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I have heard from my linear algebra professor in undergraduate studies that probability theory can be examined using linear algebra.

As a math student who enjoys linear algebra, does anyone have a good textbook that uses linear algebra to examine probability theory?

Ben Grossmann
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Griffin McShane
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    What? I've never heard of such an idea that basic concepts in probability can be viewed through a linear algebra lens. Measure theory, sure... absolutely, that's how its formalized, but linear algebra? There will certainly be some problem types who can borrow from linear algebra techniques (E.g. markov chains) but the vast majority of elementary exercises are totally unrelated to the ideas specific to linear algebra. – JMoravitz Mar 02 '21 at 15:59
  • @BenGrossmann "the vector space of probability distributions" Care to elaborate on that? How do probability distributions form a vector space? With respect to what operation? With respect to what field? I fail to see how such a set is even closed under scalar multiplication. – JMoravitz Mar 02 '21 at 16:03
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    @JMoravitz the space of random variables with finite variance form a vector space. I agree though that measure theory, in particular $\sigma$-algebras feel the most natural to learn probability theory properly, though maybe unnecessary as a first pass. – Gregory Mar 02 '21 at 16:05
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    related: https://math.stackexchange.com/questions/1721932/learning-probability-via-linear-algebra – symplectomorphic Mar 02 '21 at 16:07
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    I have a feeling you are thinking of linear models, there are very cool books on this topic, you should check our Visualizing linear models by W.D Brinda – Samael Manasseh Mar 02 '21 at 18:08
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    Fristedt and Gray has a chapter on spaces of random variable. In particular, it discusses how random variables in a probability space form a Hilbert space. (For a historical instance of this, I quite like this 1937 article by de Finetti: http://www.brunodefinetti.it/Opere/AboutCorrelations.pdf.) – Semiclassical Mar 03 '21 at 02:51
  • Relevant: Whitney, Stephen. “Cartesian Statistics: Data Analysis by Linear Algebra and Analytic Geometry.” CIENCIA ergo-sum [En línea], 4.1 (1997): 86-93. Web. 18 ene. 2024 https://dialnet.unirioja.es/descarga/articulo/5128787.pdf – beroal Jan 18 '24 at 23:45

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One book that explores probability from a linear algebraic point of view is Hilbert Space Methods in Probability and Statistical Inference by Christopher G. Small and Don L. McLeish. From the introduction:

In the traditional approach to probability and statistics, the starting point is the concept of the sample space and a class of subsets called events. The Hilbert space approach shifts the starting point away from the sample space and replaces it with a Hilbert space whose elements can be variously interpreted as random variables, estimating functions, or other quantities depending on the data and the parameter.

But be aware that this book is almost 30 years old (published 1994), and I don't know to what extent practicing probabilists or mathematical statisticians work within this framework today. (Also note that basic linear and matrix algebra is the natural language for multivariate probability and statistics. The Hilbert space view here is more general.)

symplectomorphic
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