This new arXival by Lorenzo Tomaselli, Valérie Ventura, and Larry Wasserman (from CMU) considers simulation-based inference under model misspecification (as we did for ABC in our 2020 Series B paper). Which is almost always the case. In the paper, SBI is defined as producing N parameters and N samples from the prior and the corresponding sampling distribution, respectively, and then doubling the resulting samples by permuting at random the parameters θ. This means that the second half is distributed from the product of the prior and of the marginal, hence that the classification odds ratio is equal to the likelihood, hence providing an estimation method (andlikelihood trick) à la Geyer. From this estimate, an ABC p-value can be derived, but it is incorrect as such when the model is misspecified. Hence the use of the Hellinger discrepancy, the power divergence and the kernel distance (or MMD) as alternatives to the misspecified MLE.
The paper then expands on approximating density ratios by virtue of a reproducing kernel Hilbert space, using a Gaussian kernel. (With a nice remark on requiring only one single ratio estimator for all values of θ, albeit in the joint space.) And focus on a studentized MMD estimator (à la e-value) to build a confidence set that remains valid under model misspecification. And without regularity assumptions.
Another approach is further explored, based on exponential tilting—of which I am not a great fan, from being highly dependent on the choice of the pseudo-sufficient statistic to require an intractable normalising constant, to requiring an extra optimization, even though I appreciate the mathematical appeal of the construct. Which seems to require a sample simulation for each value of θ at the learning stage, albeit relying on the same likelihood trick. The appropriateness of the tilting can be tested by a goodness of fit test tailored for the SBI structure, which sounds rather greedy in the required simulations.
Besides the g-and-k distribution example (which, as pointed out several times on the ‘Og, is not intractable, strictly speaking!), the paper studies a mixture example, despite Larry dubbing them as evil as tequila a long while ago! (The paper also offers a section called accoutrements, which is my first encounter with this use of the term, usually found in medieval contexts!)
Note that Larry will present the paper at the OWABI webinar next 25 March!
Botond Szabó gave a 
A paper about Bayesian inference on mixtures was posted on arXiv last week, as of 13 Jan 2025. 
In 
