Inverse problem with Stan

I am facing an inverse problem with two input variables and two
output variables. I think it is well posed problem.

I am looking for STAN modeling example to attack this problem
instead of using so-called Newton algorithm.

Do you know about a package appropriate for this purpose?

Concerning the conjugacies of prior distribution and posterior distribution

By using conjugation, I think one can derive an approximate solution of the inverse problem.

Do you know about an example to analytically solve
this based on Bayesian statistics?

Thx in advance,

If by “inverse problem” you mean calculating the Bayesian posterior, then yes, Stan can do that. The manual’s full of examples, but you should start with whatever interface you want to use.

Or you can use L-BFGS (way better than the Newton algorithm in practice) to do optimization in Stan.

Thanks very much, Bob.
I have recently found “particle sworm optimization”.
I will compare this with an approach with Bayesian posterior calculation.

That’s like comparing apples and organges. Bayesian posteriors don’t give you max likelihood esitmates. You can compare it with our optimization. What you’ll find is that particle methods without gradients tend not to scale well with dimensionality, either for optimization or sampling. There is a huge amount of material on this in the optimzation literature, but a bit less so in the MCMC literature, where particle methods have resurged in popularlity despite not scaling well with dimensionality in most circumstances.