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Speaker
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Transcript
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Our talk today shows how to
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use JMP to do Bayesian
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estimation. Here's an overview
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of my talk. I'm going to start
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with a brief introduction to
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Bayesian statistical methods.
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Then I'm going to go through
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four different examples that
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happen to come from reliability,
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but the methods we're presenting
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are really much more general and
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can be applied in other areas of
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application. Then I'm going to
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turn it over to Peng and he's
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going to show you how easy it is
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to actually do these things in
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JMP. Technically, reliability
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is a probability. The
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probability of a system,
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vehicle, machine or whatever it
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is that is of interest, will
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perform its intended function
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under encountered operating
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conditions for a specified period
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of time. I highlight encountered
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here to emphasize that
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reliability depends importantly
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on the environment in which a
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product is being used. Condra
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defined reliability as quality
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over time. And many engineers
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think of reliability is being
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failure avoidance, that is, to
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design and manufacture a product
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that will not fail. Reliability
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is a highly quantitative
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engineering discipline, but
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often requires sophisticated
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statistical and probabilistic
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ideas. Over the past 30 years,
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there's been a virtual
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revolution where Bayes methods
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are now commonly used and in
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many different areas of
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application. This revolution
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started by the rediscovery of
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Markov chain Monte Carlo methods
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and was accelerated by
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spectacular improvements in
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computing power that we have
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today, as well as the
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development of relatively easy
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to use software to implement
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Bayes methods. In the 1990s we
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had BUGS. Today Stan and other
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similar packages have largely
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replaced BUGS, but the other
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thing that's happening is we're
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beginning to see more Bayesian
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methods implemented in
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commercial software. So for
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example, SAS has PROC MCMC.
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And now JMP has some very
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powerful tools that were
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developed for reliability, but
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as I said, they can be applied
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in other areas as well, and
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there's strong motivation for
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the use of Bayesian methods.
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For one thing, it provides a
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means for combining prior
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information with limited data to
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be able to make useful
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inferences. Also, there are many
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situations, particularly with
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random effects complications
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like censor data, where maximum
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likelihood is difficult to
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implement, but where Bayes
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methods are relatively easy to
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implement. There's one little
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downside in the use of Bayes
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methods. You have to think a bit
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harder about certain things,
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particularly about
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parameterization and how to
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specify the prior distributions.
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My first example is about an
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aircraft engine bearing cage.
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These are field failure data
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where there was 1,703 aircraft
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engines that contained this
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bearing cage. The oldest ones
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had 2,220 hours of operation. The
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design life specification for
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this bearing cage was that no
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more than 10% of the units would
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fail by 8,000 hours of operation.
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However, 6 units had failed and
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this raised the question
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do we
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have a serious problem here? Do
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we need to redesign this bearing
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cage to meet that reliability
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condition? This is an event plot
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of the data. The event plot
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illustrates the structure
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of the data, and in particular,
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we can see the six failures
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here. In addition to that, we
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have right censored
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observations, indicated here by
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the arrows. So these are units
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that are still in service and
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they have not failed yet, and
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the right arrow indicates that
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all we know is if we wait long
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enough out to the right, the units
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will eventually fail. Here's a
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maximum likelihood analysis of
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those data, so the probability
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plot here suggests that the
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Weibull distribution provides a
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good description of these data.
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However, when we use the
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distribution profiler to
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estimate fraction failing at
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8,000 hours, we can see that the
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confidence interval is enormous,
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ranging between about 3% all the
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way up to 100%. That's not very
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useful. So likelihood methods
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work like this. We specify the
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model and the data.
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That defines the likelihood and
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then we use the likelihood to
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make inferences. Bayesian
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methods are similar, except we
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also have prior information
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specified. Bayes theorem
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combines the likelihood with the
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prior information, providing a
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posterior distribution, and then
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we use the posterior
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distribution to make inferences.
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Here's the Bayes analysis of the
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bearing cage. The priors are
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specified here for the B10
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or time at which 10% would fail.
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We have a very wide interval
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here. The range, effectively 1,000
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hours up to 50,000 hours.
