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Are there any distributions which represent uncertainty about $\hat y$ as a $t$ distribution. In other words: $ p(y\mid x, ~\text{model}) \sim \mathbf T(\hat y, \text{df}, \sigma)$ where $\bf T$ is a non standard t distribution.

I suppose this could just be done by assigning the assumption to a simple regression. But does it ever come up naturally in any probability distributions?

Snowy Baboon
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  • Look into link functions as used in generalized linear models, e.g., https://www.statisticshowto.com/link-function/ – user78229 Jul 01 '23 at 12:53
  • Does this https://stats.stackexchange.com/questions/120776/why-should-we-use-t-errors-instead-of-normal-errors answer your question? – kjetil b halvorsen Jul 01 '23 at 16:11

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