# How to define my probabilistic input distributions?

**URL:** <https://uqworld.org/t/how-to-define-my-probabilistic-input-distributions/131>\
**Category:** Chair's Blog\
**Created:** [April 15, 2019, 2:46pm UTC](https://uqworld.org/t/how-to-define-my-probabilistic-input-distributions/131 "2019-04-15T14:46:39Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![bsudret](https://uqworld.org/user_avatar/uqworld.org/bsudret/32/224_2.png) [@bsudret](https://uqworld.org/u/bsudret)\
**Post date:** [April 15, 2019, 2:46pm UTC](https://uqworld.org/t/how-to-define-my-probabilistic-input-distributions/131/1 "2019-04-15T14:46:40Z")

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## Introduction

A key step in an uncertainty quantification (UQ) study is to define the distributions that will be used to represent the uncertainty in the input parameters of a computational model of interest. Referring to the global framework for UQ that is discussed [here](https://uqworld.org/t/getting-started-with-uncertainty-quantification-uq), this step is called _Quantification of the sources of uncertainty_ (Step B).

## Probabilistic input for model exploration

As soon as a computational model has been developed, one is often interested in investigating what happens when we change some parameters from their nominal (“best estimate”) values to other smaller or larger values. This type of analysis is known as _model exploration_. In this context, we usually assign to each parameter a reasonable lower bound x\_l and upper bound x\_u based on expert knowledge on the model, literature data, etc.

When only the bounds are known and no additional information on more/less likely values is available, the suitable distribution for the parameter is a **uniform distribution** over the interval defined by the two bounds:

X \sim \mathcal{U}(x; \; [x\_l, x\_u])

This distribution satisfies the [principle of maximum entropy](https://en.wikipedia.org/wiki/Maximum_entropy_probability_distribution) when only bounds are known.

> _ **Remark** _  
> Model exploration can be carried out by computing the _Sobol’ global sensitivity indices_ of the model response, which characterize the contribution of each input parameter (or combination of parameters) to the output variance. This allows for identifying _important parameters_ (_i.e._ the ones whose variation influence the model output) from the _unimportant parameters_ (_i.e_ the ones that do not influence the output, and thus could simply be fixed to a single value without changing the nature of the analysis). Sobol’ indices can also provide valuable information about the _interactions_ between the input parameters.

## Probabilistic input for aleatory uncertainty (No data available)

In certain applications, it is not possible to assign a single value to a parameter because this parameter is known to _naturally vary_. This is typically the case for parameters describing _material properties_ in engineering applications (_e.g._ material elastic behavior, strength, toughness, thermal conductivity, electromagnetic permittivity, etc.) or _environmental conditions_ (_e.g._ temperature, climatic loads, etc.).

In such situations, it is important to select distributions that are consistent with physical constraints, such as positivity, boundedness, special behavior in extreme values (tail dependence), etc. As an example:

- Material properties that are _positive in nature_ are usually well-represented by [lognormal distributions](https://en.wikipedia.org/wiki/Log-normal_distribution). Gaussian distributions should be avoided, especially in cases where the coefficient of variation\[1\] is large, which would give frequent negative non-physical realizations and troubles in the uncertainty propagation phase. Truncated Gaussian may be used, however it may lead to unrealistic shapes of the resulting distribution.
- [Gaussian distributions](https://en.wikipedia.org/wiki/Normal_distribution) are often used for modelling the uncertainties of elements’ dimensions (lengths) in engineering systems. This allows one to represent the _scattering_ of the real dimensions of the system during the manufacturing process because of tolerances. In this case, the coefficient of variation is usually so low that there is no issue with positiveness.
- If a parameter is bounded in nature (_e.g._ the Poisson’s ratio in elasticity), a bounded distribution such as the [beta distribution](https://en.wikipedia.org/wiki/Beta_distribution) is to be used instead.
- If a parameter represents some maximum value over a time period (_e.g._ climatic loads, the water level in a river, etc.), it can be modeled with one of the so-called [extreme value distributions](https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution) such as Gumbel or Weibull.

> _ **Remark** _  
> It is important to realize that selecting a suitable distribution based on the current level of knowledge is always possible, even if there is no measured data available.

## Probabilistic input for aleatory uncertainty (Sufficient data available)

In cases where there is evidence of the natural variability of a parameter because of observed scattered values \mathcal{X} = \{x\_1, \, \cdots, x\_n\}, **statistical inference** can be used to build a probabilistic model, as soon as reasonable number of points is available (typically, n \ge 30).

- Numerical summaries and graphics are used to represent the data, _i.e._ empirical CDFs (cumulative distribution functions), histograms, “kernel smoothing” densities, etc. _Scatterplots_ can be used to plot the correlation between pairs of parameters.

