Assume X\sim Weibull(\alpha,\beta) with observations \mathcal{X}=\left\{x_{1},x_{2},\cdots\right\}. Prior distributions for \alpha and \beta are as follows:

The Bayesian inversion module by default assumes that the data are realizations of random variables distributed according to normal distributions. In your case, where \mathcal{X} are realizations of X\sim\text{Weibull}(\alpha,\beta), you need to specify a user-defined likelihood function (have a look at the Bayesian inversion user manual or Example 6 of the Bayesian inversion module).

You already know the priors for \alpha and \beta and the likelihood function is just a product of Weibull PDFs at the data \mathcal{X}.

Hi @paulremo
Thank you very much for your helpful explanation.

According to your comment, I wrote the custom likelihood function as follows:

function logL = uq_WeibullLogLik(params, y)
measurements = y;
nReal = size(params,1); % number of queried realizations
Ny = size(measurements);
logL = zeros(nReal,1);
for ii = 1:nReal
logLikeli = 0;
for jj = 1:Ny
logLikeli = logLikeli + log(wblpdf(measurements(jj,:),...
modelParams(ii,1),modelParams(ii,2)));
end
logL(ii) = logLikeli ;
end
end

Looks good ! But I think modelparams should be params and for efficiency you should avoid loops in Matlab whenever possible and write the function in vector form instead. I think the following should work:

function logL = uq_WeibullLogLik(params, y)
measurements = y;
nReal = size(params,1); % number of queried realizations
Ny = size(measurements, 2); % should be 1xNy, please double check
logL = zeros(nReal,1);
for ii = 1:nReal
logLikeli = log(wblpdf(measurements, modelParams(ii,1), modelParams(ii,2)));
logL(ii) = sum(logLikeli);
end