Introduction:

Michael Betancourt is the chief research scientist at Symplectomorphic, LLC. He received his B.S. from the California Institute of Technology and his Ph. D. from the Massachusetts Institute of Technology; both were in physics. Betancourt is an Applied Statistician whose work focuses on bridging the gap between statistical theory and practice. For example, he has collaborated with epidemiologists and pharmacologists to help maximize the utility of their data. He primarily researches the intersection between differential geometry and probability theory. Additionally, he is a core developer of Stan, where he tests tools that he has made to support practical Bayesian inference.

*1** **Please describe your specific field of work and what it is that sparked your interest. Are there any other careers you have considered and what events have led to where you are today? *

At the moment I consider myself an applied statistician and work full time as a statistical consultant. In my consulting work I advise clients how to build and apply sophisticated statistical models that extract insights from messy data and principled domain expertise. At the same time my research develops the methodologies and algorithms that enable those analyses. Much of this work is implemented in Stan (https://mc-stan.org), an open-source platform for Bayesian inference for which I am a developer.

The path that took me to where I am today is hardly a linear one. My initial aspiration was experimental physics, but during my PhD I became more and more interested in statistical inference that exploited domain expertise and the algorithms that support those inferences. In particular I became interested in the theoretical foundations of an algorithm called Hamiltonian Monte Carlo which required teaching myself differential geometry. I was fortunate enough to continue this interest into a postdoctoral research position where, with the help of many patient colleagues, I learned enough geometry and probability theory to start making useful contributions.

*2** **How would you describe your interaction with math during high school?*

I loved math in high school, but in hindsight I think that my relationship with it was largely superficial. Math was just calculation to solve a given problem, and the act of doing math was playing around to find the tricks that enabled the evaluation of those calculations. This interaction persisted through my undergraduate and graduate education in experimental physics, where the calculations were limited to repeating textbook exercises in various forms.

While there was plenty to enjoy about this relationship, it was ultimately limiting. In particular it didn’t prepare me for circumstances where the problem itself had yet to be formalized, where the challenge is not to figure out how to calculate something but rather what to calculate in the first place.

*3** **Would you consider yourself a minority in you field in any way? If so, how did you overcome any hardships?*

I am Mexican American but I pass as white and so I have been privileged enough to avoid most of the difficulties faced by communities underrepresented in mathematics and science. These days I do my best to help underrepresented communities in applied statistics overcome those hardships, and try to establish better systems that minimize those hardships in the first place.

*4** **Where do you see yourself 10 years from today? *

My current situation wouldn’t even have been a possibility in any prediction I would have made 10 years ago, and a career in statistics consulting definitely reinforces just how large forecasting uncertainties are! Consequently I try to avoid planning too far ahead into the future; I just hope that wherever this current path takes me I’m able to continue contributing to the tools that facilitate better science.

*5** **What advice would you offer to youth today looking into pursuing mathematics? As one myself, I am grappling with how to cause tangible impacts in other people’s lives. From my limited background, it seems as if impacts associated with many math-related careers seem rather abstract than tangible. I wish to hear your thoughts as an applied statistician / consultant.*

I believe that in order to have potent impacts one has to identify the needs of applied work first, and then identify the mathematical tools that are well-suited for those needs. In my opinion most academic work focuses on developing a mathematical tool first and only then looking for an application. The end result is tools applied to the wrong problems and frustrated practitioners driven away from collaboration.

Consequently my first piece of advice is to prioritize the problem instead of yourself. Don’t compromise an application by forcing a tool that is convenient for you but ill-suited for the problem at hand. This might mean limiting the scope of the application until the problems reduce to something your tools can handle, and it might mean stepping back to learn or develop new tools to accommodate the specific structure of that application. It might mean finding the right collaborator to complement your existing tools, and it might even mean acknowledging that you are unequipped to tackle that problem at the moment.

In order to make a proper match between formal tools and applications, one needs a deep understanding of not only the application but also the mathematical concepts from which tools can be constructed. In order to generalize methods from one application to another one needs to be able to do more than just reproduce common calculations and derivations. One needs to appreciate the structure of mathematical objects, how they transform, and the structure of those transformations! One needs to understand how mathematical structures relate to real world phenomena. Unfortunately this conceptual understanding falls right in the middle of most introductory courses, which focus on superficial analogies, and most advanced courses, which focus on technical rigor and proofs at the expense of motivation and intuition.

My second piece of advice is then to complement your education with strong conceptual foundations in mathematics and at least one applied field.

If you are pursuing a more applied education then supplement your classes by reviewing more theoretical references. Ignore the technical details and focus instead on what objects are being introduced and why. Familiarize yourself with mathematical themes like invariances, classification, and representations and connect them to the calculations you learn in applied classes. When learning linear algebra, for example, work towards moving beyond vectors and matrices as arrays of real numbers with some mechanical operations and into vectors as geometric objects with matrices as representations of linear transformations. Learn enough that you can formalize your models and questions to the point where you can start to communicate with mathematicians who might be able to direct you to the right answers.

On the other hand if you are pursuing a more theoretical education, but are interested in applied that education in practice, then expose yourself to applications as much as possible. Take mathematical modeling classes offered in specific fields to learn how mathematical methods are commonly used. Learn what mathematical properties are productive in specific applications and why. Work through applied computational references and familiarize yourself with the basic techniques for implementing mathematical calculations on computers, and how those implementations fail. Deconstruct algorithms common to a field and connect them to the more formal calculations they are trying to implement. Learn enough that you can talk with a domain expert about their models, identifying what mathematics might be appropriate and the general strategies for implementing those mathematics in practice.

Above all, talk to as many people across the applied-theory spectrum as possible. Successful collaborations are puzzles that can be solved only when the right complementary pieces are brought together.