Choice of proposal distributions in Bayesian calibration

Methodology
1

Hi all,

I am running a Bayesian calibration using Independant Metropolis Hastings algorithm.
My problem has 6 inputs variables and 2 quantities of interest as outputs.
I use a Gibbs sampler.

I have some trouble tuning my MH algorithm in order to get good acceptance rates of the chains and to get nice posterior distributions.
I assume that the choice of the proposal distributions (for each of my 6 input variables) is of great importance here.

Do you have advice on how i can choose these distributions ?

For the sake of completeness, here are the PRIOR distributions I use :

ot.Normal(1.096e5,2e3)
ot.Normal(0.23,1.5e-3)
ot.Uniform(0.1,1.0) 
ot.Uniform(0.047,0.75)
ot.Uniform(0.7,0.9)
ot.Uniform(0.29,0.7)

Also, I use a Kriging surrogate model as forward model in the MH algorithm.
I also specify that the experimental values I use to perform the calibration are within the model response using prior distributions.

Thanks by advance for any help or clue on this topic !
Best regards

Elie

2

Hi @elieso ! With independent Metropolis-Hastings, your proposal distribution needs to be “close” to the posterior distribution. The closer it is, the better the acceptance rate. If you have no idea what the posterior distribution looks like, it will be hard to choose a proposal distribution.

I have several questions for you:

  1. Have you tried random walk Metropolis-Hastings?
  2. Have you tried to visualize the log-PDF of the posterior distribution using cross-cuts? Here is an example : you need to replace m.model with a function which calls the computeLogPosterior method and you need to define a reference point (maybe simply use the mean of the prior distribution, that way you only need to replace m.distributionX with your prior distribution).
  3. Could you share more of your code? As always, it is easier to give appropriate answers with a better understanding of your use case.
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