# Obtaining a better parameter marginal fit

**URL:** <https://uqworld.org/t/obtaining-a-better-parameter-marginal-fit/1863>\
**Category:** Community Q&A and How To\
**Tags:** input\
**Created:** [April 14, 2023, 5:53pm UTC](https://uqworld.org/t/obtaining-a-better-parameter-marginal-fit/1863 "2023-04-14T17:53:02Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![Truong\_Le](https://uqworld.org/user_avatar/uqworld.org/truong_le/32/1153_2.png) [@Truong\_Le](https://uqworld.org/u/Truong_Le)\
**Post date:** [April 14, 2023, 5:53pm UTC](https://uqworld.org/t/obtaining-a-better-parameter-marginal-fit/1863/1 "2023-04-14T17:53:02Z")

</div>

I have run into a (probably ignorant) issue defining a new input when using the statistical inference module.

I have some data that looks to be lognormal and I am hoping to propregate the distribution for further bayesian inference.

![untitled](https://uqworld.org/uploads/default/original/2X/b/b97477696942e485f8ae877bfaffeb9fc897afb9.jpeg)

The red histogram is the output of the Bayesian inference and the blue is the infered marginal.

The relevant snippet of code is provided below. The data is uploaded as a zip file.

Any help would be greatly appreciated 🙂

I would like to add that I have tried all the inferences criterion, and defining various marginal families to no greater success than the default.

Regards,

Truong

[UQLAB\_QUERY2.zip](https://uqworld.org/uploads/short-url/aHI6EZ3ju6wFoYKJdgX4N0m1FpZ.zip) (1.5 MB)

```matlab
% Inference of the samples created in MCMC
PostSample3D = myBayesianAnalysis_surrogateModel.Results.PostProc.PostSample ; 
PostSample2D = reshape(permute(PostSample3D, [2 1 3]), size(PostSample3D, 2), [])';
iOpts.Inference.Data = PostSample2D(:,2) ; 
iOpts.Marginals.Type = 'auto';
iOpts.Copula.Type = 'Independent';
iOpts.Inference.Criterion = 'KS';
mydist_infer = uq_createInput(iOpts);
uq_print(mydist_infer)

%Creation of samples from the two post-inputs 
Xtest_inf= uq_getSample(mydist_infer , 10000, 'LHS');

% % hold on to the existing plot and add a histogram of the second dataset
figure
histogram(Xtest_inf,'BinMethod','auto')

hold on
histogram(PostSample2D(:,2))

xlabel('\theta_1')
ylabel('Count')

```

---

<div class="post-metadata">

**Author:** ![olaf.klein](https://uqworld.org/user_avatar/uqworld.org/olaf.klein/32/468_2.png) [@olaf.klein](https://uqworld.org/u/olaf.klein)\
**Post date:** [April 17, 2023, 7:24am UTC](https://uqworld.org/t/obtaining-a-better-parameter-marginal-fit/1863/2 "2023-04-17T07:24:06Z")

</div>

Dear Truong,

it seems to me that UQLab tries to represent your data by a normal distributed random variable, thanks to the automatic choice for the marginal family thanks to ` iOpts.Marginals.Type = 'auto';` .  
Maybe you should try to define the marginal family to be log-normal distributions by ` iOpts.Marginals.Type = 'Lognormal' ;`

Greetings  
Olaf

---

<div class="post-metadata">

**Author:** ![Truong\_Le](https://uqworld.org/user_avatar/uqworld.org/truong_le/32/1153_2.png) [@Truong\_Le](https://uqworld.org/u/Truong_Le)\
**Post date:** [April 17, 2023, 8:00am UTC](https://uqworld.org/t/obtaining-a-better-parameter-marginal-fit/1863/3 "2023-04-17T08:00:56Z")

</div>

Hi Olaf,

I am able to get good agreement by using non-parametric inference. I have yet to see what the implications are for further Bayesian inference.

I still cannot figure out why I cannot use ‘lognormal’ as a marginal type.  
Also there is an issue with this line: `iOpts.Marginals(1).Type = {'Gaussian', 'Exponential', 'Weibull'}; `  
Works fine as is but doesnt work when I change one of the entries to `'ks'`. It might be that it doesnt like parameteric mixed with non-parametric?

Maybe someone on the programming team can help.

