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This answer provided some good general advice, but in my specific case I want to create a model of my prior beliefs about the variance of a normally-distributed random variable:

$$x \sim \mathcal{N}(0, \sigma_x^2)$$

I want an appropriate parameterized function $f$ for my beliefs about $\sigma_x^2$:

$$\sigma_x^2 \sim f(\hat{\sigma_x}^2, ...)$$

where $\hat{\sigma_x}^2$ is my prior 'best guess'.

In the answer linked above it suggests using the Gamma distribution for positive real parameters. However, I read on wikipedia.org that the inverse-chi-squared distribution can be used in Bayesian inference:

as the prior and posterior distribution for an unknown variance of the normal distribution.

But there is also the scaled inverse chi-squared distribution and the Inverse-gamma distribution so now I am confused which is the best one to use.

My application is not rocket science so whichever is the most common and/or most appropriate in general.

Bill
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  • The answer is straightforward : there is no "best prior" unless one enjoys such precise information as to have no choice about the prior. A Bayesian analysis is conditional upon a (chosen rather than given) prior. – Xi'an Oct 26 '21 at 03:15
  • Well, there are certainly very poor ones! Such as a normal distribution in this case. Suppose my estimate of the std. deviation is s and I am sure that the probability of the true std. deviation is between 0.5s and 2s is 50%. Is there a pdf for the variance (or std dev.) that would satisfy these conditions? This would actually be a useful prior that I can explain. – Bill Oct 26 '21 at 03:32

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