Choice of proposal distributions in Bayesian calibration

Methodology
3

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:

Here I can show you the prior distributions I use :

# 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) :

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