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Is there a calculus based approach to statistics? Like understanding linear regression simply as a minimization problem, but ALL THE WAY ie extending to other s

Is there a calculus based approach to statistics? Like understanding linear regression simply as a minimization problem, but ALL THE WAY ie extending to other statistical techniques?

fny 9y ago
It's called probability theory, but that's not going to resolve these issues. If you're interested, there's a great 100-level course online from Joe Blitzstein at Harvard: https://projects.iq.harvard.edu/stat110/home https://projects.iq.harvard.edu/stat110/home
- catnaroek 9y ago
  It's called “probability theory”, but is it actually probability theory? How can you have a course in “probability theory” that doesn't mention the Lebesgue integral even once?
  - rocqua 9y ago
    I took a course in "probability theory" that never mentioned Lebesgue integrals. Those came in measure and integration later. The course was about distributions, how product distributions interact, the central limit theorem, etc. That is, it was about turning stochastic models into predictions. Later we had a course "Statistics" that is about matching seen results to stochastic models.
    - catnaroek 9y ago
      > The course was about distributions, How can you study these without knowing the Lebesgue integral? > how product distributions interact How can you do this without the Fubini-Tonelli theorem? etc. etc. etc.
      - rocqua 9y ago
        By talking about the operations without giving the rigorous backgrounds. Heck, you can do quite a bit with rieman integrals, and for e.g. binomial distributions you just need sums.
        catnaroek 9y ago
        This is exactly what leaves many people feeling like mathematics is a bunch of unmotivated tricks and rabbits pulled out of a magician's hat.
        rocqua 9y ago
        This was during a Bsc in mathematics. Like I said we later got the theoretical grounding. Starting with measure theory would be what leaves people feeling that mathematics is useless formal bickering.
lliiffee 9y ago
To build on fny's answer, there is one school of statistics (Bayesian statistics) where there basically is a "right" way to do analysis for any problem, provided you make the necessary assumptions (likelihood and prior) correctly. However, the most common statistical concepts (e.g. p-values or confidence intervals) are not in the Bayesian school
- mattkrause 9y ago
  Credible interval or highest posterior density intervals are arguably much closer to what most people think a confidence interval are. I’m not sure there is a single right way to solve any given problem isn a Bayesian way, but it does force you to think more about the problem at hand and make your assumptions explicit.
ronald_raygun 9y ago
Statistics/Probability theory is really deeply tied to calculus/real analysis . Basically any quantity you can think about in statistics or probability is really an integral. The trick is that they aren't Riemann integrals, but they are a more advanced type called Lebesgue integrals, which has to do with measure theory. The main difference between the two is that a Riemann integral count every area of space the same ([0,1] counts just as much towards your integral as [1,2]), but Lebesgue integrals use what is called a measurable function, which maps f(set) -> positive number. You can use this function to weight different parts of your integral differently. Now what you can do is make measurable functions that map f(set) -> how likely that set happens (which is exactly the probability)
analog31 9y ago
The stats course that I took in college was certainly calc based. There were two stats options: "Stats for scientists" was a one semester course, mostly involving plugging numbers into formulas. There was lots of hypothesis testing. "Math stats" was for math majors. It was two semesters, and focused mainly on proofs. I took math stats. I also ran a tutoring session for the scientists. A problem is that the basic stats course is taken by a lot of students who wouldn't have gotten through calculus. So, building stats on top of calculus would have created two forbidding layers of abstraction instead of one. On the other hand, I graduated from college in 1986, and we did all of our calculus by hand. I wonder if a potential compromise today would be to teach stats by exploration using random numbers. You're still doing integration, albeit numerically, but maybe it wouldn't seem so forbidding. And by playing with random numbers, you can learn the hard way what erroneous conclusions you can draw from them.

Is there a calculus based approach to statistics? Like understanding linear regression simply as a minimization problem, but ALL THE WAY ie extending to other statistical techniques?

fny 9y ago

It's called probability theory, but that's not going to resolve these issues. If you're interested, there's a great 100-level course online from Joe Blitzstein at Harvard: https://projects.iq.harvard.edu/stat110/home https://projects.iq.harvard.edu/stat110/home

catnaroek 9y ago
It's called “probability theory”, but is it actually probability theory? How can you have a course in “probability theory” that doesn't mention the Lebesgue integral even once?
- rocqua 9y ago
  I took a course in "probability theory" that never mentioned Lebesgue integrals. Those came in measure and integration later. The course was about distributions, how product distributions interact, the central limit theorem, etc. That is, it was about turning stochastic models into predictions. Later we had a course "Statistics" that is about matching seen results to stochastic models.
  - catnaroek 9y ago
    > The course was about distributions, How can you study these without knowing the Lebesgue integral? > how product distributions interact How can you do this without the Fubini-Tonelli theorem? etc. etc. etc.
    - rocqua 9y ago
      By talking about the operations without giving the rigorous backgrounds. Heck, you can do quite a bit with rieman integrals, and for e.g. binomial distributions you just need sums.
      - catnaroek 9y ago
        This is exactly what leaves many people feeling like mathematics is a bunch of unmotivated tricks and rabbits pulled out of a magician's hat.
        rocqua 9y ago
        This was during a Bsc in mathematics. Like I said we later got the theoretical grounding. Starting with measure theory would be what leaves people feeling that mathematics is useless formal bickering.

lliiffee 9y ago

To build on fny's answer, there is one school of statistics (Bayesian statistics) where there basically is a "right" way to do analysis for any problem, provided you make the necessary assumptions (likelihood and prior) correctly. However, the most common statistical concepts (e.g. p-values or confidence intervals) are not in the Bayesian school

mattkrause 9y ago
Credible interval or highest posterior density intervals are arguably much closer to what most people think a confidence interval are. I’m not sure there is a single right way to solve any given problem isn a Bayesian way, but it does force you to think more about the problem at hand and make your assumptions explicit.

ronald_raygun 9y ago

Statistics/Probability theory is really deeply tied to calculus/real analysis . Basically any quantity you can think about in statistics or probability is really an integral. The trick is that they aren't Riemann integrals, but they are a more advanced type called Lebesgue integrals, which has to do with measure theory. The main difference between the two is that a Riemann integral count every area of space the same ([0,1] counts just as much towards your integral as [1,2]), but Lebesgue integrals use what is called a measurable function, which maps f(set) -> positive number. You can use this function to weight different parts of your integral differently. Now what you can do is make measurable functions that map f(set) -> how likely that set happens (which is exactly the probability)

analog31 9y ago

The stats course that I took in college was certainly calc based. There were two stats options: "Stats for scientists" was a one semester course, mostly involving plugging numbers into formulas. There was lots of hypothesis testing. "Math stats" was for math majors. It was two semesters, and focused mainly on proofs. I took math stats. I also ran a tutoring session for the scientists. A problem is that the basic stats course is taken by a lot of students who wouldn't have gotten through calculus. So, building stats on top of calculus would have created two forbidding layers of abstraction instead of one. On the other hand, I graduated from college in 1986, and we did all of our calculus by hand. I wonder if a potential compromise today would be to teach stats by exploration using random numbers. You're still doing integration, albeit numerically, but maybe it wouldn't seem so forbidding. And by playing with random numbers, you can learn the hard way what erroneous conclusions you can draw from them.