Informative priors for standard deviation (or variance) Suppose I want to perform Bayesian estimation of the mean $\mu$ and standard deviation $\sigma$ of a Gaussian distribution. Is there a standard way to specify an informative prior over $\sigma$, assuming I have prior info that it lies "close" to some given $\gamma$ in log-space?
In addition to having an initial guess for $\sigma$, I want to avoid two problems with using the Jeffrey's prior (A Bayesian perspective on estimating mean, variance, and
standard-deviation from data, Travis Oliphant). With the Jeffrey's prior, one does not have a well-defined MAP unless $n>3$. And if the sum of squared deviances is 0, one's MAP estimate of $\sigma$ will also be 0.


I want to avoid two problems with using [the Jeffrey's prior][1] (A Bayesian perspective on estimating mean, variance, and
standard-deviation from data, Travis Oliphant). With the Jeffrey's prior, one does not have a well-defined MAP unless $n>3$. And if the sum of squared deviances is 0, one's MAP estimate of $\sigma$ will also be 0.

Why would avoiding those two problems be a problem? Do you have $n\leq 3$ or the sum of residuals equal to zero? These two situations are theoretical cases in which case statistics are not of much use anyway.

Suppose I want to perform Bayesian estimation of the mean $\mu$ and standard deviation $\sigma$ of a Gaussian distribution. Is there a standard way to specify an informative prior over $\sigma$...

You ask for a standard way to specify an informative prior but there is not a single standard. However, there are several standards. One example would be to use the conjugate prior, which makes computations easy. In this case, estimating the mean and variance, this would be the normal inverse gama distribution.
