# inv-gamma distribution as prior for multivariate normal distribution

it's known the conjugate priors for multivariate normal distribution are the normal & inverse-whishart distributions. but i'm interest in very specific case where the correlation matrix is $R$ multiply by random variable $\kappa^2$ with inverse-gamma prior summed with $\sigma^2 I$ . I'm trying to get the closed-form posterior from an inverse gamma prior and a likelihood based on a multivariate normal distribution expecting to get inverse gamma posterior but I don't have any success . I appreciate it if you could help me get the posterior. the model:

$$Y | \kappa \sim \mathcal{N}_j(0,( \kappa^2 R + \sigma^2 I))$$ $$\kappa \sim \text{InverseGamma} (A_0, B_0)$$

Where $A_0$, $B_0$, and $\sigma^2$ are known constants and $R$ is $j$ by $j$ known matrix we can even say it is rank(1) matrix . $I$ is a $j$ by $j$ identity matrix.

• I wouldn't be surprised if there is no closed form posterior. – user3903581 Apr 3 '17 at 13:45

Since the distribution of $Y$ given $\kappa$ is an exponential family, as a sample $(y_1,\ldots,y_n)$ from $\mathcal{N}_j(0,( \kappa^2 R + \sigma^2 I)^{-1})$ has density $$|\kappa^2 R + \sigma^2 I|^{n/2}\exp-\frac{1}{2}\text{tr}\left\{ (\kappa^2 R + \sigma^2 I)\sum_{i=1}^n Y_iY_i^t\right\}$$ i.e., $$|\kappa^2 R + \sigma^2 I|^{n/2}\exp-\frac{1}{2}\left\{ \kappa^2 \text{tr}\left(R\sum_{i=1}^n Y_iY_i^t\right) + \sigma^2 \text{tr}\sum_{i=1}^n Y_iY_i^t\right\}$$ there exists a conjugate family on $\kappa$ but not of the inverse Gamma variety.
Since the distribution of $Y$ given $\kappa$ is a curved exponential family, as a sample $(y_1,\ldots,y_n)$ from $\mathcal{N}_j(0,( \kappa^2 R + \sigma^2 I)^{-1})$ has density $$|\kappa^2 R + \sigma^2 I|^{-n/2}\exp-\frac{1}{2}\text{tr}\left\{ (\kappa^2 R + \sigma^2 I)^{-1}\sum_{i=1}^n Y_iY_i^t\right\}$$ which factorises through the sufficient statistic $\sum_{i=1}^n Y_iY_i^t$, there exists a conjugate family on $\kappa$ but not of the inverse Gamma variety, since this family is actually $j(j+1)/2$ dimensional.
• actually the density suppose to by like that $$|\kappa^2 R + \sigma^2 I|^{-n/2}\exp-\frac{1}{2}\text{tr}\left\{ (\kappa^2 R + \sigma^2 I)^{-1}\sum_{i=1}^n Y_iY_i^t\right\}$$ it is possible to use woodbary decomposition but still what is the family of $\kappa$? thanks – yosef soussana Apr 3 '17 at 18:54