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):

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:

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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