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Everybody would agree that B10
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is somewhere in that range. For
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the Weibull shape parameter,
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however, we're going to use an
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informative prior distribution
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based upon the engineers'
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knowledge of the failure
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mechanism and their vast
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previous experience with that
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mechanism. They can say with
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little doubt that the Weibull
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shape parameter should be
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between 1.5 and 3, and
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here's where we specify that
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information. So instead of
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specifying information for the
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traditional Weibull parameters,
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we've reparameterized, where now
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the B10 is one of the
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parameters, and here's the
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specified range. And then we
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have the informative prior for
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the Weibull shape parameter
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specified here. And then JMP
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will generate samples from the
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joint posterior, leading to
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these parameter estimates and
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confidence intervals shown here.
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Here's a graphical depiction of
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the Bayes analysis. The black
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points here are a sample from
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the prior distribution, so again
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very wide for the .1 quantile
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and somewhat constrained for the
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Weibull shape parameter beta. On
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the right here, we have the
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joint posterior, which in effect
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is where the likelihood contours
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and the prior sample
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overlap. And then those draws
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from the joint posterior are
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used to compute estimates and
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Bayesian credible intervals. So
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here's the same profiler that we
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saw previously where the
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confidence interval was not
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useful. After bringing in the
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information about the Weibull
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shape parameter, now we can see
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that the confidence interval
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ranges between 12% and 83%,
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clearly illustrating that we
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have missed the target of 10%.
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So what have we learned here?
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With a small number of failures,
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there's not much information
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about reliability. But engineers
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often have information that can
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be used, and by using that prior
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information, we can get improved
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precision and more useful
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inferences. And Bayesian methods
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provide a formal method for
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combining that prior information
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with our limited data. Here's
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another example. Rocket motor is
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one of five critical components
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in a missile. In this particular
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application, there were
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approximately 20,000 missiles
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in the inventory. Over time, 1,940
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of these missiles had been fired
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and they all worked, except in
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three cases, where there was
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catastrophic failure. And these
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were older missiles, and so
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there was some concern that
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there might be a wearout failure
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mechanism that would put into
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jeopardy the roughly 20,000
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missiles remaining in inventory.
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The failures were thought to be
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due to thermal cycling, but
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there was no information about
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the number of thermal cycles. We
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only have the age of the
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missile when it was fired.
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That's a useful surrogate, but
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the effect of using a surrogate
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like that is you have more
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variability in your data. Now
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in this case, there were no
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directly observed failure times.
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When a rocket is called upon to
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operate and it operates
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successfully, all we know is
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that they had not failed at the
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age of those units when they
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were asked to operate. And for
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the units that failed
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catastrophically, again, we
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don't know the time that those
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units failed. At some point
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before they were called upon to
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operate. They had failed, so all
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we know is that the failure was
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before the age at which it was
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fired. So as I said, there was
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concern that there is a wear out
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failure mechanism kicking in
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here that would put into
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jeopardy the amount of remaining
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life for the units in the
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stockpile. So here's the table
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of the data. Here we have the
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units that operated
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successfully, and so these are
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right censored observations, but
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these observations here
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are the ones that failed and as
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I said, at relatively higher
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ages. This is the event plot of
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the data and again, we can see
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the right censored observations
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here with the arrow pointing to
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the right, and we can see the
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left censored observations with
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the arrow pointing to the left
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indicating the region of
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uncertainty. But even with those
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data we can still fit a Weibull
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distribution. And here's the
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probability plot showing the
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maximum likelihood estimate and
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confidence bands. Here's more
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information from the maximum
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likelihood analysis. And here we
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have the estimate of fraction
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failing at 20 years, which was
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the design life of these rocket
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motors, and again the interval
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is huge, ranging between 3% and
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100%. Again, not very useful.
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But the engineers,
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knowing what the failure
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mechanism was, again had
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information about the Weibull
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shape parameter. The maximum
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likelihood estimate was
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extremely large and the
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engineers were pretty sure that
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that was wrong, especially with
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the extra variability in the
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data that would tend to drive
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the Weibull shape parameter to a
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lower value. As I showed you on
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the previous slide, confidence
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interval for fraction failing at
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20 years was huge. So once again,
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we're going to specify a prior
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distribution and then use that
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in a Bayes analysis. Again, the
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prior for B10, the time at which
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10% will fail, is chosen to be
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extremely wide. We don't really
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want to assume anything there,
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and everybody would agree that
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that quantity is somewhere
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between five years and 400
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years. But for the Weibull shape
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parameter, we're going to assume
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that it's between one and five.
