A Bayesian perspective on omitted-variable bias (and other covariate-selection bias problems) As I know OVB, from a frequentist education, when you leave a variable $(z)$ out of your control set $(X)$ that is correlated with both your independent variable of interest (treatment $T$) and your dependent variable of interest ($Y$), your coefficient estimate will be biased because the explanatory power of the missing variable $z$ is distributed to the coefficients of included variables $(\hat\beta_i X_i$).
How does a Bayesian perspective view OVB?    For instance, if we use a data-based, rather than theory-based, variable-selection algorithm -- LR, AIC, BIC -- is it  hard to conceive of OVB?   Furthermore, how would one formally integrate the awareness of $z$ into our conditional probability statement? I mean, in Bayesian inference we want to estimate $P( \text{model} \mid \text{data}) = P(\theta \mid X)$. If we acknowledge some important but unobserved $z$, would we write $P(\theta \mid X, z)$?
Furthermore, how would a Bayesian perspective interpret other classes of covariate-selection bias problems? I thinking about covariate-selection issues as elaborated by Pearl and others including,


*

*Pearl: Controling for an instrument can augment bias in linear models, and introduce and augment bias in non-linear models: http://arxiv.org/pdf/1203.3503.pdf

*Greenland: Conditioning on pre-treatment variables may induce bias (M-bias, aka Collider-Stratification bias): http://journals.lww.com/epidem/Abstract/2003/05000/Quantifying_Biases_in_Causal_Models__Classical.9.aspx (Sorry, paywalled)
 A: In general, Bayesian estimation is not very concerned with unbiasedness of estimators since the model is always misspecified. There definitely exist proofs about conditions for unbiased estimation in Bayesian frameworks. I just don't think practitioners care very much about that and try to avoid using fitting procedures that would be susceptible to this kind of thing at all. 
And sometimes doing tricky things just to get an "unbiased" estimator can come at the expense of other exploitable problem structure (e.g. when pooling is used to get an unbiased estimator, you are trading usable category-level variance in exchange for guarantees of unbiasedness under implausible assumptions. Whether that is a useful trade-off or not should be considered at the level of specific applied inference problems, rather than as a generic thing to do with any model. Here is a post by Andrew Gelman about that.)
For the problem at hand, I believe Bayesian practitioners look more generally at model fit assessment and model misspecification. It's more about whether you are missing an appreciable or significant effect size for the omitted variable, and less about whether the omission has sprayed effect size onto other variables. 
One way to address this is to perform posterior predictive checks on your model. If you do this with a procedure like continuous model expansion (section 5.2 of this paper), then the posterior predictive checks should give you evidence about the best model specification (or better yet, the best distribution over some set of model specifications), rather than forcing you to make an unnatural choice like "The model with Variable Z is 'better' than the model without Variable Z" (which are almost always misunderstood or misinterpreted later by readers).
A: It's not properly true that Bayesian models are always misspecified!
Try by yourself... You will realize that even with a wrong prior you can find conditions for having an unbiased posterior estimator.
