Embedded bias initial implementation

[danielandresarcones]

This is the initial implementation of embedded bias approaches to be integrated in probeye. It provides a working implementation for the computation of inverse problems with embedded bias.

• Currently the bias embedding must be implemented manually in the forward model. Ideally, this should not be the case and the bias should be specified in the parameters and variables description.
• The additions take care of the MCMC solver with stochastic forward model response and introduce four embedded likelihood formulations: ABC Moment matching, Global Moment Matching, Relative Global Moment Matching and Independent Normal. The details on the implementations will eventually be published.
• Kennedy and O'Hagan-type formulations are not implemented in this PR and have been eliminated from the branched code.
• Tests and one simple example for the documentation have been added.
• chaospy has been added as a dependency. Probably should be an optional one, but kept as compulsory for now.
• mypy was updated to the latest version

[danielandresarcones]

This update also implements the proposed solution of Issue #90 .

[danielandresarcones]

Tests cover is not 100% on the new implementations as there probably will be changes in the definition of the bias in the near future. The lack of cover is precisely on that front, the checks that some parameter definitions are correctly imported to the models. Nevertheless, they can be tested if considered necessary.

[aradermacher]

It should be b_dist instead of bias_dist

[danielandresarcones]

I missed that totally, thanks!

[aradermacher]

Thanks Daniel,
I had problems running the example file, because of a typo (see comment)
Furthermore, during creating a new env I run into an error with python 3.13. Changing to 3.10 solved it.

[danielandresarcones]

Do you remember what whas the error with python 3.13? I think probeye had some problems with it at some point, not necessarily the embedded bias implementation only.

[aradermacher]

Do you remember what whas the error with python 3.13? I think probeye had some problems with it at some point, not necessarily the embedded bias implementation only.

Yes it was something with the required packages (but I didn't saved the message sorry). So I think it's a general problem of probeye and not with your implementation.