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Oct 12, 2016 at 14:42 comment added whuber Because the question asks about the difference between a canonical link and an arbitrary link, it is a duplicate, so I voted to close it as such.
Oct 12, 2016 at 14:40 comment added ilanman I didn't mean for canonical to equate to correct. I added that to clarify that there can be many different types of links. No one correct link. As well, the OP seems to be confused with what a link function actually does. My answer is an attempt to add clarity around that.
Oct 12, 2016 at 14:26 comment added whuber You seem to equate "canonical" with "correct." That's not what "canonical" means. There's a mathematical reason that Wikipedia article associates a unique, natural (i.e., canonical) link function with each of the distribution families it lists. Consequently your answer doesn't appear to address the question.
Oct 12, 2016 at 13:43 history edited ilanman CC BY-SA 3.0
added c-log log link
Oct 12, 2016 at 13:37 history edited ilanman CC BY-SA 3.0
added 39 characters in body
Oct 12, 2016 at 13:22 history edited kjetil b halvorsen CC BY-SA 3.0
deleted 4 characters in body
Oct 12, 2016 at 13:18 history edited ilanman CC BY-SA 3.0
made a bunch of clarifications
Oct 12, 2016 at 2:14 history edited ilanman CC BY-SA 3.0
added 110 characters in body
Oct 12, 2016 at 2:05 history answered ilanman CC BY-SA 3.0