Level: Intermediate
William Q. Meeker, Professor of Statistics and Distinguished Professor of Liberal Arts and Sciences, Iowa State University
Peng Liu, JMP Principal Research Statistician Developer, SAS
The development of theory and application of Monte Carlo Markov Chain methods, vast improvements in computational capabilities and emerging software alternatives have made it possible for the wide use of Bayesian methods in many areas of application.
Motivated by applications in reliability, JMP now has powerful and easy-to-use capabilities for using Bayesian methods to fit different single distributions (e.g., normal, lognormal, Weibull, etc.) and linear regression. Bayesian methods, however, require the specification of a prior distribution. In many applications (e.g., reliability) useful prior information is typically available for only one parameter (e.g., imprecise knowledge about the activation energy in a temperature-accelerated life test or about the Weibull shape parameter in analysis of fatigue failure data). Then it becomes necessary to specify noninformative prior distributions for the other parameter(s). In this talk, we present several applications showing how to use JMP Bayesian capabilities to integrate engineering or scientific information about one parameter and to use a principled way to specify noninformative or weakly informative prior distributions for the other parameters.
Speaker | Transcript |
Our talk today shows how to | |
use JMP to do Bayesian | |
estimation. Here's an overview | |
of my talk. I'm going to start | |
with a brief introduction to | |
Bayesian statistical methods. | |
Then I'm going to go through | |
four different examples that | |
happen to come from reliability, | |
but the methods we're presenting | |
are really much more general and | |
can be applied in other areas of | |
application. Then I'm going to | |
turn it over to Peng and he's | |
going to show you how easy it is | |
to actually do these things in | |
JMP. Technically, reliability | |
is a probability. The | |
probability of a system, | |
vehicle, machine or whatever it | |
is that is of interest, will | |
perform its intended function | |
under encountered operating | |
conditions for a specified period | |
of time. I highlight encountered | |
here to emphasize that | |
reliability depends importantly | |
on the environment in which a | |
product is being used. Condra | |
defined reliability as quality | |
over time. And many engineers | |
think of reliability is being | |
failure avoidance, that is, to | |
design and manufacture a product | |
that will not fail. Reliability | |
is a highly quantitative | |
engineering discipline, but | |
often requires sophisticated | |
statistical and probabilistic | |
ideas. Over the past 30 years, | |
there's been a virtual | |
revolution where Bayes methods | |
are now commonly used and in | |
many different areas of | |
application. This revolution | |
started by the rediscovery of | |
Markov chain Monte Carlo methods | |
and was accelerated by | |
spectacular improvements in | |
computing power that we have | |
today, as well as the | |
development of relatively easy | |
to use software to implement | |
Bayes methods. In the 1990s we | |
had BUGS. Today Stan and other | |
similar packages have largely | |
replaced BUGS, but the other | |
thing that's happening is we're | |
beginning to see more Bayesian | |
methods implemented in | |
commercial software. So for | |
example, SAS has PROC MCMC. | |
And now JMP has some very | |
powerful tools that were | |
developed for reliability, but | |
as I said, they can be applied | |
in other areas as well, and | |
there's strong motivation for | |
the use of Bayesian methods. | |
For one thing, it provides a | |
means for combining prior | |
information with limited data to | |
be able to make useful | |
inferences. Also, there are many | |
situations, particularly with | |
random effects complications | |
like censor data, where maximum | |
likelihood is difficult to | |
implement, but where Bayes | |
methods are relatively easy to | |
implement. There's one little | |
downside in the use of Bayes | |
methods. You have to think a bit | |
harder about certain things, | |
particularly about | |
parameterization and how to | |
specify the prior distributions. | |
My first example is about an | |
aircraft engine bearing cage. | |
These are field failure data | |
where there was 1,703 aircraft | |
engines that contained this | |
bearing cage. The oldest ones | |
had 2,220 hours of operation. The | |
design life specification for | |
this bearing cage was that no | |
more than 10% of the units would | |
fail by 8,000 hours of operation. | |
However, 6 units had failed and | |
this raised the question | do we |
have a serious problem here? Do | |
we need to redesign this bearing | |
cage to meet that reliability | |
condition? This is an event plot | |
of the data. The event plot | |
illustrates the structure | |
of the data, and in particular, | |
we can see the six failures | |
here. In addition to that, we | |
have right censored | |
observations, indicated here by | |
the arrows. So these are units | |
that are still in service and | |
they have not failed yet, and | |
the right arrow indicates that | |
all we know is if we wait long | |
enough out to the right, the units | |
will eventually fail. Here's a | |
maximum likelihood analysis of | |
those data, so the probability | |
plot here suggests that the | |
Weibull distribution provides a | |
good description of these data. | |
However, when we use the | |
distribution profiler to | |
estimate fraction failing at | |
8,000 hours, we can see that the | |
confidence interval is enormous, | |
ranging between about 3% all the | |
way up to 100%. That's not very | |
useful. So likelihood methods | |
