Thanks for the tag @adlauretig. BTW the more updated version can be found here A practical introduction to Bayesian estimation of causal effects: Parametric and nonparametric approaches - Oganisian - 2021 - Statistics in Medicine - Wiley Online Library arxiv version has some minor errors and doesn’t incorporate reviewer comments.
Violations of the no unmeasured confounding assumptions can be expressed in terms of non-identifiable parameters. They are non-identifiable in the sense that the likelihood doesn’t inform the parameter at all - we can never rule out or “detect” unmeasured confounding with observed data.
So the general approach is to 1) formalize the structure of your violation in terms of such non-identifiable parameters. 2) place a prior on those parameters that reflect your belief about the direction/magnitude of that violation. 3) find the posterior of the causal estimand. This posterior has uncertainty in the violation baked in via the prior on the non-identifiable parameter.
This approach is mathematically identical to the Bayesian approach for sensitivity analyses in missing data. There, the equivalent assumption is missing-at-random, but otherwise the math is the same. See Daniels & Hogan " Missing data in longitudinal studies: Strategies for Bayesian modeling and sensitivity analysis" I believe chapter 8. and “handbook of missing data methodology” chapter 5.