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Again, we know it's greater than
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one because it's a wear out
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failure mechanism, and the
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engineers were sure that it
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wasn't anything like the number
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8 that we had seen in the
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maximum likely estimate.
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And indeed, five is also a very
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large Weibull shape parameter.
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Once again, JMP is called upon
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to generate draws from the
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posterior distribution. And here are
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plot similar to the ones that we
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saw in the bearing cage example.
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The black points here, again, are
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a sample from the prior
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distribution. Again very wide in
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terms of the B10, but somewhat
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constrained for the beta, so the
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beta is an informative prior
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distribution. And again the
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contour plots represent the
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information in the limited data.
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In our posterior, once again, is
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where we get overlap between the
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prior and the likelihood and we
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can see it here. So once again
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we have a comparison between the
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maximum likelihood interval,
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which is extremely wide, and the
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interval that we get for the
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same quantity using the Bayes
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inference, which incorporated the
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prior information on the Weibull
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shape parameter. And now the
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interval ranges between .002 and
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.98 or about .1, about 10%
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failing, so that might be
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acceptable. Some of the things
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that we learned here, even
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though there were no actual
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failure times, we can still get
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reliability information from the
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data, but with very few failures
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there isn't much information
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there. But we can use the
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engineer's knowledge about the
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Weibull shape parameter to
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supplement the data to get
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useful inferences and JMP makes
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this really easy to do. My last
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two examples are about
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accelerated testing. Accelerated
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testing is a widely used
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technique to get information
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about reliability of components
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quickly when designing a
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product. The basic idea is to
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test units at high levels of
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variables like temperature or
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voltage to make things fail
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quickly and then to use a model
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to extrapolate back down to the
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use conditions. Extrapolation is
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always dangerous and we have to
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keep that in mind. That's the
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reason we would like to have our
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model be physically motivated.
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So here's an example of an
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accelerated life test
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on a laser. Units were tested at
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40, 60 and 80 degrees C, but the
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use condition was 10 degrees C.
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That's the nominal temperature
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at the bottom of the Atlantic
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Ocean, where these lasers were
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going to be used in a new
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telecommunications system. The
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test lasted 5,000 hours, a little
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bit more than six months. The
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engineers wanted to estimate
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fraction failing at about 30,000
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hours. That's about 3.5 years
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and again, at 10 degrees C.
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Here's the results of the
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analysi. In order to
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appropriately test and build the
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model, JMP uses these three
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different analyses. The first
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one fits separate distributions
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to each level of temperature.
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The next model does the same
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thing, except that it constrains
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the shape parameter Sigma to be
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the same at every level of
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temperature. This is analogous
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to the constant Sigma assumption
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that we typically make in
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regression analysis. And then
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finally, we fit the regression
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model, which in effect, is a
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simple linear regression
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connecting lifetime to
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temperature. And to supplement
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this visualization of these
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three models, JMP does
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likelihood ratio tests to test
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whether there's evidence that
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the Sigma might depend on
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temperature and then to test
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whether there's evidence of lack
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of fit in the regression model.
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And from the large P values
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here, we can see that there's no
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evidence against this model.
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Another way to plot the results
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of fitting this model
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is to plot lifetime versus
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temperature on special scales. A
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log rhythmic scale for hours of
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operation in what's known as an
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Arrhenius scale for temperature.
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Corresponding to the Arrhenius
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model, which describes how
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temperature affects reaction
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rates, and thereby lifetime. And
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this is the results of the
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maximum likelihood estimation
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for our model. The JMP
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distribution profiler gives us
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an estimate of the fraction
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failing at 30,000 hours.
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And we can see it ranges between
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.002 and about .12, or 12%
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failing. The engineers in
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applications like this, however,
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often have information about
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what's known as the effective
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activation energy of the failure
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mechanism, and that corresponds
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to the slope of the regression
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line in the Arrhenius model. So
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we did a Bayes analysis and in
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that analysis, we made an
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assumption about the effective
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activation energy. And that's
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going to provide more precision
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for us. So what we have here is
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a matrix scatterplot of the
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joint posterior distribution
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after having specified prior
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distributions for the
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parameters, weakly informative
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for the .1 quantile at 40
|
|
|
degrees C. Again, everybody
|
|
|
would agree that that number is
|
|
|
somewhere between 100 and
|
|
|
32,000. Also weakly informative
|
|
|
for the lognormal shape
|
|
|
parameter. Again, everybody
|
|
|
would agree that that number is
|
|
|
somewhere between .05
|
|
|
and 20. But for the slope of the
|
|
|
regression line, we have an
|
|
|
informative prior ranging
|
|
|
between .6 and .8, based upon
|
|
|
previous experience with the
|
|
|
failure mechanism. And that leads
|
|
|
to this comparison, where now
|
|
|
on the right-hand side here, the
|
|
|
interval for fraction failing at
|
|
|
30,000 hours is much narrower
|
|
|
than it was with the maximum
|
|
|
likelihood estimate. In
|
|
|
particular, the upper bound now
|
|
|
is only about 4% compared with
|
|
|
12% for the maximum likelihood
|
|
|
estimates. So lessons learned.
