Bayes' Theorem

The simple form of Bayes' theorem represented in blue neon signage at the offices of HP Autonomy in Cambridge, England. (Photo courtesy of Matt Buck)

Tobias is interested in identification and causality, and using Bayesian methods to obtain reliable estimates of the effect of a variable “x” on an outcome “y” when faced with observational (non-experimental) data.

He has applied such methods to estimate, for example, the effect of education on earnings, obesity on labor market outcomes, payday loan regulations on borrower behavior, and dropping out of high school on cognitive aptitude. He has also co-authored a popular textbook on Bayesian methods, which offers recipes to practitioners for estimating these types of models.

Tobias explains to his students that these types of questions are not uniquely Bayesian.

"There is a rich, important and fascinating classical literature devoted to these issues, but empirical applications inevitably involve assumptions,” Tobias says. “If these assumptions are correct, one can obtain a reliable estimate of the desired causal impact.

"The assumptions made in practice, however, in order to make the problem tractable and clean, are often controversial and are not directly testable. When these assumptions are relaxed, the objects you are trying to estimate are no longer well-identified by the data. Entertaining weaker types of assumptions makes the problem significantly more complicated — and more interesting.”

Tobias sees Bayesian methods as offering a very natural and useful way for incorporating different types of assumptions in settings like this, since these assumptions simply represent different prior beliefs on the part of the researcher.

“I think many non-Bayesians — certainly the majority of empirical researchers — have convinced themselves that their results are completely objective and solely data-driven,” Tobias says.

“What we see in a published paper, however, is simply the very end result of a long process and a variety of choices that the investigator has already made yet not fully documented. The subsequent results conditioned on those decisions are packaged as objective. Empirical work is personal. Bayesian methods provide a formal role for incorporating those views.”

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