Simulating from a normal with "unknown" variance Suppose I want to performing sampling from a normal distribution with an unspecified variance, and I want a way to sample so that I am in some sense "averaging out the possible values of the variance." In other words, I want to use the simulation results elsewhere, and hear results  are highly dependent on the variance, and so I want to avoid "favoring" one choice of variance over another. 
I was thinking maybe I could put a prior on the variance and derive the marginal posterior of the variance given the draws, find the MAP estimate of the posterior, say, $m$, and then simulate again from a $\mathcal{N}(0,m)$ distribution?
If not, how else could I do this?
 A: As you point out, the Bayesian approach to dealing with an unknown variance is to use a prior distribution for this value, and then derive the resulting marginal distribution of the value to be sampled.  This gives you a "mixture distribution" which is a mixture of normal distributions with different variances, with the "weights" for that mixture being determined by the prior distribution.
You might be interested in a well-known mixture distribution of this kind, which is the Student's T distribution.  If we take $X|\lambda \sim \text{N}(0,\tfrac{1}{\lambda})$ and $\lambda \sim \text{Ga}(\tfrac{\varphi}{2},\tfrac{\varphi}{2})$ then we obtain the marginal distribution:
$$\text{St}(t|\varphi) 
= \int \limits_0^\infty \text{N}(t|0,\tfrac{1}{\lambda}) \text{Ga}(\lambda|\tfrac{\varphi}{2},\tfrac{\varphi}{2}) \ d \lambda.$$
This is a distributional form that is a "mixture" of normal distributions, where the variance parameter has an inverse gamma distribution.
A: Take a look at something called stochastic volatility in financial asset modeling. What you could do is generate the variances according to one of the many models and then use the series of variances to sample from a normal distribution with "unknown" variance. I should say that this assumes a temporal evolution in variance but it might suit you or give you an idea.
The SV models were developed to emulate some stylized facts in empirical data. For example, the GARCH model produces volatility clustering. That is, points of high variance tends to be followed by high variance (and likewise for low variance).
It sounds like the Heston model would be acceptable by your specifications. You would basically have your volatility follow a Wiener process but you still need to specify some parameters like the long run variance and the variance of variance.
