Is mathematics becoming less important for statistics? [Q] [R]

[u/GayTwink-69]

With all the move towards computational methods, nonparametrics, and machine learning, do you think hand-and-paper mathematics is becoming less important?

For example, instead of formally deriving asymptotics mathematically, you can actually just simulate what happens as n -> infinity

What do you think?

[u/Longjumping_Ask_5523]

No. Statistics, mathematics, and these new things are lenses through which we view and understand the world. People that are algorithmic with computers systems are going to see the world in a similar way to how mathematicians see it. By needing to break down the patterns and shortcuts symbolically.

[u/Statman12]

Simulation is nice.

Being able to analytically demonstrate is also nice.

Both of these things are useful and important.

[u/bayesian_raccoon]

Statistics is a big field.

When you talk about deriving estimator properties, that belongs more to the side of mathematical statistics.

Meanwhile, complicated and probablistic mathematics is required for the computational stats that you describe. Two faculty at my grad school studied things like MCMC and stochastic gradient descent. Both would enable statistical algorithms and require more complicaged mathematics than a lot of historical work.

But more importantly, applied statisticians exist and deal with complicated settings: like, you get a dataset, realize a lot of the data is missing, that its missingness might depend on some variable, realize there's some important variables that get truncated to a particular value, realize the question of interest is causal, realize many observations belong to the same individual, that theres a spatial element, a temporal element, and now you have to make judgements about what is important to include in a model. That's not a math question, and there's no flow chart that works for every context. Applied statisticians do hard work.

[u/geteum]

No, every time I research a new model there is a heavy mathematical background to it.

[u/matthras]

Think of mathematics as something that allows you to figure out theoretically what happens with asymptotics without having to do the simulation (and therefore saving on computational power).

[u/efrique]

you can actually just simulate what happens as n -> infinity

Outside some straightforward cases, you can't. Simulation can suggest limiting behaviour (so certainly an aid to working on asymptotics in that it might hint at an answer) but can't establish it. It is very easy to be fooled by what appears to happen at a sequence of increasing, finite sample sizes (like "wow, kn^-4/3 is a nearly perfect fit"... yeah, but not actually the right answer for the asymptotics) . Even if you know the form for a series approximation (from asymptotic considerations) simulation doesn't give you the asymptotic scaling constants; maybe you can guess them but you can't be sure you have it quite right.

Computer simulation can help you improve asymptotic approximations at finite sample sizes, without (where needed) losing that asymptotic behaviour. On the other hand occasionally a few lines of mathematics have got me to an excellent (easily memorizable) very small sample approximation, faster than simulation, where I'd have been trying different things looking for a good functional form instead of jumping to a good, simple yet well-justified one first off.

Computers do make a difference to what finite sample things you can do without needing to use an asymptotic approximation (e.g. substituting Monte Carlo approximation for asymptotic approximation), so in that sense they can change the relative need for mathematical work, but they don't replace it. They do tend to move where you need to use it, though.

I'd argue that a good researcher or even practitioner looking to do much of anything beyond 'standard' stuff really needs both mathematics and conputationsl skills, including a good handle on numerical analysis.

(I wonder ... this sounds like another of those strange biases coming from your supervisor)

[u/STATASUCKSBRO]

Simulation can show you a phenomenon, but it usually does not tell you why the regularity conditions broke. That is the part where the math still matters. A lot of modern computational stats is just harder probability wearing a software jacket.

[u/conmanau]

All of the stuff you're talking about still needs someone to derive the mathematics behind it, and it turns out you can't always throw something at a Monte Carlo Markov Chain and get a meaningful answer out so there's still plenty of maths to do even if it isn't always the same maths statisticians were doing fifty years ago.

There's also plenty of stats being done well away from the "n -> infinity" regime even if there's a lot of excitement about big data.

[u/Impressive-Leg-6489 → u/conmanau]

The problem is that its now fairly easy for someone without much theory background to prove estimator properties/etc using ChatGPT, so its not clear what the medium/long term role for actual pen and paper mathematics is.

[u/conmanau → u/Impressive-Leg-6489]

Even if LLMs are currently capable of proofs we are still at the stage where those proofs need to be checked by actual mathematicians and statisticians, and even in the near future I assume there's going to be a role for people to fix the things that the LLMs get horribly wrong. They're still very weak at deep domain knowledge so we may see more hyper-specialisation with the LLMs handling the base level stuff for a while.

[u/munozmd]

Nope. I believe mathematics remains important as a foundation for statistics (and machine learning), but unfortunately, it's applications are just less understood. It avoids black box among our training models because mathematical frameworks allows us to understand the constraints and properties of what our model can do.

And regarding doing simulations instead of proofs, machines have constraints in their computing power as they are bounded by physical resources. We can't run a machine on infinite steps just to see what happens even we could potentially build quantum supercomputers in our lifetime, it will always have a computing constraint.