Question 8 - exponential families

Hello,

I don't understand why the uniform interval doesn't belong to the exponential family.

If we take \(h(y) = p_u(y|a,b)\), \(\phi(y)\)=0 and \(A(\nu)=0\)
then, \(\int h(y) \exp(0+0) dy =1\)
So it belongs to the exponential family, no?

Because of the indicator function in it

You can't just throw \(a\) and \(b\) like that in \(h\), they are parameters of the model and it effectively makes \(h\) a function of \(y\), \(a\), and \(b\).

You can make an argument that \(p_\mathcal{U}(y|a,b)\) can be written as \(e^{-\ln (b - a)}\) if \(a \le y \le b\) and \(1e^0\) otherwise. You can identify all the given functions, however, you would have to do the check \(a \le y \le b\) for them because they are calculated differently depending on it, and no function takes both \(y\) and \(a, b\) in at the same time.

What do you mean by "indicator function", I don't find that term in the course. You mean link function I guess? And you mean the relationship between \(\eta\) and {a,b} is not defined?

Indicator function - https://en.wikipedia.org/wiki/Indicator_function (i.e. takes an equation for example and if that equation is true it returns 1, and if the equation is false it returns a 0, something similar like in C if you are from the CS field)

As I am sending links like a maniac I should just send you one of the chapters of this super cool professor from Berkely - Michael Jordan (he is not a famous basketball player haha :D ), i.e. this is his explanation of the exponential family of distributions (you will see that PDF is not completely finished, but is it MORE than enough):

https://people.eecs.berkeley.edu/~jordan/courses/260-spring10/other-readings/chapter8.pdf

Anyways he is super cool and I read a lot from him also Lex Friedman did a podcast with him which is also very cool and inspiring:
https://www.youtube.com/watch?v=EYIKy_FM9x0

Okay, I will stop with links haha :D If you need additional help call me via email: milos.novakovic@epfl.ch

Ok, I got it, thank you!

@ Milos Novakovi: thank you for the links!

No problemo mate! :))

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