work like this. We specify the | |
model and the data. | |
That defines the likelihood and | |
then we use the likelihood to | |
make inferences. Bayesian | |
methods are similar, except we | |
also have prior information | |
specified. Bayes theorem | |
combines the likelihood with the | |
prior information, providing a | |
posterior distribution, and then | |
we use the posterior | |
distribution to make inferences. | |
Here's the Bayes analysis of the | |
bearing cage. The priors are | |
specified here for the B10 | |
or time at which 10% would fail. | |
We have a very wide interval | |
here. The range, effectively 1,000 | |
hours up to 50,000 hours. | |
Everybody would agree that B10 | |
is somewhere in that range. For | |
the Weibull shape parameter, | |
however, we're going to use an | |
informative prior distribution | |
based upon the engineers' | |
knowledge of the failure | |
mechanism and their vast | |
previous experience with that | |
mechanism. They can say with | |
little doubt that the Weibull | |
shape parameter should be | |
between 1.5 and 3, and | |
here's where we specify that | |
information. So instead of | |
specifying information for the | |
traditional Weibull parameters, | |
we've reparameterized, where now | |
the B10 is one of the | |
parameters, and here's the | |
specified range. And then we | |
have the informative prior for | |
the Weibull shape parameter | |
specified here. And then JMP | |
will generate samples from the | |
joint posterior, leading to | |
these parameter estimates and | |
confidence intervals shown here. | |
Here's a graphical depiction of | |
the Bayes analysis. The black | |
points here are a sample from | |
the prior distribution, so again | |
very wide for the .1 quantile | |
and somewhat constrained for the | |
Weibull shape parameter beta. On | |
the right here, we have the | |
joint posterior, which in effect | |
is where the likelihood contours | |
and the prior sample | |
overlap. And then those draws | |
from the joint posterior are | |
used to compute estimates and | |
Bayesian credible intervals. So | |
here's the same profiler that we | |
saw previously where the | |
confidence interval was not | |
useful. After bringing in the | |
information about the Weibull | |
shape parameter, now we can see | |
that the confidence interval | |
ranges between 12% and 83%, | |
clearly illustrating that we | |
have missed the target of 10%. | |
So what have we learned here? | |
With a small number of failures, | |
there's not much information | |
about reliability. But engineers | |
often have information that can | |
be used, and by using that prior | |
information, we can get improved | |
precision and more useful | |
inferences. And Bayesian methods | |
provide a formal method for | |
combining that prior information | |
with our limited data. Here's | |
another example. Rocket motor is | |
one of five critical components | |
in a missile. In this particular | |
application, there were | |
approximately 20,000 missiles | |
in the inventory. Over time, 1,940 | |
of these missiles had been fired | |
and they all worked, except in | |
three cases, where there was | |
catastrophic failure. And these | |
were older missiles, and so | |
there was some concern that | |
there might be a wearout failure | |
mechanism that would put into | |
jeopardy the roughly 20,000 | |
missiles remaining in inventory. | |
The failures were thought to be | |
due to thermal cycling, but | |
there was no information about | |
the number of thermal cycles. We | |
only have the age of the | |
missile when it was fired. | |
That's a useful surrogate, but | |
the effect of using a surrogate | |
like that is you have more | |
variability in your data. Now | |
in this case, there were no | |
directly observed failure times. | |
When a rocket is called upon to | |
operate and it operates | |
successfully, all we know is | |
that they had not failed at the | |
age of those units when they | |
were asked to operate. And for | |
the units that failed | |
catastrophically, again, we | |
don't know the time that those | |
units failed. At some point | |
before they were called upon to | |
operate. They had failed, so all | |
we know is that the failure was | |
before the age at which it was | |
fired. So as I said, there was | |
concern that there is a wear out | |
failure mechanism kicking in | |
here that would put into | |
jeopardy the amount of remaining | |
life for the units in the | |
stockpile. So here's the table | |
of the data. Here we have the | |
units that operated | |
successfully, and so these are | |
right censored observations, but | |
these observations here | |
are the ones that failed and as | |
I said, at relatively higher | |
ages. This is the event plot of | |
the data and again, we can see | |
the right censored observations | |
here with the arrow pointing to | |
the right, and we can see the | |
left censored observations with | |
the arrow pointing to the left | |
indicating the region of | |
uncertainty. But even with those | |
data we can still fit a Weibull | |
distribution. And here's the | |
probability plot showing the | |
maximum likelihood estimate and | |
confidence bands. Here's more | |
information from the maximum | |
likelihood analysis. And here we | |
have the estimate of fraction | |
failing at 20 years, which was | |
the design life of these rocket | |
motors, and again the interval | |
is huge, ranging between 3% and | |
100%. Again, not very useful. | |
But the engineers, | |
knowing what the failure | |
mechanism was, again had | |
information about the Weibull | |
shape parameter. The maximum | |
likelihood estimate was | |
extremely large and the | |
engineers were pretty sure that | |
that was wrong, especially with | |
the extra variability in the | |
data that would tend to drive | |
