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In a lecture this week my professor stated that

exponential families have convenient mathematical properties due to their natural parameterization such as the natural parameter space being convex.

Question: What does it mean that "the natural parameter space is convex"?


Some "thoughts": Does this suggest maximum likelihood estimators of the parameters always exist? What other mathematical properties of this result are useful?

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    $\begingroup$ This would probably make for a good discussion with your professor, showing interest in these remarks. However for the purpose of Math.SE content more context is needed to make it accessible to a broad range of Readers. You got a response limited to the issue of what it means for the parameter space to be convex, but the rest of the Question is pretty broad. Math.SE Questions should be more narrowly focused. You can get there by breaking up this material into well researched chunks, starting with what you understand well enough to check answers for. $\endgroup$ – hardmath Feb 4 '18 at 16:28
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    $\begingroup$ See this fairly recent Question at Cross Validated.SE about references for maximum likelihood estimation on exponential families. $\endgroup$ – hardmath Feb 4 '18 at 16:31
  • $\begingroup$ I will probably come back and expand my answer later today or tomorrow. $\endgroup$ – Michael Hardy Feb 4 '18 at 18:09
  • $\begingroup$ This question only makes sense if one knows what the terms "exponential family" and "natural parameter space" mean, which are specialized terms typically known only by workers in mathematical statistics. $\endgroup$ – kimchi lover Feb 5 '18 at 3:28
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    $\begingroup$ This post has three separate parts: 1. "What does it mean that "the natural parameter space is convex"?" 2. "Does this suggest maximum likelihood estimators of the parameters always exist?" 3. "What other mathematical properties of this result are useful?" Now, 1. is utterly trivial (saying that the space of parameters is convex means that it is... well, convex) and it is the only part addressed in the accepted answer below (which basically states the parenthesis in this sentence). What happened to 2. and 3.? Is 1. on its own, really the kind of stuff questions on the site should be made of? $\endgroup$ – Did Feb 5 '18 at 10:59
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That the natural parameter space is convex means that if $\alpha,\beta$ are two different points in the natural parameter space, then every point between $\alpha$ and $\beta$ is also within the natural parameter space.

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