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I've been recently learning about GLM's after learning about ordinary linear regression. In simple linear regression, I believe that an error term is used in order to account for randomness in the actual response when we predict values, like the equation below:

Yi= β0+ βxi+ ϵi

Where ϵi follows a normal distribution with expectation 0 and variance sigma^2. However I believe that this assumes that our Y variable follows a normal distribution.

I was wondering what would be the equivalent error term (if any) for GLM's that we model assuming Y follows another distribution (such as Poisson or Gamma), and if it relates anyway to the "link" function used in GLMS.

Thanks!

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For a GLM, instead of errors $\epsilon$, we model uncertainty with probabilities.

Link functions constrain our output from all real numbers (linear regression) to whatever we specify it to be. This is because with GLMs, not every distribution works with all real numbers.

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  • $\begingroup$ Hi John! Thank you for explaining link functions - I think i get that part now. Just with the errors component, would you be able to elaborate more? Say if I am modelling count using a GLM assuming it follows a Poisson Distribution with a log link, how can we model uncertainty with probabilities? - do we look at the prediction /confidence interval?] $\endgroup$
    – Sushiix
    Commented Nov 22, 2020 at 10:27
  • $\begingroup$ The uncertainty is inherent in the confidence interval. Example: We believe with 95% confidence that the true coefficient is between x and y." If we were to take repeated samples from the same population, and run the same regression on each data set we draw, then the true population parameter would be covered by 95% of these models’ 95% confidence intervals. $\endgroup$
    – John Stud
    Commented Nov 22, 2020 at 16:00

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