Bayesian-Weibull Analysis - Revision history

Lisa Hacker at 00:13, 9 March 2023

2023-03-09T00:13:32Z

← Older revision		Revision as of 00:13, 9 March 2023
Line 4:		Line 4:
	::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\!</math>		::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\!</math>

	In this model, <math>\eta\,\!</math> is assumed to follow a noninformative prior distribution with the density function <math> \varphi (\eta )=\dfrac{1}{\eta } \,\!</math>. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on <math>\eta\,\!</math>. Specifically, since <math>\eta\,\!</math> is always positive, we can assume that ln(<math>\eta\,\!</math>) follows a uniform distribution, <math>U( - ∞, + ~~∞).~~\,\!</math> Applying Jeffrey's rule as given in Gelman et al. [[Appendix:_Life_Data_Analysis_References\|[9]]] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields <math> \varphi (\eta )=\dfrac{1}{\eta }\,\!</math>.		In this model, <math>\eta\,\!</math> is assumed to follow a noninformative prior distribution with the density function <math> \varphi (\eta )=\dfrac{1}{\eta } \,\!</math>. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on <math>\eta\,\!</math>. Specifically, since <math>\eta\,\!</math> is always positive, we can assume that ln(<math>\eta\,\!</math>) follows a uniform distribution, <math>U( -\infty, +\infty \,\!</math>). Applying Jeffrey's rule as given in Gelman et al. [[Appendix:_Life_Data_Analysis_References\|[9]]] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields <math> \varphi (\eta )=\dfrac{1}{\eta }\,\!</math>.

	The prior distribution of <math>\beta\,\!</math>, denoted as <math> \varphi (\beta )\,\!</math>, can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:		The prior distribution of <math>\beta\,\!</math>, denoted as <math> \varphi (\beta )\,\!</math>, can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

Lisa Hacker at 00:06, 9 March 2023

2023-03-09T00:06:55Z

← Older revision		Revision as of 00:06, 9 March 2023
Line 1:		Line 1:
	~~<noinclude>~~{{template:~~Banner~~ Weibull ~~Articles}~~}~~{{Navigation box~~}}		{{template:LDABOOK\|8.3\|Bayesian-Weibull Analysis}}
	~~''This article appears in the [[The_Weibull_Distribution#Bayesian-Weibull_Analysis\|Life Data Analysis Reference book]]''.~~		The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in [[Parameter Estimation]]. This model considers prior knowledge on the shape (<math>\beta\,\!</math>) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of <math>\beta\,\!</math> and <math>\eta\,\!</math> are independent, we obtain the following posterior ''pdf'':

	~~</noinclude>~~The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in [[Parameter Estimation]]. This model considers prior knowledge on the shape (<math>\beta\,\!</math>) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of <math>\beta\,\!</math> and <math>\eta\,\!</math> are independent, we obtain the following posterior ''pdf'':

	::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\!</math>		::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\!</math>

Richard House: /* Bounds on Time for Bayesian-Weibull */

2012-09-25T19:58:39Z

Bounds on Time for Bayesian-Weibull

← Older revision		Revision as of 19:58, 25 September 2012
Line 117:		Line 117:

	=== Bounds on Time for Bayesian-Weibull===		=== Bounds on Time for Bayesian-Weibull===
	Following the same procedure described for bounds on Reliability, the bounds of time <math>t\,\!</math> can be calculated, given <math>R\,\!</math>. The Bayesian one-sided upper bound estimate for <~~span class="texhtml"~~>''t''(''R'')</~~span~~> is given by:		Following the same procedure described for bounds on Reliability, the bounds of time <math>t\,\!</math> can be calculated, given <math>R\,\!</math>. The Bayesian one-sided upper bound estimate for <math>t(R)\,\!</math> is given by:

	::<math> \int\nolimits_{0}^{T_{U}(R)}f(T\|Data,R)dT=CL \,\!</math>		::<math> \int\nolimits_{0}^{T_{U}(R)}f(T\|Data,R)dT=CL \,\!</math>
Line 126:		Line 126:
	::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\!</math>		::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T_{U}\exp (-\dfrac{\ln (-\ln R)}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\!</math>

	The above equation can be solved for <~~span class="texhtml"~~>''T~~''<sub>''~~U~~''</sub>~~(''R'')</~~span~~>. The Bayesian one-sided lower bound estimate for <~~span class="texhtml"~~>''T''(''R'')</~~span~~> is given by:		The above equation can be solved for <math>{{T}_{U}}(R)\,\!</math>. The Bayesian one-sided lower bound estimate for <math>T(R)\,\!</math> is given by:

	::<math> \int\nolimits_{0}^{T_{L}(R)}f(T\|Data,R)dT=1-CL \,\!</math>		::<math> \int\nolimits_{0}^{T_{L}(R)}f(T\|Data,R)dT=1-CL \,\!</math>
Line 134:		Line 134:
	::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\!</math>		::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{T_{L}\exp (\dfrac{-\ln (-\ln R)}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\!</math>

	The above equation can be solved for <~~span class="texhtml"~~>''T~~''<sub>''~~L~~''</sub>~~(''R'')</~~span~~>. The Bayesian two-sided lower bounds estimate for <~~span class="texhtml"~~>''T''(''R'')</~~span~~> is:		The above equation can be solved for <math>{{T}_{L}}(R)\,\!</math>. The Bayesian two-sided lower bounds estimate for <math>T(R)\,\!</math> is:

	::<math> \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T\|Data,R)dT=CL \,\!</math>		::<math> \int\nolimits_{T_{L}(R)}^{T_{U}(R)}f(T\|Data,R)dT=CL \,\!</math>

Richard House: /* Bounds on Reliability for Bayesian-Weibull */

2012-09-25T19:55:55Z

Bounds on Reliability for Bayesian-Weibull

Richard House: /* Posterior Distributions for Functions of Parameters */

2012-09-25T19:52:25Z

Posterior Distributions for Functions of Parameters

← Older revision		Revision as of 19:52, 25 September 2012
Line 53:		Line 53:
	'''Reliability'''		'''Reliability'''

	In order to calculate the median value of the reliability function, we first need to obtain posterior ''pdf'' of the reliability. Since <~~span class="texhtml"~~>''R''(''T'')</~~span~~> is a function of <~~span class="texhtml"~~>β</~~span~~>, the density functions of <~~span class="texhtml"~~>β</~~span~~> and <~~span class="texhtml"~~>''R''(''T'')</~~span~~> have the following relationship:		In order to calculate the median value of the reliability function, we first need to obtain posterior ''pdf'' of the reliability. Since <math>R(T)\,\!</math> is a function of <math>\beta\,\!</math>, the density functions of <math>\beta\,\!</math> and <math>R(T)\,\!</math> have the following relationship:

	::<math> \begin{align} f(R\|Data,T)dR = & f(\beta \|Data)d\beta)\\		::<math> \begin{align} f(R\|Data,T)dR = & f(\beta \|Data)d\beta)\\

Richard House at 19:48, 25 September 2012

2012-09-25T19:48:24Z

← Older revision		Revision as of 19:48, 25 September 2012
Line 2:		Line 2:
	''This article appears in the [[The_Weibull_Distribution#Bayesian-Weibull_Analysis\|Life Data Analysis Reference book]]''.		''This article appears in the [[The_Weibull_Distribution#Bayesian-Weibull_Analysis\|Life Data Analysis Reference book]]''.

	</noinclude>The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in [[Parameter Estimation]]. This model considers prior knowledge on the shape (<math>\beta\,\!</math>) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of <~~span class="texhtml"~~>β</~~span~~> and <~~span class="texhtml"~~>η</~~span~~> are independent, we obtain the following posterior ''pdf'':		</noinclude>The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in [[Parameter Estimation]]. This model considers prior knowledge on the shape (<math>\beta\,\!</math>) parameter of the Weibull distribution when it is chosen to be fitted to a given set of data. There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available. For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions. A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say (and you would be right). The Bayesian-Weibull model in Weibull++ (which is actually a true "WeiBayes" model, unlike the 1-parameter Weibull that is commonly referred to as such) offers an alternative to the 1-parameter Weibull, by including the variation and uncertainty that might have been observed in the past on the shape parameter. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of <math>\beta\,\!</math> and <math>\eta\,\!</math> are independent, we obtain the following posterior ''pdf'':

	::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\!</math>		::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } \,\!</math>

	In this model, <math>\eta\,\!</math> is assumed to follow a noninformative prior distribution with the density function <math> \varphi (\eta )=\dfrac{1}{\eta } \,\!</math>. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on <~~span class="texhtml"~~>η.</~~span~~> Specifically, since <math>\eta\,\!</math> is always positive, we can assume that ln(<math>\eta\,\!</math>) follows a uniform distribution, <math>U( - ∞, + ∞).\,\!</math> Applying Jeffrey's rule as given in Gelman et al. [[Appendix:_Life_Data_Analysis_References\|[9]]] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields <math> \varphi (\eta )=\dfrac{1}{\eta }\,\!</math>.		In this model, <math>\eta\,\!</math> is assumed to follow a noninformative prior distribution with the density function <math> \varphi (\eta )=\dfrac{1}{\eta } \,\!</math>. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on <math>\eta\,\!</math>. Specifically, since <math>\eta\,\!</math> is always positive, we can assume that ln(<math>\eta\,\!</math>) follows a uniform distribution, <math>U( - ∞, + ∞).\,\!</math> Applying Jeffrey's rule as given in Gelman et al. [[Appendix:_Life_Data_Analysis_References\|[9]]] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields <math> \varphi (\eta )=\dfrac{1}{\eta }\,\!</math>.

