Template:Normal distribution bayesian confidence bounds

Bayesian Confidence Bounds

Bounds on Parameters

From Chapter 5, we know that the marginal posterior distribution of [math]\displaystyle{ \mu }[/math] can be written as:

[math]\displaystyle{ \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} }[/math]

where:

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

[math]\displaystyle{ \varphi (\mu ) }[/math] is a uniform distribution from - [math]\displaystyle{ \infty }[/math] to + [math]\displaystyle{ \infty }[/math] , the non-informative prior of [math]\displaystyle{ \mu . }[/math]

Using the above prior distributions, [math]\displaystyle{ f(\mu |Data) }[/math] can be rewritten as:

[math]\displaystyle{ 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]

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

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

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

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

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

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

The same method can be used to obtained the bounds of [math]\displaystyle{ \sigma }[/math].

Bounds on Time (Type 1)

The reliable life for the normal distribution is:

[math]\displaystyle{ T=\mu +\sigma {{\Phi }^{-1}}(1-R) }[/math]

The one-sided upper bound on time is:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(T\le {{T}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu +\sigma {{\Phi }^{-1}}(1-R)\le {{T}_{U}}) }[/math]

Eqn. (1SCBT) can be rewritten in terms of [math]\displaystyle{ \mu }[/math] as:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le {{T}_{U}}-\sigma {{\Phi }^{-1}}(1-R)) }[/math]

From the posterior distribution of [math]\displaystyle{ \mu \ \ : }[/math]

[math]\displaystyle{ 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]

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:

[math]\displaystyle{ CL=\underset{}{\overset{}{\mathop{\Pr }}}\,(R\le {{R}_{U}})=\underset{}{\overset{}{\mathop{\Pr }}}\,(\mu \le T-\sigma {{\Phi }^{-1}}(1-{{R}_{U}})) }[/math]

From the posterior distribution of [math]\displaystyle{ \mu \ \ : }[/math]

[math]\displaystyle{ 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]

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