# Normal Conjugate Prior, Known Mean and Unknown Variance?

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 $$\beta$$ ?

• 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. – Xi'an Oct 21 '20 at 18:18
• $\sigma_0$, could be for example standard deviation of past data. – Michael D Oct 22 '20 at 7:16