Uncertainty Propagation and Second Order Probability

[#1]

In my field data is expensive to collect (or simulate deterministically), so we typically do our uncertainty propagation using metamodels (or surrogate models, response surface, etc.).  JMP excels in the generation of these models.  However, I have been unable to find any clear way to do uncertainty propagation using second order probability, as summarized in this diagram below:

I can see how to define input probabilities in the profiler simulator, but I'm not sure if there is a straightforward way to define some inputs as "epistemic" with uniform inputs that are sampled first, and then all the other inputs are sampled according to their own distributions N times to create a single CDF on the system response (which is then repeated M times to create the probability box on the system response). 

Anybody have any ideas how to accomplish inside the graphical JMP environment without diving into the world of JSL?  I have typically accomplished this using Python, in which I am fluent.   

[#2 · Accepted solution]

My understanding of what you're trying to accomplish is that you want N unique aleatory variable samples for each M epistemic variable samples.

Here's what I would do if JSL isn't an option:

1. Create a table, name it epistemic
   • Create a column
     • Name it "Sample"
     • Initialize Data with a sequence
       • Number of runs = 1
       • From = 1
       • To = M
       • Step = 1
   • For each epistemic variable you have:
     • Create a column with an appropriate name
     • Add a formula to that column
       
       • Use an appropriate Random function (Uniform?)
2. Create another data table, name it aleatory
   • Create a column
     • Name it "Sample"
     • Initialize Data with a sequence
       • Number of runs = 1
       • From = 1
       • To = N
       • Step = 1
       • Repeat = M   <= This is important!
   • For each aleatory variable you have:
     • Create a column with an appropriate name
     • Add a formula to that column
       
       • Use an appropriate Random function
   • While in this table (aleatory) click on Tables => Update
     • Select the epistemic table
     • Check "match columns"
       
       • Select Sample from both tables and click the "match" button
     • Under Add Columns from Update table select "selected"
       • Only select your epistemic variable columns, ignore "sample" column
     • Click the "OK" button
   • You can now remove the Sample column from the aleatory table if you like.
3. You can now close the epistemic table if you want.
4. The aleatory table should repeat your epistemic sample M times while the aleatory samples are not repeated and are sampled N times for each epistemic sample.

If your goal is to repeat the aleatory sampling for each epistemic sample (same aleatory sampling goes into each epistemic sample) then it's easier:

1. Create a table, name it epistemic
   
   • Add M rows
   • For each epistemic variable you have:
     • Create a column with an appropriate name
     • Add a formula to that column
       
       • Use an appropriate Random function (Uniform?)
2. Create another data table, name it aleatory
   • Add N rows 
   • For each aleatory variable you have:
     • Create a column with an appropriate name
     • Add a formula to that column
       
       • Use an appropriate Random function
3. In the aleatory  table go to Table => Join
   • Select epistemic table on the left
   •  Under "matching specification" select "Cartesian Join"
   • Click the "OK" button

Hope this helps

[#3]

That's excellent.  The first method is what I was after, and it's a nice quick solution.  I verified that your instructions worked as desired (edit: with one small change, see reply below). 

One last step, however, is that I need to plot all of the CDFs for each sample group.  So I take the response and plot distributions of some "response" by "Sample."  For M = 100, I end up with 100 distinct PDF and CDF plots.  Is there an easy way to send all of the probability scores to the table simulataneously without simply clicking each of the M plots and manually saving the Prob Scores?  

By the way, JSL is an option, but I'm looking to see if there is an elegant GUI solution.

Thank you for your help.

[#4]

One small correction for step 2:

Create another data table, name it aleatory

• Create a column
  • Name it "Sample"
  • Initialize Data with a sequence
    • Number of runs = 1
    • From = 1
    • To = N
    • Step = 1
    • Repeat = M   <= This is important!

Should be (changes in bold):

Create another data table, name it aleatory

• Create a column
  • Name it "Sample"
  • Initialize Data with a sequence
    • Number of runs = 1
    • From = 1
    • To = M
    • Step = 1
    • Repeat = N   

Because each epistemic sample should be repeated with new aleatory samples N times, not M times.