Exercise 9.1 - Conjugate prior for univariate Gaussian in exponential family form

Recall that the 1d Gaussian distribution is:

Rewrite it into the standard exponential family form:

With $\lambda =\frac{1}{{\sigma }^2}$, denote:

Now consider the likelihood w.r.t. a dataset $𝒟={\left\{x_n\right\}}_{n=1}^N$:

The prior distribution of $𝜃$ should satisfy the following variational form:

The first term, in which $\lambda$ is the target variable, takes the form of a Gamma distribution since:

The second term is just another Gaussian distribution since the sufficient statistics are $\mu$ and ${\mu }^2$. Combine these two observations together, we have:

The transformations between variables are:

In case the variance of the prior for $\mu$ is written in the information form, the precision can be written by: $\lambda \cdot (2\alpha -2)$.