Recommend reading for multivariate probability/statistics

[monmon_4]

Im working my way through a thesis paper that has a lot to do with this. I have a strong background in linear algebra which I've noticed is important as the determinant of the covariance matrix comes up a lot in density functions.

Other than the linear algebra though, I've only taken an introductory course in statistics. Could anyone recommend some good reading on multivariate statistics/probability that can help me get through this paper? I'd like to understand concepts like the gamma distribution, elliptical distributions, and exponential densities, all in a multivariate setting.

I have a copy of an introduction to multivariate statistical analysis by Anderson but I'm wondering if there exists anything else that might be better for an introduction.

 

[Stephen Tashi]

My guess is that you'll get better advice if you emphasize what statistical tests and methods the paper uses rather that what multivariate distributions it uses.

 

[chiro]

Hey monmon_4 and welcome to the forums.

Following on from what Stephen Tashi said, you should be aware that some distributions are used primarily for statistics and some are used for more-so for probability modelling.

For the probability modelling, the distributions can be put into the context of a real-world process with specific properties. The statistical distributions on the other hand may not necessarily have this kind of interpretation in this way even though they are based on known assumptions and have a very specific use in things like statistical inference.

 

[monmon_4]

I understand my question may be a bit vague to get a proper answer. Perhaps it would be easier if I attach the first chapter of the paper I am reading to give a better idea of what I'm dealing with.

Pages 6-8 are what I'm currently reading. I know what a multivariate probability distribution but I feel I'm lacking some of the intuition to fully appreciate the material.

 

[chiro]

I understand my question may be a bit vague to get a proper answer. Perhaps it would be easier if I attach the first chapter of the paper I am reading to give a better idea of what I'm dealing with.

Pages 6-8 are what I'm currently reading. I know what a multivariate probability distribution but I feel I'm lacking some of the intuition to fully appreciate the material.

If you have no statistics background, you might be in a bit of a pickle.

The big things I would learn if I were you if you have no multi-variate distribution experience is to learn about covariance, joint distributions, conditional expectation and moment calculation for these distributions. When you understand co-variance and the matrix expression, you'll end up understanding a lot of the multi-variate stuff.

For Bayesian, the first thing is to understand Bayes theorem and then extend the idea to likelihood, priors, and subsequently posteriors. Then after this you can look at how to apply it to problems like Binomial with prior and Poisson with prior. From this example you can then move to regression and inference for the Bayesian case which will give you methods similar to least squares for classical regression, but for the more general Bayesian case.

In terms of the multivariable normal and student-t, the multivariable normal should be simple to understand intuitively, and the student-t is very similar once you understand covariance and the other stuff mentioned above.

For the multivariable stuff with continuous distributions, you will need to have the calculus skills to calculate stuff and prove stuff with the multi-variable stuff. These include calculating actual probabilities, moments, and also for calculating parameters and PDF's/CDF's is as well (amongst other things).

 

[monmon_4]

If you have no statistics background, you might be in a bit of a pickle.

The big things I would learn if I were you if you have no multi-variate distribution experience is to learn about covariance, joint distributions, conditional expectation and moment calculation for these distributions. When you understand co-variance and the matrix expression, you'll end up understanding a lot of the multi-variate stuff.

For Bayesian, the first thing is to understand Bayes theorem and then extend the idea to likelihood, priors, and subsequently posteriors. Then after this you can look at how to apply it to problems like Binomial with prior and Poisson with prior. From this example you can then move to regression and inference for the Bayesian case which will give you methods similar to least squares for classical regression, but for the more general Bayesian case.

In terms of the multivariable normal and student-t, the multivariable normal should be simple to understand intuitively, and the student-t is very similar once you understand covariance and the other stuff mentioned above.

For the multivariable stuff with continuous distributions, you will need to have the calculus skills to calculate stuff and prove stuff with the multi-variable stuff. These include calculating actual probabilities, moments, and also for calculating parameters and PDF's/CDF's is as well (amongst other things).

Thanks for taking the time to help me out. I feel comfortable with the material and notation, I would just like a better intuition for a few of the concepts like the ones you've just mentioned.

Can you recommend a good introductory book for me to look at?

 

[chiro]

Thanks for taking the time to help me out. I feel comfortable with the material and notation, I would just like a better intuition for a few of the concepts like the ones you've just mentioned.

Can you recommend a good introductory book for me to look at?

Any comprehensive probability/statistics book should cover these. The one I used was Mathematical Statistics with Applications by Wackerly, Scheaffer and the other guy although you could pretty much use a tonne of books that are all equally good. Some actuarial societies recommend the book by Hogg so you could check that out as well.

As for the Bayesian stuff, I didn't learn this necessarily through a textbook so I can't help you there. Anything that covers Bayesian inference in a detailed way should suffice.

Also I'd recommend Introduction to Probability Models by Sheldon Ross if you want to get into some of the more complex but still highly used and necessary probability models that cover things including the markovian models amongst others.