eqn. (EQNREF BayesCLT ) is solved numerically for T_U. The same method can be applied to calculate one sided lower bounds and two-sided bounds on time.