	The prior distribution of <math>\beta\,\!</math>, denoted as <math> \varphi (\beta )\,\!</math>, can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:		The prior distribution of <math>\beta\,\!</math>, denoted as <math> \varphi (\beta )\,\!</math>, can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

Richard House at 18:27, 25 September 2012

2012-09-25T18:27:08Z

Show changes

Richard House at 18:17, 5 September 2012

2012-09-05T18:17:11Z

← Older revision		Revision as of 18:17, 5 September 2012
Line 6:		Line 6:
	::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } </math>		::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } </math>

	In this model, <math>\eta\,\!</math> is assumed to follow a noninformative prior distribution with the density function <math> \varphi (\eta )=\dfrac{1}{\eta } </math>. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on <span class="texhtml">η.</span> Specifically, since <math>\eta\,\!</math> is always positive, we can assume that ln(<math>\eta\,\!</math>) follows a uniform distribution, <math>U( - ∞, + ∞).\,\!</math> Applying Jeffrey's rule as given in Gelman et al.[[Appendix:_Life_Data_Analysis_References\|[9]]] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields <math> \varphi (\eta )=\dfrac{1}{\eta }. </math>		In this model, <math>\eta\,\!</math> is assumed to follow a noninformative prior distribution with the density function <math> \varphi (\eta )=\dfrac{1}{\eta } </math>. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on <span class="texhtml">η.</span> Specifically, since <math>\eta\,\!</math> is always positive, we can assume that ln(<math>\eta\,\!</math>) follows a uniform distribution, <math>U( - ∞, + ∞).\,\!</math> Applying Jeffrey's rule as given in Gelman et al. [[Appendix:_Life_Data_Analysis_References\|[9]]] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields <math> \varphi (\eta )=\dfrac{1}{\eta }. </math>

	The prior distribution of <math>\beta\,\!</math>, denoted as <math> \varphi (\beta ) \,\!</math>, can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:		The prior distribution of <math>\beta\,\!</math>, denoted as <math> \varphi (\beta ) \,\!</math>, can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

Richard House at 18:16, 5 September 2012

2012-09-05T18:16:31Z

← Older revision		Revision as of 18:16, 5 September 2012
Line 6:		Line 6:
	::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } </math>		::<math> f(\beta ,\eta \|Data)=\dfrac{L(\beta ,\eta )\varphi (\beta )\varphi (\eta )}{ \int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta } </math>

	In this model, <math>\eta\,\!</math> is assumed to follow a noninformative prior distribution with the density function <math> \varphi (\eta )=\dfrac{1}{\eta } </math>. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on <span class="texhtml">η.</span> Specifically, since <math>\eta\,\!</math> is always positive, we can assume that ln(<math>\eta\,\!</math>) follows a uniform distribution, <math>U( - ∞, + ∞).\,\!</math> Applying Jeffrey's rule [[Appendix: ~~Weibull References~~\|[9]]] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields <math> \varphi (\eta )=\dfrac{1}{\eta }. </math>		In this model, <math>\eta\,\!</math> is assumed to follow a noninformative prior distribution with the density function <math> \varphi (\eta )=\dfrac{1}{\eta } </math>. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on <span class="texhtml">η.</span> Specifically, since <math>\eta\,\!</math> is always positive, we can assume that ln(<math>\eta\,\!</math>) follows a uniform distribution, <math>U( - ∞, + ∞).\,\!</math> Applying Jeffrey's rule as given in Gelman et al.[[Appendix:_Life_Data_Analysis_References\|[9]]] which says "in general, an approximate non-informative prior is taken proportional to the square root of Fisher's information," yields <math> \varphi (\eta )=\dfrac{1}{\eta }. </math>

	The prior distribution of <math>\beta\,\!</math>, denoted as <math> \varphi (\beta ) \,\!</math>, can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:		The prior distribution of <math>\beta\,\!</math>, denoted as <math> \varphi (\beta ) \,\!</math>, can be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:

Richard House at 18:14, 5 September 2012

2012-09-05T18:14:16Z

← Older revision		Revision as of 18:14, 5 September 2012
Line 12:		Line 12:
	:*Collect the times-to-failure data.		:*Collect the times-to-failure data.
	:*Specify a prior distribution for <math>\beta\,\!</math> (the prior for <math>\eta\,\!</math> is assumed to be <math>1/\beta\,\!</math>).</span>		:*Specify a prior distribution for <math>\beta\,\!</math> (the prior for <math>\eta\,\!</math> is assumed to be <math>1/\beta\,\!</math>).</span>
	:*Obtain the posterior''pdf'' from the above equation.		:*Obtain the posterior ''pdf'' from the above equation.