|
|
|
Accelerated tests provide
|
|
|
reliability information quickly,
|
|
|
and engineers often have
|
|
|
information about the effect of
|
|
|
activation energy. And that can
|
|
|
be used to either improve
|
|
|
precision or to reduce cost by
|
|
|
not needing to test so many
|
|
|
units. And once again, Bayesian
|
|
|
methods provide an appropriate
|
|
|
method to combine the engineers'
|
|
|
knowledge with the limited data.
|
|
|
My final example concerns an
|
|
|
accelerated life test of
|
|
|
an integrated circuit device. Units
|
|
|
were tested at high temperature
|
|
|
and the resulting data were
|
|
|
interval censored. That's
|
|
|
because failures were discovered
|
|
|
only during inspections that
|
|
|
were conducted periodically. In
|
|
|
this test, however, there were
|
|
|
only failures at the two high
|
|
|
levels of temperature. The goal
|
|
|
of the test was to estimate the
|
|
|
.01 quantile at 100 degrees C.
|
|
|
This is a table of the data
|
|
|
where we can see the failures at
|
|
|
250 and 300 degrees C.
|
|
|
And no failures all right
|
|
|
censored at the three lower
|
|
|
levels of temperature. Now when
|
|
|
we did the maximum likelihood
|
|
|
estimation, in this case, we saw
|
|
|
strong evidence that the Weibull
|
|
|
shape parameter depended on
|
|
|
temperature. So the P value is
|
|
|
about .03. That turns out
|
|
|
to be evidence against the
|
|
|
Arrhenius model, and that's
|
|
|
because the Arrhenius model should
|
|
|
only scale time. But if you
|
|
|
change the shape parameter by
|
|
|
increasing temperature, you're
|
|
|
doing more than scaling time.
|
|
|
And so that's a problem, and it
|
|
|
suggested that at 300 degrees C,
|
|
|
there was a different failure
|
|
|
mechanism. And indeed, when the
|
|
|
engineers followed up and
|
|
|
determined the cause of failure
|
|
|
of the units at 250 and 300,
|
|
|
they saw that there was a
|
|
|
different mechanism at 300. What
|
|
|
that meant is that we had to
|
|
|
throw those data away. So what
|
|
|
do we do then? Now we've only
|
|
|
got failures at 250 degrees C
|
|
|
and JMP doesn't do very well
|
|
|
with that. It's surprisingly,
|
|
|
actually runs and gives
|
|
|
answers, but the confidence
|
|
|
intervals are enormously wide
|
|
|
here, as one would expect. But
|
|
|
the engineers knew what the
|
|
|
failure mechanism was and they
|
|
|
had had previous experience and
|
|
|
so they can bring that
|
|
|
information about the slope into
|
|
|
the analysis using Bayes
|
|
|
methods. So again, here's the
|
|
|
joint posterior and the width of
|
|
|
the distribution in the
|
|
|
posterior for beta 1 is
|
|
|
effectively what we assumed
|
|
|
when we put in a prior
|
|
|
distribution for that parameter.
|
|
|
So again, here's the
|
|
|
specification of the prior
|
|
|
distributions, where we used
|
|
|
weakly informative for the
|
|
|
quantile and for Sigma, but
|
|
|
informative prior distribution
|
|
|
for the slope beta 1. And I can
|
|
|
get an estimate of the time at
|
|
|
which 1% fail. So the lower end
|
|
|
point of the confidence interval
|
|
|
for the time at which 1% will fail
|
|
|
is more than 140,000 hours.
|
|
|
So that's about 20 years, much
|
|
|
longer than the technological
|
|
|
life of these products in which
|
|
|
this integrated circuit will be
|
|
|
used. So what did we learn here?
