Applications of Bayesian Methods Using JMP® (2021-EU-45MP-786)

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.

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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.

Presenter

William Q. Meeker

Applications of Bayesian Methods Using JMP® (2021-EU-45MP-786)

Presenter

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