Below is the code for verification:

```matlab
%%

% Inference of the samples created in MCMC
PostSample3D = myBayesianAnalysis_surrogateModel.Results.PostProc.PostSample ; 
PostSample2D = reshape(permute(PostSample3D, [2 1 3]), size(PostSample3D, 2), [])';
iOpts.Inference(1).Data = PostSample2D(:,2) ; 
iOpts.Marginals(1).Type = {'Gaussian', 'Exponential', 'Weibull'}; % works
% iOpts.Marginals(1).Type = 'ks'; % works
% iOpts.Marginals(1).Type = {'Gaussian', 'ks', 'lognormal'}; % Doesn't work
iOpts.Copula(1).Type = 'Independent';
iOpts.Inference(1).Criterion = 'KS';
mydist_infer = uq_createInput(iOpts);
uq_print(mydist_infer)

%Creation of samples from the two post-inputs 
Xtest_inf= uq_getSample(mydist_infer , 15000, 'LHS');

% % hold on to the existing plot and add a histogram of the second dataset
figure
histogram(Xtest_inf,55)
%histfit(PostSample2D(:,2),55,'loglogistic') %Checking matlab's default
hold on
histogram(PostSample2D(:,2),55)

xlabel('\theta_1')
ylabel('Count')

```

---

<div class="post-metadata">

**Author:** ![olaf.klein](https://uqworld.org/user_avatar/uqworld.org/olaf.klein/32/468_2.png) [@olaf.klein](https://uqworld.org/u/olaf.klein)\
**Post date:** [April 21, 2023, 9:53am UTC](https://uqworld.org/t/obtaining-a-better-parameter-marginal-fit/1863/4 "2023-04-21T09:53:34Z")

</div>

Dear Truong,

by investing the UQLab-code in `uq_infer_marginals.m` I deduced that one had to use the string `'LogNormal'` to refer to lognormal distribution and that neither `'lognormal'` or `'Lognormal'` can  
be used instead.

But, also correctly requesting a lognormal approximation seems not to provide a better approximation, see  
 ![LogNorrmalApprox](https://uqworld.org/uploads/default/original/2X/6/6e54cd44bed80e130de63935b60defc71e0a0f7b.png)

Comparing the histogram for lognormal samples with a plot for the lognormal PDF using  
the parameters in the output of `uq_print(mydist_infer)` it seems that the computed samples  
represent the corresponding lognormal distribution with the given parameters.

So, it seems to me that the approximation of your data points generated by a lognormal distribution  
resulting by using the `‘KS’ criterion is not better then the one by a normal distribution in auto mode.  
Maybe an other criterion may help, but this is not my field of expertise,sorry.

Below pleas find my variation of your code.

Greetings  
Olaf

```matlab
% Inference of the samples created in MCMC
PostSample3D = myBayesianAnalysis_surrogateModel.Results.PostProc.PostSample ; 
PostSample2D = reshape(permute(PostSample3D, [2 1 3]), size(PostSample3D, 2), [])';
iOpts.Inference(1).Data = PostSample2D(:,2) ; 
% iOpts.Marginals(1).Type = {'Gaussian', 'Exponential', 'Weibull'}; % works
% iOpts.Marginals(1).Type = 'ks'; % works
% iOpts.Marginals(1).Type = {'Gaussian', 'ks', 'Lognormal'}; % Doesn't work
% iOpts.Marginals(1).Type = 'Lognormal'; % Doesn't work
iOpts.Marginals(1).Type = 'LogNormal'; % works

iOpts.Copula(1).Type = 'Independent';
iOpts.Inference(1).Criterion = 'KS';
mydist_infer = uq_createInput(iOpts);
uq_print(mydist_infer)

%Creation of samples from the two post-inputs 
Xtest_inf= uq_getSample(mydist_infer , 15000, 'LHS');

% % hold on to the existing plot and add a histogram of the second dataset
figure
histogram(Xtest_inf,55)
%histfit(PostSample2D(:,2),55,'loglogistic') %Checking matlab's default
hold on
histogram(PostSample2D(:,2),55)

xlabel('\theta_1')
ylabel('Count')

% Comapring plot
figure
histogram(Xtest_inf,55)
xlabel('\theta_1')
ylabel('Count')

figure
x=min(PostSample2D(:,2)):0.01:max(PostSample2D(:,2));
% values Parameter values from output of uq_print(mydist_infer)
y = lognpdf(x,4.531e+00, 8.173e-02 );
plot (x,y)

figure 
histogram(log(PostSample2D(:,2)),55)

xlabel('log(\theta_1)')
ylabel('Count')

figure 
histogram(log(log(PostSample2D(:,2))),55)

xlabel('log(log(\theta_1))')
ylabel('Count')

```

`