Gibbs sampling: Conjugate prior for a two component known-unknown variance?

Question

If I have a normally distributed variable $\mathcal{N}(\mu,\frac{1}{\tau})$ with fixed $\mu$ then the conjugate prior for an unknown $\tau$ is then $\mathcal{Ga}(\frac{n}{2}+\alpha, \beta + \sum_i \frac{(x_i - \mu)^2}{2})$.

If my precision is instead $\tau = \frac{1}{\sigma_a^2 + \sigma_b^2}$, where $\sigma_a^2$ is known and $\sigma_b^2$ is unknown, is there a form of the conjugate prior for $\sigma_a^2$?

It seems as though this should be simple but I have struggled with it. The application of this is to sample $\sigma_b^2$ in a Gibbs sampler.

Answer 1

Since $\tau = \frac{1}{\sigma_a^2 + \sigma_b^2}$ is in one-to-one correspondence with $\sigma_a^2$, setting a prior on $\tau$ is equivalent to setting a prior on $\sigma^2_a$. The Gamma prior of $\tau$ continues to be conjugate in this setting, except that it is now restricted to $(0,\tau_a^{-2})$.