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I am trying to find a loss function for a beta distributed variable, other than the log-likelihood. Since the beta distribution belongs to the exponential family, I wonder if it is also possible to define the unit deviance, something like the gamma deviance defined here (I have very limited experience with GLMs).

I want to train different machine learning models for a beta distributed variable, and want a loss function that takes into account the shape of the distribution. The log likelihood depends on two parameters, that's why I am trying to find a simplification like that done with the gamma deviance.

D1X
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    What is your purpose? Loss usually depends on the decision problem, and not on the distribution ... – kjetil b halvorsen Nov 27 '22 at 00:25
  • I want to train different machine learning models for a beta distributed variable, and want a loss function that takes into account the shape of the distribution – D1X Nov 27 '22 at 13:30
  • Please add that new info as an edit to the post! We want posts to be self-contained, and comments are easily overseen, and can be deleted! – kjetil b halvorsen Nov 27 '22 at 16:40
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    Not clear why you don't want log likelihood (not only is it the go to loss function for probabilistic models, but it also satisfies your requirement of "taking into account shape of distribution" -- which is why it's standard procedure). The Gamma deviance link you give is explicitly using log likelihoods as well. Are you asking for a mathematical simplification of likelihood fir beta distribution? ( Like MSE being the (negative) log likelihood for gaussian constant variance distribution) – Georg M. Goerg Nov 27 '22 at 17:02
  • @Georg M. Goerg Indeed I would like a simplification of likelihood for the beta distribution, since the deviance only requires the mean, whereas the log likelihood requires both parameters – D1X Nov 27 '22 at 21:54
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    @D1X I suggest you take a look at https://www.semanticscholar.org/paper/Beta-Regression-for-Modelling-Rates-and-Proportions-Ferrari-Cribari%E2%80%90Neto/09b8ab5c04d6a914ae9a7157f7de8880205165fd which reparameterizes beta distribution in terms of mean and dispersion. I m still not sure why you want a measure that depends on mu only , yet should "take into account shape of distribution" ( ie the other parameter). That seems impossible to achieve but maybe I m missing sthg. The paper has a deviance measure for large phi ( constant). That might be what you have in mind – Georg M. Goerg Nov 27 '22 at 23:51

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