Template:Normal distribution bayesian confidence bounds

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


 * $$\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}$$


 * where:

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


 * $$\varphi (\mu )$$ is a uniform distribution from - $$\infty $$  to + $$\infty $$, the non-informative prior of  $$\mu .$$

Using the above prior distributions, $$f(\mu |Data)$$  can be rewritten as:


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

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


 * $$CL=P(\mu \le {{\mu }_{U}})=\int_{-\infty }^f(\mu |Data)d\mu $$

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


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

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


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

The same method can be used to obtained the bounds of $$\sigma $$.

Bounds on Time (Type 1)
The reliable life for the normal distribution is:


 * $$T=\mu +\sigma {{\Phi }^{-1}}(1-R)$$

The one-sided upper bound on time is:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(T\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu +\sigma {{\Phi }^{-1}}(1-R)\le {{T}_{U}})$$

Eqn. (1SCBT) can be rewritten in terms of $$\mu $$  as:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le {{T}_{U}}-\sigma {{\Phi }^{-1}}(1-R))$$

From the posterior distribution of $$\mu \ \ :$$


 * $$CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{{{T}_{U}}-\sigma {{\Phi }^{-1}}(1-R)}L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }$$

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:


 * $$CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le T-\sigma {{\Phi }^{-1}}(1-{{R}_{U}}))$$

From the posterior distribution of $$\mu \ \ :$$


 * $$CL=\frac{\int_{0}^{\infty }\int_{-\infty }^{T-\sigma {{\Phi }^{-1}}(1-{{R}_{U}})}L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }{\int_{0}^{\infty }\int_{-\infty }^{\infty }L(\sigma ,\mu )\tfrac{1}{\sigma }d\mu d\sigma }$$

The same method can be used to calculate the one-sided lower bounds and the two-sided bounds on reliability.