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I know that for a Bayesian uniform/flat prior, the formula is 1/n (and n=1), as each value has an equal chance of being chosen.

However, is there an equation for when the prior is biased/informative? For example, so that the prior is skewed to the left or to the right?

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    $\begingroup$ uniform has nothing to do with uninformative $\endgroup$ – Neil G Jul 17 '17 at 13:56
  • $\begingroup$ @NeilG when I say uniform I mean flat prior $\endgroup$ – Kirsten Morehouse Jul 17 '17 at 13:59
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    $\begingroup$ Right, and that has nothing to do with being uninformative. $\endgroup$ – Neil G Jul 17 '17 at 14:15
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    $\begingroup$ Suppose I give you a weighted coin with some unknown bias probability $0\le p\le 1$, then you might argue that the uninformative belief over $p$ is uniform over $[0,1]$. Now, suppose you travel to another planet where the aliens like to measure biases in odds $0 \le o = \frac{p}{1+p}$. Then, you might similarly argue that the uninformative belief over $o$ is flat on $[0, \infty)$. But, these aren't the same distribution. So which one is uninformative? $\endgroup$ – Neil G Jul 17 '17 at 14:20
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    $\begingroup$ @Neil G: In a finite parameter set, the uniform can more easily be argued to be "non-informative" as the change-of-variable argument does not apply so naturally. $\endgroup$ – Xi'an Jul 18 '17 at 5:04
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Yes, but just as there are many ways of a curve not being flat, there are many different non-flat priors. For example, a coin's bias could have a $\operatorname{Beta}(α, β)$ prior, which is only flat when $α$ and $β$ are both 1, or it could have a triangular prior with endpoints $0$ and $1$, which can't be flat no matter how you set the remaining parameter.

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