	In other words, a distribution (the posterior ''pdf'') is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior ''pdf'' needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the ''Analysis'' page, under the ''Results As'' area, as shown next.		In other words, a distribution (the posterior ''pdf'') is obtained, rather than a point estimate as in classical statistics (i.e., as in the parameter estimation methods described previously in this chapter). Therefore, if a point estimate needs to be reported, a point of the posterior ''pdf'' needs to be calculated. Typical points of the posterior distribution used are the mean (expected value) or median. In Weibull++, both options are available and can be chosen from the ''Analysis'' page, under the ''Results As'' area, as shown next.

← Older revision		Revision as of 19:55, 25 September 2012
Line 87:		Line 87:

	=== Bounds on Reliability for Bayesian-Weibull===		=== Bounds on Reliability for Bayesian-Weibull===
	The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of <~~span class="texhtml"~~>β</~~span~~> is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for <~~span class="texhtml"~~>''R''(''T'')</~~span~~> is given by:		The confidence bounds calculation under the Bayesian-Weibull analysis is very similar to the Bayesian Confidence Bounds method described in the previous section, with the exception that in the case of the Bayesian-Weibull Analysis the specified prior of <math>\beta\,\!</math> is considered instead of an non-informative prior. The Bayesian one-sided upper bound estimate for <math>R(T)\,\!</math> is given by:

	::<math> \int\nolimits_{0}^{R_{U}(T)}f(R\|Data,t)dR=CL \,\!</math>		::<math> \int\nolimits_{0}^{R_{U}(T)}f(R\|Data,t)dR=CL \,\!</math>
Line 95:		Line 95:
	::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\!</math>		::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{t\exp (-\dfrac{\ln (-\ln R_{U})}{\beta })}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=CL \,\!</math>

	The above equation can be solved for <~~span class="texhtml"~~>''R~~''<sub>''~~U~~''</sub>~~(''t'')</~~span~~>. The Bayesian one-sided lower bound estimate for <math> \ R(t) \,\!</math> is given by:		The above equation can be solved for <math>{{R}_{U}}(t)\,\!</math>. The Bayesian one-sided lower bound estimate for <math> \ R(t) \,\!</math> is given by:

	::<math> \int\nolimits_{0}^{R_{L}(t)}f(R\|Data,t)dR=1-CL \,\!</math>		::<math> \int\nolimits_{0}^{R_{L}(t)}f(R\|Data,t)dR=1-CL \,\!</math>
Line 104:		Line 104:
	::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\!</math>		::<math> \dfrac{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{T\exp (-\dfrac{\ln (-\ln R_{L})}{\beta })}L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }{\int\nolimits_{0}^{\infty }\int\nolimits_{0}^{\infty }L(\beta ,\eta )\varphi (\beta )\varphi (\eta )d\eta d\beta }=1-CL \,\!</math>

	The above equation can be solved for <~~span class="texhtml"~~>''R~~''<sub>''~~L~~''</sub>~~(''t'')</~~span~~>. The Bayesian two-sided bounds estimate for <~~span class="texhtml"~~>''R''(''t'')</~~span~~> is given by:		The above equation can be solved for <math>{{R}_{L}}(t)\,\!</math>. The Bayesian two-sided bounds estimate for <math>R(t)\,\!</math> is given by:

	::<math> \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R\|Data,t)dR=CL \,\!</math> which is equivalent to:		::<math> \int\nolimits_{R_{L}(t)}^{R_{U}(t)}f(R\|Data,t)dR=CL \,\!</math> which is equivalent to:
Line 114:		Line 114:
	::<math> \int\nolimits_{0}^{R_{L}(t)}f(R\|Data,T)dR=(1-CL)/2 \,\!</math>		::<math> \int\nolimits_{0}^{R_{L}(t)}f(R\|Data,T)dR=(1-CL)/2 \,\!</math>

	Using the same method for one-sided bounds, <~~span class="texhtml"~~>''R~~''<sub>''~~U~~''</sub>~~(''t'')</~~span~~>and <~~span class="texhtml"~~>''R~~''<sub>''~~L~~''</sub>~~(''t'')</~~span~~> can be computed.		Using the same method for one-sided bounds, <math>{{R}_{U}}(t)\,\!</math> and <math>{{R}_{L}}(t)\,\!</math> can be computed.

	=== Bounds on Time for Bayesian-Weibull===		=== Bounds on Time for Bayesian-Weibull===