Open ardollam opened 2 years ago
The product of two Gaussian random variables is not a Gaussian random variable. This product is a new random variable, which is not Gaussian distributed. However, the product of two Gaussian densities is a Gaussian density. The Gaussian density is a function, and this is product of two functions that results in a third function, which just so happens you can identify the third function as a Gaussian density.
Dear Professor Ong:
Thanks for your reply. Can you point me to sources that clarify the distinction between Gaussian densities and Gaussian random variables?
Sincerely,
On Mon, Sep 20, 2021 at 9:44 AM Cheng Soon Ong @.***> wrote:
The product of two Gaussian random variables is not a Gaussian random variable. This product is a new random variable, which is not Gaussian distributed. However, the product of two Gaussian densities is a Gaussian density. The Gaussian density is a function, and this is product of two functions that results in a third function, which just so happens you can identify the third function as a Gaussian density.
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I feel that the confusion here is between two distinct concepts of random variables and densities. At the beginning of Chapter 6, we give a brief discussion. We also point to the excellent more detailed introduction by Michael Betancourt: https://betanalpha.github.io/assets/case_studies/probability_theory.html
The Gaussian distribution discussion follows from the distinction between random variables and densities.
Dear Professor Ong:
Thanks for pointing me to these sources. I look forward to reading them.
Sincerely,
On Thu, Sep 23, 2021 at 1:01 AM Cheng Soon Ong @.***> wrote:
I feel that the confusion here is between two distinct concepts of Random variables and densities. At the beginning of Chapter 6, we give a brief discussion. We also point to the excellent more detailed introduction by Michael Betancourt: https://betanalpha.github.io/assets/case_studies/probability_theory.html
The Gaussian distribution discussion follows from the distinction between random variables and densities.
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The product of two Gaussian random variables is not a Gaussian random variable. This product is a new random variable, which is not Gaussian distributed. However, the product of two Gaussian densities is a Gaussian density. The Gaussian density is a function, and this is product of two functions that results in a third function, which just so happens you can identify the third function as a Gaussian density.
p. 201 is ambiguous, but p. 312 unambiguously states that "the product of two Gaussian random variables is an (unnormalized) Gaussian distribution", see image below
2
Hi:
I have been enjoying reading your book and watching the Youtube videos (especially the one on probability spaces and random variables).
I believe that there is a problem, though, with the section on products of gaussian densities (see attached screen capture). I do not think that (a) the product of two gaussians is a gaussian, and also, (b) I am not sure that the product of a prior and a likelihood is necessarily the product of two gaussians. For (a), take a look at this posting on stack exchange: https://math.stackexchange.com/questions/101062/is-the-product-of-two-gaussian-random-variables-also-a-gaussian
Good luck with the book and your course.
Sincerely,
Niall Bolger