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For Normal distribution, with know mean and unknown variance.

When $\tau = 1/\sigma^2$ ~ Gamma().

In such has posterior of $\tau$ has the following distribution:

$p(\tau|\alpha, \beta, x) \sim G(\alpha + \frac{n}{2},\ \ \beta\cdot\sum_{i=1}^{n}\frac{(x_i-\mu)^2}{2})$

How it usually defined initial $\alpha$ and $\beta$ ?

I can think of one way to define alpha, which is $\alpha + n/2 = \sigma_0$ -> $\alpha = \sigma_0 - n/2$.

If it makes sense, what could be the way to define initial $\beta$ ?

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    $\begingroup$ What is $\sigma_0$? The prior (hyper-)parameters are free to chose. Each choice leads to a different posterior, this is the point of running Bayesian analysis. $\endgroup$ – Xi'an Oct 21 '20 at 18:18
  • $\begingroup$ $\sigma_0$, could be for example standard deviation of past data. $\endgroup$ – Michael D Oct 22 '20 at 7:16

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