Bayesian inversion for multiple parameter calibration (some questions on the example problem of sec 2.3 of the user manual)

UQ with UQLab Community Q&A and How To
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Hello all,

Firstly, I wanted to thank you for the UQLAB package and this community,

I’m a newcomer to the field of Bayesian inversion, and I’ve been experimenting with the “multiple model outputs” example in the section 2.3 of Bayesian user manual (the process of calibrating a beam parameters based on beam deflection at different spans) trying to to improve my understanding of the entire process.

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At first, I made some artificial deflection data based on fixed values of (b,h,p) and variability in (E,L). the Young’s Modulus values(E) are sampled from a normal distribution with a mean value and standard deviation of 50 (GPa) and 4.5 (GPa) respectively. While in the original example the beam length (L) was considered as a fixed value, I’ve considered it to follow a normal distribution with a mean value and standard deviation of 5 (m) and 0.1 (m) respectively.
A noise term sampled from a normal distribution (mean=0 , std dev = 0.5 (mm)) has been added to the artificial deflection data as well.

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I ran two models, one with an informative prior and another one with a non-informative prior
informative prior (UQ_main_Informative.m):

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non-informative prior (UQ_main_nonInformative.m):

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To reduce the number of calibrated model parameters, I’ve considered a fixed discrepancy parameter in both cases with a sigma2= 0.25 (have I chose it correctly based on the std dev of the imposed white noise ?)
according to the discussions I’ve read on here on the convergence check for MCMC, I’ve also increased the MCMC.steps to 1000 to maintain a MPSRF value below 1.1 (MPSRF=1.08).

Below you can find the output I get from the model with informative prior function (UQ_main_Informative.m):

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While the posterior mean that I get from this model matches well (within 10% error) with the expected mean value of the sampled “E” and “L” values, but the posterior std_dev show much more deviation than the expected value. I was wondering if there is anything I can do to reach better agreement between my posterior moments and the expected values?

I have noticed that the correlation between E and L is very close to 1, which was expected as the deflection formula is in the form of L^4/E. I am wondering if this constitutes an “ill-posed inversion problem” and how I could find a solution for it?

Furthermore, I am curious if this issue is related to the problem being an under-constrained inverse problem. If so, how should I determine the appropriate number of deflections at different spans based on the number of calibrated parameters?

Here’s the output I get from the model with non-informative uniform prior function
(UQ_main_nonInformative.m):

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As it could be see, in this case the posterior simply doesn’t look like the expected distribution at all. (is there any way to improve the outcome ?)

I’m sorry for this long post and so many questions, you can find the script and the forward model in here (google drive)

Thanks a lot for your time and consideration in advance,

Mohammad

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Dear Mohammend,

welcome to the UQLAB user community.

I am sorry, but I have bad news. If you start from a probability distribution for parameter values, generate a set of samples for parameter values as model input and add some noise afterwards to the generated model output, you can use the generated samples as the observations considered for Bayesian inversion, but the posterior density you will get from Bayesian inversion (at least the one out-of the box with UQLab) will not reproduce your original distribution. (I tried the same and was quite surprised on the result.) Instead, the result of Bayesian inversion will be the result of finding one true input parameter value for the model such that all considered samples are equal to the output of the model at this true input parameter with different vales for the added noise. and the posterior density will represent the information one can extract on this true value from the observations. Your are exactly in the situation I describe in the warning in the Remark 3.8 in my recent publication On forward and inverse uncertainty quantification for a model for a magneto mechanical device involving a hysteresis opera.

And if you would compute the mean of the deflection values over all samples and would
afterwards replace in all sample the computed value by this mean value and run Bayes again, you should get almost the same posterior distribution. In your situation, std_dev of the posterior distribution is independent of the variation of the sample values but depends only on the chosen value for the discrepancy, at least according to my computations.

There are maybe some ways to solve the identification problem you have in mind,
but I have not been able to do it.

Maybe more next week, now I have to finish.

Greetings

  Olaf
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See also the discussion in
Bayesian inversion: how should the discrepancy model be calculated?

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