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I'm quite new to all this and am just trying to understand the concepts of GLM - getting very confused between distribution assumptions and the actual regression function. Let's keep it simple with one explanatory variable.

Looking at e.g. Poisson:

Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters.

Now for a start, by "variable Y has a Poisson distribution" do we just mean the distribution of $\epsilon_i$ rather than $Y_i$? Clearly a line $y=exp(x)$ looks nothing like a Poisson distribution, and I think that's where my confusion is coming from.

I just don't see why a linear relationship needs to be modelled using a normal distribution, and why an exponential relationship needs to be modelled using a Poisson distribution. Why can't we just stick with the normal distribution?

Also, Wikipedia suggests the Poisson regression as a typical use for "count of occurrences in fixed amount of time/space". Again, if I want to regress something that looks like $y=exp(x)$, where's the relationship to the Poisson distribution and why is count a typical occurence?

I really hope my thoughts make sense to someone - please put some order into my thoughts!

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    $\begingroup$ The quote is not quite right. Y is not assumed to have a poisson distribution. The conditional distributions Y | X are assumed to be poisson. $\endgroup$ – Matthew Drury Jan 14 '17 at 19:46
  • $\begingroup$ That makes a lot more sense to me. Got it from Wiki... en.wikipedia.org/wiki/Poisson_regression $\endgroup$ – Oli Jan 14 '17 at 20:04
  • $\begingroup$ But again, if our distribution assumption is only for Y|X, why is the shape of X->Y important? $\endgroup$ – Oli Jan 14 '17 at 20:05
  • $\begingroup$ @Oli the shape of X->Y is why you're taking the log-link. There are other distributions which could be used instead of poisson (normal, gamma, etc) at that point. $\endgroup$ – Michael Oberst Jan 15 '17 at 2:59
  • $\begingroup$ @Oli in effect , while "Poisson...assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters.", there are other GLMs that make the same assumption - but they make different assumptions on the conditional distribution $\endgroup$ – Michael Oberst Jan 15 '17 at 3:00

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