# Conjugate prior for multivariate with known mean and covariance known to a constant

I have a linear trend model (evolving mean and slope) embedded in a larger state space time series model that I would like to constrain to be a spline.

With that assumption, the mean and trend innovations $(\theta_1,\theta_2)$ are bivariate normal and the covariance between them is constrained to be a known 2x2 matrix $A$ multiplied by an unknown constant $\tau^2$, $$(\theta_1,\theta_2)\sim\mathcal{N}_2(0_2,\tau^2\,A)$$i.e., $$\pi(\theta_1,\theta_2|\tau)\propto\tau^{-2}\exp\left\{-(\theta_1,\theta_2)\,A^{-1}\left(\matrix{\theta_1\\\theta_2}\right)\big/2\tau^2\right\}$$ I am using a Gibbs sampler and would like to get a full conditional distribution for that constant and thus for the covariance or precision. I'd like to work with a conjugate prior at least as a start. This does not seem like a Wishart result, because I am drawing a scalar not a full matrix. Is it inverse gamma? Thanks.

• If you take a look at the above conditional of $(\theta_1,\theta_2)$ given $\tau$ or $\tau^{-2}$, what form of conjugate prior distribution on $\tau^{-2}$ does it suggest? can you show $\tau$ is independent of the data given $(\theta_1,\theta_2)$? – Xi'an Apr 15 '15 at 6:35
• It suggests inverse gamma to me, but I don't do this kind of derivation/calculation much. If I understand your last question the answer is of course ... this is trivial in a Gibbs sampling context. In fact I've re-edited the observed data out of the question. – Eli S Apr 16 '15 at 8:09
• You end up with an expression $\tau^{-2}\exp\{-\alpha\tau^{-2}$ for the density of $(\theta_1,\theta_2)$ taken as a function of $\tau^{-2}$ so indeed this is an inverse gamma density (up to a constant) that suggests using an inverse gamma density on $\tau^2$ as your conjugate prior. – Xi'an Apr 16 '15 at 8:22