 ![jointpdf](https://uqworld.org/uploads/default/original/1X/afd696a229c0fcf965a23fd21e768b552060fb31.png)

- Based on visual inspection, assumptions are made on the distribution families that could fit the data. This means selecting a parametric family such as Gaussian, lognormal, Gamma, Beta, etc.
- Under the assumption that parameter X follows a distribution f\_{X} parametrized by a vector of _hyperparameters_ \boldsymbol{\theta}:

X \sim f\_{X}(x; \; \boldsymbol{\theta}),

the goal of statistical inference is to find the best estimator \hat{\boldsymbol{\theta}} so that f\_X fits the data in \mathcal{X}. The most common methods to fit the parameters are:

- the [method of moments](https://en.wikipedia.org/wiki/Method_of_moments_(statistics)), in which the empirical moments of the sample \mathcal{X} are set equal to the moments of the distribution f\_X, and the resulting equations solved for \hat{\boldsymbol{\theta}};
- [maximum likelihood estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation), in which the _likelihood_ of the sample set is maximized as a function of \boldsymbol{\theta}.

When several candidate distributions are fitted, they can be compared using a model selection criterion such as the [Akaike information criterion](https://en.wikipedia.org/wiki/Akaike_information_criterion) (AIC) or the [Bayesian information criterion](https://en.wikipedia.org/wiki/Bayesian_information_criterion) (BIC).

## What to do in case of limited data?

There are practical situations in which data is scarce, _i.e._ where only a handful of data points is available, which does not allow us to use standard statistical inference. In this case, Bayesian statistics may be used to combine _prior knowledge_ on the distribution of X and its hyperparameters together with the scarce data. Prior knowledge can be represented by following the same procedure as for the case of _no data available_, i.e. by applying the maximum entropy principle to the available knowledge of the problem.

In the Bayesian paradigm, the hyperparameters \boldsymbol{\theta} themselves are treated as random variables, which are given a distribution that reflects the prior knowledge. For instance:

> X follows a Gaussian distribution with mean value \mu, which belongs to a given range [m\_l, \, m\_u]:
> 
> X \sim \mathcal{N}(x;\; \mu, \sigma^2) \quad \textsf{where} \quad \mu \sim\mathcal{U}(\mu; [m\_l, \, m\_u])

The data gathered in \mathcal{X} allows one to _update_ the distribution of \mu and get a _posterior distribution_, which can then be used to define the distribution of X.

## Conclusions

Whatever the available amount of information and data, it can be used to build a probabilistic model for further uncertainty propagation. The construction of such a probabilistic model should not be overlooked: the subsequent results in terms of output distribution, important parameters or probability of threshold exceed will depend on this choice. The “good practices” summarized above can be used as a starting point. In any case, uninformed choices such as “I use a Gaussian distribution with 10% coefficient of variation for all parameters”, which is often seen in practice, should be absolutely avoided.

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_[Back to the Chair’s Blog index](https://uqworld.org/t/about-the-chairs-blog-category)_

### Notes

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1. The coefficient of variation of a random variable is defined as the ratio of its standard deviation to its mean value (when the latter is non zero).

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**Author:** ![YaHengYang](https://uqworld.org/user_avatar/uqworld.org/yahengyang/32/1870_2.png) [@YaHengYang](https://uqworld.org/u/YaHengYang)\
**Post date:** [July 14, 2025, 9:42am UTC](https://uqworld.org/t/how-to-define-my-probabilistic-input-distributions/131/2 "2025-07-14T09:42:16Z")

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Thank you for the clear explanation.

I have a quick follow-up question: if a truncated distribution is required for a specific application, for example, when two input parameters follow bounded log-normal distributions with a specified target correlation, can a copula still be used to capture the correlation between these parameters? Or is the alternative way described in the paper necessary to ensure that the sampling satisfy the target correlation?  
[Evaluating correlation coefficient for Nataf transformation - ScienceDirect](https://www.sciencedirect.com/science/article/pii/S0266892014000241)

Thank you!

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**Author:** ![bsudret](https://uqworld.org/user_avatar/uqworld.org/bsudret/32/224_2.png) [@bsudret](https://uqworld.org/u/bsudret)\
**Post date:** [July 14, 2025, 4:57pm UTC](https://uqworld.org/t/how-to-define-my-probabilistic-input-distributions/131/3 "2025-07-14T16:57:13Z")

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Dear YaHeng

Assigning a Pearson’s correlation together with prescribed non Gaussian marginal is a ill-posed problem (you have to introduce a pseudo-correlation using the paper you mentioned):

- if you have data, just compute the Spearman’s correlation coefficient and assign a Gaussian copula accordingly
- if you have no data, but assign some correlation based on expert knowledge, again why not pretend you assign Spearman’s correlation?

As a conclusion, doing so, it’s not necessary to compute a pseudo linear correlation that does not have good properties.  
Best regards  
Bruno