# Choice of proposal distributions in Bayesian calibration

**URL:** <https://openturns.discourse.group/t/choice-of-proposal-distributions-in-bayesian-calibration/337>\
**Category:** Methodology\
**Created:** [January 12, 2024, 1:56pm UTC](https://openturns.discourse.group/t/choice-of-proposal-distributions-in-bayesian-calibration/337 "2024-01-12T13:56:46Z")\
**Posts on this page:** 1\
**Showing post:** 3

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**Author:** ![elieso](https://avatars.discourse-cdn.com/v4/letter/e/f04885/32.png) [@elieso](https://openturns.discourse.group/u/elieso)\
**Post date:** [February 29, 2024, 2:58pm UTC](https://openturns.discourse.group/t/choice-of-proposal-distributions-in-bayesian-calibration/337/3 "2024-02-29T14:58:15Z")

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Hi,

Thanks for these information !  
I have indeed no idea what the shapes of the posterior distributions will look like…

I have 6 uncertain variables to calibrate, and my surrogate model outputs 2 quantities of interest.

This problem is related to the topic i’ve created before here:

> [@Use Kriging as forward model in Bayesian calibration](https://openturns.discourse.group/t/use-kriging-as-forward-model-in-bayesian-calibration/262/13):
>
> Hi @elieso and sorry for the delay, I have not had time to browse the forum for a while. If your Kriging model has output dimension 2, let’s say \boldsymbol{Y}(x) = (Y\_1(x), Y\_2(x))^T where x is the N\_d-dimensional input, then \boldsymbol{Y}(x) follows a bivariate normal distribution. Therefore your conditional distribution is going to be defined with: conditional = ot.Normal(2) And your link function will need to output the parameters of a bivariate normal distribution. To see what these p…

Here I can show you the prior distributions I use :

```auto
# variable X0
distribName	= 'Normal'
    P_mean = 1.0e5  	
    P_std = 2000  	
    PdistParams = [P_mean,P_std] 
# variable X1
    distribName = 'Normal'
    lev_mean = 0.27
    lev_std = 3.5e-2 # good 1.5e-2
    levDistParam = [lev_mean,lev_std] 
# variable X2
    distribName = 'LogNormal'     
    m_mu = 0.1           
    m_sigma = 1.0           
    mDistParam = [m_mu,m_sigma] 

# variable X3 
    alphadistribName	= 'Uniform'
    alpha_min = 0.047 
    alpha_max = 0.75
    alphaDistParams = [alpha_min,alpha_max] 
# variable X4
    betaDistribName = 'Uniform'
    beta_min = 0.7 
    beta_max = 0.9 
    betaDistParams = [beta_min,beta_max] 
# variable X5 
    gammaDistribName	= 'Uniform'
    gamma_min = 0.29 
    gamma_max = 0.7
    gammaDistParams = [gamma_min,gamma_max] 

```

The Random MH algorithm and Gibbs sampler definition i tried is the following (note that i tried to take proposals distributions having the same mean as the priors) :

```auto
propStd = 1 
proposal =[ot.Normal(1e5,propStd),    
                       ot.Normal(0.27	,propStd),      
                       ot.Normal(0.1	,propStd),     
                       ot.Normal(0.3985	,propStd),      
                       ot.Normal(0.8	,propStd),      
                       ot.Normal(0.495	,propStd)]      
                    
mh_coll = [
        ot.RandomWalkMetropolisHastings(priorDistrib,initialState,proposal,[i])  
        for i in range(0,Nd)
            ]    
        
for mh in mh_coll:
    mh.setLikelihood(conditional,Yexp,linkFunction) 

sampler = ot.Gibbs(mh_coll) 
sampleSize = int(5e4) 
sampler.setBurnIn(int(sampleSize/5))

```

Also, to be complete, the linkFunction is a Kriging model mapping the 6 uncertain inputs to the 2 quantities of interest. Of course, the experimental data used to calibrate are within the bounds of the quantities of interest when propagated with these prior distributions.

I get really small acceptance rates with this set of parameters…

Thanks again !

Best regards,  
Elie

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