the Weibull shape parameter to a | |
lower value. As I showed you on | |
the previous slide, confidence | |
interval for fraction failing at | |
20 years was huge. So once again, | |
we're going to specify a prior | |
distribution and then use that | |
in a Bayes analysis. Again, the | |
prior for B10, the time at which | |
10% will fail, is chosen to be | |
extremely wide. We don't really | |
want to assume anything there, | |
and everybody would agree that | |
that quantity is somewhere | |
between five years and 400 | |
years. But for the Weibull shape | |
parameter, we're going to assume | |
that it's between one and five. | |
Again, we know it's greater than | |
one because it's a wear out | |
failure mechanism, and the | |
engineers were sure that it | |
wasn't anything like the number | |
8 that we had seen in the | |
maximum likely estimate. | |
And indeed, five is also a very | |
large Weibull shape parameter. | |
Once again, JMP is called upon | |
to generate draws from the | |
posterior distribution. And here are | |
plot similar to the ones that we | |
saw in the bearing cage example. | |
The black points here, again, are | |
a sample from the prior | |
distribution. Again very wide in | |
terms of the B10, but somewhat | |
constrained for the beta, so the | |
beta is an informative prior | |
distribution. And again the | |
contour plots represent the | |
information in the limited data. | |
In our posterior, once again, is | |
where we get overlap between the | |
prior and the likelihood and we | |
can see it here. So once again | |
we have a comparison between the | |
maximum likelihood interval, | |
which is extremely wide, and the | |
interval that we get for the | |
same quantity using the Bayes | |
inference, which incorporated the | |
prior information on the Weibull | |
shape parameter. And now the | |
interval ranges between .002 and | |
.98 or about .1, about 10% | |
failing, so that might be | |
acceptable. Some of the things | |
that we learned here, even | |
though there were no actual | |
failure times, we can still get | |
reliability information from the | |
data, but with very few failures | |
there isn't much information | |
there. But we can use the | |
engineer's knowledge about the | |
Weibull shape parameter to | |
supplement the data to get | |
useful inferences and JMP makes | |
this really easy to do. My last | |
two examples are about | |
accelerated testing. Accelerated | |
testing is a widely used | |
technique to get information | |
about reliability of components | |
quickly when designing a | |
product. The basic idea is to | |
test units at high levels of | |
variables like temperature or | |
voltage to make things fail | |
quickly and then to use a model | |
to extrapolate back down to the | |
use conditions. Extrapolation is | |
always dangerous and we have to | |
keep that in mind. That's the | |
reason we would like to have our | |
model be physically motivated. | |
So here's an example of an | |
accelerated life test | |
on a laser. Units were tested at | |
40, 60 and 80 degrees C, but the | |
use condition was 10 degrees C. | |
That's the nominal temperature | |
at the bottom of the Atlantic | |
Ocean, where these lasers were | |
going to be used in a new | |
telecommunications system. The | |
test lasted 5,000 hours, a little | |
bit more than six months. The | |
engineers wanted to estimate | |
fraction failing at about 30,000 | |
hours. That's about 3.5 years | |
and again, at 10 degrees C. | |
Here's the results of the | |
analysi. In order to | |
appropriately test and build the | |
model, JMP uses these three | |
different analyses. The first | |
one fits separate distributions | |
to each level of temperature. | |
The next model does the same | |
thing, except that it constrains | |
the shape parameter Sigma to be | |
the same at every level of | |
temperature. This is analogous | |
to the constant Sigma assumption | |
that we typically make in | |
regression analysis. And then | |
finally, we fit the regression | |
model, which in effect, is a | |
simple linear regression | |
connecting lifetime to | |
temperature. And to supplement | |
this visualization of these | |
three models, JMP does | |
likelihood ratio tests to test | |
whether there's evidence that | |
the Sigma might depend on | |
temperature and then to test | |
whether there's evidence of lack | |
of fit in the regression model. | |
And from the large P values | |
here, we can see that there's no | |
evidence against this model. | |
Another way to plot the results | |
of fitting this model | |
is to plot lifetime versus | |
temperature on special scales. A | |
log rhythmic scale for hours of | |
operation in what's known as an | |
Arrhenius scale for temperature. | |
Corresponding to the Arrhenius | |
model, which describes how | |
temperature affects reaction | |
rates, and thereby lifetime. And | |
this is the results of the | |
maximum likelihood estimation | |
for our model. The JMP | |
distribution profiler gives us | |
an estimate of the fraction | |
failing at 30,000 hours. | |
And we can see it ranges between | |
.002 and about .12, or 12% | |
failing. The engineers in | |
applications like this, however, | |
often have information about | |
what's known as the effective | |
activation energy of the failure | |
mechanism, and that corresponds | |
to the slope of the regression | |
line in the Arrhenius model. So | |
we did a Bayes analysis and in | |
that analysis, we made an | |
assumption about the effective | |
activation energy. And that's | |
going to provide more precision | |
for us. So what we have here is | |
a matrix scatterplot of the | |
joint posterior distribution | |
after having specified prior | |
distributions for the | |
parameters, weakly informative | |
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. |