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So here is another question regarding bayesian updating, which I try to understand.

In my scenario, I sequentially process single pieces of information from an unknown distribution. I am interested in estimating the mean of this distribution.

According to my readings, best would be, to know the variance of the observed distribution. Well, it would be better, to know the mean, but I don't know either, and (appart from maybe the need for the process of bayesian updating) I am also not interested in knowing the variance.

So my questions:

  1. Do I need to estimate this variance, if it is not in my desired output? What is the downside on largly over/underestimating this variance, by just stating e.g. I assume it is something 50ish?
  2. If so, how would I do that, given that I process a single draw at a time.
  3. Would it help, if I introduced a short-term memory holding a few (maybe 1 or 2 of the past values) to estimate the variance of the samples?

Now, what I read was, that I would need to estimate the variance using the inverse gamma distribution with two parameters $\alpha, \beta$ - which I assume I somehow need to have? What is the intuitive undestanding of those parameters in my scenario?

derM - not here for BOT dreams
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    As a Bayesian, one need put priors on all unknown parameters. – Xi'an Mar 14 '24 at 13:30
  • Just because you’re not interested in it means it’s known. In Bayesian statistics, everything unknown, collectively, gets a prior distribution put on it before you see any data. After you see the data you get a posterior distribution. If you are only interested in one of those two parameters, then you take the joint distribution and integrate out the quantity you’re not interested in. – Taylor Mar 14 '24 at 13:32
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    So basically, instead of saying it is something 50ish, I say I assume it is gamma-distributed, and choose alpha and beta, such that the mean of my gamma-distribution is 50, and then I syncronously update my gamma-distribution and my normal distribution with each observation? – derM - not here for BOT dreams Mar 14 '24 at 13:50

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