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I have a process which may be assumed Gaussian - for now I'm assuming it to be zero-mean. The variance is unknown.

I will estimate variance in the usual unbiased way to give the sample variance and thus the ratio of the sample variance to the true variance follows the Chi Squared distribution.

In parallel i have developed an equation for a quantity of interest (a risk) which is a function of the pdf of the true population variance.

What inference approach do/should i take to give me a pdf of this unknown population variance?

Thanks in advance.

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Since you're looking for a pdf of a fixed parameter given some data, you could go with Bayesian analysis: You can use either an improper prior (Jeffrey's prior) or an informative prior if you have some idea of where the variance is. Then you multiply this by the likelihood of observing the sample statistic given a particular value of the variance. Finally, you normalise this to get a posterior distribution.

If this is too mathematically tedious, you can also create a "confidence distribution" for the true variance, which is just the left-sided p-value of your test statistic as a function of the null hypothesis for variance. Take the derivative of this to get the pdf.

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