|
|
|
Well, in some applications we
|
|
|
have interval censoring because
|
|
|
failures are discovered only
|
|
|
when there's an inspection. We
|
|
|
need appropriate statistical
|
|
|
methods for handling such data,
|
|
|
and JMP has those methods. If
|
|
|
you use excessive levels of an
|
|
|
accelerating variable like
|
|
|
temperature, you can generate
|
|
|
new failure modes that make the
|
|
|
information misleading. So we
|
|
|
had to throw those units away.
|
|
|
But even with failures at only
|
|
|
one level of temperature, if we
|
|
|
have prior information
|
|
|
about the effective activation
|
|
|
energy, we can combine that
|
|
|
information with the limited
|
|
|
data to make useful inferences.
|
|
|
Finally, some concluding
|
|
|
remarks, improvements in
|
|
|
computing hardware and software
|
|
|
have greatly advanced our ability
|
|
|
to analyze reliability and other
|
|
|
data. Now we can also use Bayes
|
|
|
methods, providing another set of
|
|
|
tools for combining information
|
|
|
with limited data and JMP has
|
|
|
powerful tools for doing this.
|
|
|
So, although these Bayesian
|
|
|
capabilities were developed for
|
|
|
the reliability part of JMP,
|
|
|
they can certainly be used in
|
|
|
other areas of application. And
|
|
|
here are some references,
|
|
|
including the 2nd edition of
|
|
|
Statistical Methods Reliability,
|
|
|
which should be out probably in
|
|
|
June of 2021. OK, so now I'm
|
|
|
going to turn it over to Peng
|
|
|
and he's going to show you how
|
|
|
easy it is to do these analyses.
|
|
|
Thank you, professor.
|
|
|
Before I start my
|
|
|
demonstration, I would like
|
|
|
to show this slide about
|
|
|
Bayesian analysis workflow
|
|
|
in life distribution and
|
|
|
Fit Life by X.
|
|
|
First, you need to fit a
|
|
|
parametric model using maximum
|
|
|
likelihood. I assume you already
|
|
|
know how to do this in these two
|
|
|
platforms. Then you need to
|
|
|
review or find model
|
|
|
specification graphical user
|
|
|
interface for Bayesian
|
|
|
estimation within the report
|
|
|
from the previous step. For
|
|
|
example, this is screenshot of
|
|
|
Weibull model in Life
|
|
|
Distribution. You need to go to the
|
|
|
red triangle menu
|
|
|
and choose Bayesian estimates
|
|
|
to reveal the graphical
|
|
|
user interface for the
|
|
|
Bayesian analysis.
|
|
|
In Fit Life by X, please see
|
|
|
the screenshot of a lognormal
|
|
|
result. And the graphical user
|
|
|
interface for the Bayesian
|
|
|
inference is on the last step.
|
|
|
After finding the graphical user
|
|
|
interface for Bayesian analysis,
|
|
|
you will need to supply the
|
|
|
information about the priors.
|
|
|
You need to decide the priors
|
|
|
dispersion for individual
|
|
|
parameters. You need to supply
|
|
|
the information for the
|
|
|
hyperparameters and additional
|
|
|
information such as the
|
|
|
probability for the quantile.
|
|
|
In addition to that, we
|
|
|
need to provide the number
|
|
|
of posterior samples.
|
|
|
Also need to provide a random
|
|
|
seed in case you want to
|
|
|
replicate your result in the
|
|
|
future. Then you can click Fit
|
|
|
Model. This will generate a
|
|
|
report of the model.
|
|
|
You can fit multiple models
|
|
|
in case you want to study the
|
|
|
sensitivity of different
|
|
|
Bayesian models given
|
|
|
different prior distribution.
|
|
|
The result of a Bayesian model,
|
|
|
including the following things,
|
|
|
first is a method of sampling.
|
|
|
And then is a copy of the priors.
|
|
|
Then had a posterior estimates
|
|
|
of the parameters.
|
|
|
And then there are some scatterplots,
|
|
|
either for the prior or
|
|
|
the posteriors.
|
|
|
In the end, we have two
|
|
|
profilers. One for distribution
|
|
|
and the one for the quantile.
|
|
|
Using these results, you
|
|
|
can make further inferences
|
|
|
such as failure prediction.
|
|
|
Now look at the demonstration.
