# 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,

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).

• Is it really that Bayesian estimation doesn't care very much about bias, or that it just doesn't care exclusively about bias? If a Bayesian approach focuses more on MSE than on bias, then, ceteris paribus, higher bias is undesirable. – Dr. Beeblebrox Aug 6 '13 at 13:32
• I could be mistaking, but my understanding is that the goal is to never break down the analysis along dimensions like MSE or other specified error terms. What you care about are the ultimate inferences that come from the model and how they inform the adoption of policies (where policies is broadly interpreted to be whatever beliefs or actions one should adopt as inferred from the data). This is a major distinction between frequentist education in statistics, which tends to focus on a quantity that can be optimized to identify a model, vs. Bayesian which wants a nuanced distribution over models – ely Aug 6 '13 at 13:51
• That is to say, if you're performing some inference for a goal or to answer a question, you (as the statistician) should be aware of the outcomes that matter and how they differ from one another. I suppose that could be a function of a simple error term, and in those applied inference problems you would care about getting an unbiased answer. In another problem where distinguishing between group-level effects was extremely important, and knowing the average effect within a group was a lot less important, you might choose models for completely different properties. – ely Aug 6 '13 at 13:53
• I think there is one case where Bayesian may care about unbiasedness estimation: when estimating causal effects. See many papers by Rubbin. – Manoel Galdino Aug 12 '13 at 13:59
• I agree. I'm just saying that Bayesian inference is not a priori concerned with unbiasedness properties of estimators. If such properties are useful for the substantive inference demanded by the applied problem, then of course it would make sense to try to get guarantees about unbiasedness. But if an inference procedure performs well without any particular good or bad properties related to its bias, a Bayesian will still use it while a frequentist may disregard it on a priori grounds that it doesn't provide a particular guarantee about unbiasedness. – ely Aug 12 '13 at 14:48

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.