The difference between mathematical statistics, computational statistics, and applied statistics
If you are someone who recently visited webpages of the statistics department of a top north American universities, you might have come across these three terms,
- Mathematical Statistics
- Computational Statistics
- Applied Statistics.
This is especially relevant if you are looking into the information on the graduate programs. If you have a background in Statistics, you are probably very much familiar with these terms. However, I have come across many people coming into the masters programs in Statistics from different backgrounds (mostly with STEM background). For these folks the differences between mathematical statistics, computational statistics, and applied statistics might not be so obvious. In this post I will try to give a very high level idea about the differences between these three areas.
Mathematical Statistics
As the name suggests, mathematical statistics is concerned about the application of mathematics to help us understand the properties of statistical techniques. Understanding these properties allow us to identify strengths and weaknesses of different techniques, which can be very important when we want to consider a technique for a given practical problem. Not just that. mathematical statistics provides a strong foundation to develop new techniques.
What you are expected to do?
In mathematical statistics, you will be dealing with a lot of mathematical problems. Generally, you are expected to be strong with probability theory, measure theory, asymptotic analysis, real analysis, linear algebra, and calculus. You will frequently encounter terms like probability distributions, law of large numbers, convergence, sigma algebra (if pursuing higher level courses), etc.
Computational Statistics
Computational Statistics deals with problems that Mathematical Statistics is not able to handle or hits a brick wall. There are many problems for which we can not derive any closed form elegant solution. These kinds of problem might require computational techniques. Think about Linear Regression analysis for a moment. There is an elegant mathematical solution to the problem of estimating the parameters of the regression model. I am talking about this equation.
But unfortunately, we do not have such elegant solutions for many other types of problems. Expectation Maximization is one of them. Estimating the weights of the deep learning model is another. In these cases, the problem is so complicated that we can not proceed after a certain point with only mathematical approach. We need iterative algorithms that can help us estimating the parameter values. In some sense, Computational Statistics has some similarities with Computer Science.
What you are expected to do?
Common topics like Newton Raphson, Gradient Descent, Expectation-Maximization, Machine Learning, Monte Caroo simulations, Markov Chain Monte Carlo (MCMC) all fall under the umbrella of computational statistics. So you will be required to have strong mathematical skills (probability distributions, linear algebra, calculus) as well as coding skills. You have to remember that in these kinds of problems, the mathematical skills can take you up to a certain point in solving the problem, but in order to reach a point that is useful, you would need to apply computational techniques. Generally speaking, most people who are working in Computational Statistics are good in either R or Python (or both). In some cases, they use other lower level languages to speed things up.
Applied Statistics
Applied Statistics is concerned about applying statistical tools to solve real life problems. So it is about setting up an unstructured problem so that it is easier to apply existing techniques (in some cases new techniques might be developed). There are many disciplines (e.g., psychology, medicine) that require statistical expertise. Even a lot of the tech companies use statistical methods to help them make decisions (online AB testing comes to mind).
What you are expected to do?
Since we are talking about real life problems, it can be very unstructured. Topics like experimental design, causal inference, biostatistics are some of the areas that you need to master. The practitioners are required to be good with understanding problems, asking questions, analyzing data, and solid command of programming language to conduct their statistical analysis. So you probably will not be developing a new machine learning algorithm, but you will be using machine learning models to solve a problem. As an applied statistician you will have to know which model to use for a given problem. You might have to design experiments to validate hypothesis.
Conclusion
In todays world, it is very difficult to draw boundaries between different disciplines. It is not to say that a mathematical statistician will not be good in applied statistics, or vice-versa. But it is certain that mathematical statistician will be more focused on the mathematical side of the things compared to someone who deals with computation or applies statistics. But it is certain that all these three sub-disciplines would require unique sets of skills.