|
|
|
We will demonstrate with the
|
|
|
last example that the professor
|
|
|
mentioned that ??? presentation.
|
|
|
It's the IC device there.
|
|
|
We have two columns for the time
|
|
to event
|
HoursL and HoursU to
|
|
|
represent the censoring situation.
|
|
|
We have a count
|
|
|
for individual observation and
|
|
|
temperature for individual
|
|
|
observation. We exclude the last
|
|
|
four observations because they
|
|
|
are associated with a different
|
|
|
failure mode. We want to exclude
|
|
|
these observations from the
|
|
|
analysis. Now start to specify.
|
|
|
???.
|
|
|
We put hours into Y
|
|
|
We put Count into frequency.
|
|
|
We put Degrees C into X.
|
|
|
We use the Arrhenius Celsius
|
|
|
for our relationship.
|
|
|
We use lognormal for our
|
|
|
distribution.
|
|
|
Then click OK.
|
|
|
The result is the maximum
|
|
|
likelihood influence
|
|
|
for the lognormal. We go to the
|
|
|
Bayesian estimates, and
|
|
|
start to specify our priors like
|
|
|
Professor did in his
|
|
|
presentation. We choose a
|
|
|
lognormal for the quantile.
|
|
|
It's 250 degrees C.
|
|
|
And its B10 life. So
|
|
|
probability is 0.1.
|
|
|
The two ends of the
|
|
|
lognormal distribution is 100
|
|
|
and 10,000.
|
|
|
Now specify the slope.
|
|
|
Distribution is lognormal.
|
|
|
And two ends of the
|
|
|
distribution is .65
|
|
|
and .85 because it's informative
|
|
|
to require the range is narrow.
|
|
|
Now we specify the prior
|
|
|
distribution for Sigma,
|
|
|
which is a lognormal
|
|
|
and it had a wide range;
|
|
|
it's .05 and 5.
|
|
|
We decided to draw 5,000
|
|
|
posterior samples. And assign an
|
|
|
arbitrary random seed. And then
|
|
|
we click...click on Fit Model.
|
|
|
And what the report generates for this
|
|
|
specification. The method is
|
|
|
simple rejection. And here's a
|
|
|
copy of our proir specification.
|
|
|
The posterior estimates
|
|
|
summarize our posterior samples.
|
|
|
You can export the posterior samples
|
|
|
by either clicking this Export
|
|
|
Monte Carlo samples
|
|
|
or choose it from the menu,
|
|
which is here
|
Export Monte
|
|
|
Carlo Samples.
|
|
|
Posterior samples are illustrated
|
|
|
in these scatterplots.
|
|
|
We have two scatterplots here.
|
|
|
The first to use
|
|
|
the same paramaterization as the prior
|
|
|
specification, which is...which
|
|
|
use quantile, slope and signal.
|
|
|
The other scatter plot....the
|
|
|
second scatter plot use a
|
|
|
traditional parameterization,
|
|
|
which includes the intercept of
|
|
|
regression, a slope of the
|
|
|
regression and Sigma.
|
|
|
In the end to make ???, we
|
|
|
can you we can look at the
|
|
|
profiler. Here let's look
|
|
|
at the second profiler,
|
|
|
quantile profile, so we can
|
|
|
find the same result
|
|
|
as what Professor had shown in
|
|
|
one of the previous slides.
|
|
|
Enter 0.1...
|
|
|
0.01 for probability. So this is
|
|
|
1%. We enter 100 degree C
|
|
|
for DegreesC.
|
|
|
And we adjust
|
|
|
the axes.
|
|
|
So now we see a similar
|
|
|
profiler.
|
|
|
It has that was already in
|
|
|
the previous slide.
|
|
|
And we can read off the Y axis
|
|
|
to get the result we want, which
|
|
|
is the time that 1% of the
|
|
|
device will fail at 100 degrees
|
|
|
C. So this concludes my
|
|
|
demonstration. And let
|
|
|
me move on.
|
|
|
This slide explains
|
|
|
about...explains JMP implementation
|
|
|
of sampling algorithm.
|
|
|
We have seen that the simple
|
|
|
rejection has shown up in the
|
|
|
previous example and this is the
|
|
|
first stage of our
|
|
|
implementation. The simple
|
|
|
rejection algorithm is tried and
|
|
|
true method to your samples,
|
|
|
but it can be impractical
|
|
|
if rejection rate is high.
|
|
|
So if the rejection
|
|
|
rate is high, we...our
|
|
|
implementations switch to the
|
|
|
second stage, which is a Random
|
|
|
Walk Metropolis-Hastings
|
|
|
algorithm. The second algorithm
|
|
|
is efficient, but in case...in
|
|
|
situations it can fail
|
|
|
undetectably if the likelihood
|
|
|
is irregular. For example, the
|
|
|
likelihood is rather flat. We
|
|
|
designed this implementation
|
|
|
because we have a situation,
|
|
|
there are very few failures or
|
|
|
even no failures. In that
|
|
|
situation the likelihood is
|
|
|
relatively high, but ???
|
|
|
situation, we use simple
|
|
|
rejection algorithm and the
|
|
|
rejection rate is not that bad
|
|
|
and this method will suffice.
|
|
|
When we have more and more
|
|
|
failures, the likelihood it
|
|
|
becomes more regular. So it has
|
|
|
a peak in the middle. In that
|
|
|
situation, the simple
|
|
|
rejecction rate...the simple
|
|
|
rejection method becomes
|
|
|
impractical because of the high
|
|
|
rejection rate. But the Random
|
|
|
Walk algorithm becomes more and
|
|
|
more promising to succeed
|
|
|
without failure. So this is our
|
|
|
implementation and explanation of
|
|
|
why we do that.
|
|
|
This slide explains how do we
|
|
|
specify truncated normal prior
|
|
|
in these two platforms. Because
|
|
|
truncated normal is not a
|
|
|
building prior distribution in
|
|
|
these two platforms.
|
|
|
First, look at what is
|
|
|
truncated normal. Here we
|
|
|
give example of truncated normal
|
|
|
with two ends at 5 and 400. The
|
|
|
two ends are illustrated by
|
|
|
this L and this R.
|
|
|
But truncated normal is nothing
|
|
|
but a normal by discarding
|
|
|
all the values that are
|
|
|
negative, which is represented
|
|
|
by this through curve as the
|
|
|
equivalent normal distribution
|
|
|
for this particular truncated
|
|
|
normal distribution.
|
|
|
If we want to specify this
|
|
|
truncated normal or we need to
|
|
|
define is equivalent...equivalent
|
|
|
normal distribution with two ends
|
|
|
that had that had the same new
|
|
|
and Sigma parameters as those of
|
|
|
these truncated normal
|
|
|
distributions. So we provide a
|
|
|
script to do this.
|
|
|
In this script, the
|
|
|
calculation is the
|
|
|
following. We find the new
|
|
|
and Sigma from the truncated
|
|
|
normal.
|
|
|
So we get can get the
|
|
|
equivalent normal
|
|
|
distribution.
|
|
|
And we that ??? Sigma of this
|
|
|
normal distribution we can find
|
|
|
out the two ends of this normal
|
|
|
distribution.
|
|
|
So to specify the truncated
|
|
|
normal with this two end
|
|
|
value to specify equivalent
|
|
|
normal distribution with two
|
|
|
end...these two end values.
|
|
|
This is how we specify
|
|
|
truncated normal in these
|
|
|
two platforms using
|
|
|
equivalent normal
|
|
|
distribution. And here's the
|
|
|
content of the script.
|
|
|
All you need to do is by calling
|
|
|
a function, which here is the
|
|
|
reverse parameter tnorm to
|
|
|
normal value. What do you need
|
|
|
to provide are the two ends of
|
|
|
the truncated normal
|
|
|
distribution and it will give
|
|
|
the two ends of the equivalent
|
|
|
normal distribution and you can
|
|
|
use those two numbers to specify
|
|
|
the prior distribution.
|
|
|
So this concludes my
|
|
|
demonstration. In my
|
|
|
demonstration I showed how to
|
|
|
start Bayesian analysis in Life
|
|
|
distribution and Fix Life by
|
|
|
X, how to enter prior
|
|
|
information,
|
|
|
and what's the content of
|
|
|
Bayesian result. Also I explained
|
|
|
what our implementation of the
|
|
|
sampling and why do you do
|
|
|
that. And in the end I explain
|
|
|
how do we specify a truncated
|
|
|
normal prior using an
|
|
|
equivalent normal prior in
|
|
|
these two...in these two
|
|
|
platforms. Thank you.
|
